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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndProcSet %%BeginProcSet: cmr8.pfb %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 60 /exclamdown put dup 61 /equal put dup 62 /questiondown put dup 63 /question put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndProcSet %%BeginProcSet: cmr12.pfb %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 192 /AE put dup 193 /OE put dup 194 /Oslash put dup 195 /suppress put dup 196 /dieresis put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 16 /dotlessi put dup 17 /dotlessj put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 21 /breve put dup 22 /macron put dup 23 /ring put dup 24 /cedilla put dup 25 /germandbls put dup 26 /ae put dup 27 /oe put dup 28 /oslash put dup 29 /AE put dup 30 /OE put dup 31 /Oslash put dup 32 /suppress put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 36 /dollar put dup 37 /percent put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 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cleartomark %%EndProcSet %%BeginProcSet: cmmi8.pfb %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 165 /Xi put dup 166 /Pi put dup 167 /Sigma put dup 168 /Upsilon put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 192 /upsilon put dup 193 /phi put dup 194 /chi put dup 195 /psi put dup 196 /tie put dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 36 /pi1 put dup 37 /rho1 put dup 38 /sigma1 put dup 39 /phi1 put dup 40 /arrowlefttophalf put dup 41 /arrowleftbothalf put dup 42 /arrowrighttophalf put dup 43 /arrowrightbothalf put dup 44 /arrowhookleft put dup 45 /arrowhookright put dup 46 /triangleright put dup 47 /triangleleft put dup 48 /zerooldstyle put dup 49 /oneoldstyle put dup 50 /twooldstyle put dup 51 /threeoldstyle put dup 52 /fouroldstyle put dup 53 /fiveoldstyle put dup 54 /sixoldstyle put dup 55 /sevenoldstyle put dup 56 /eightoldstyle put dup 57 /nineoldstyle put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /flat put dup 92 /natural put dup 93 /sharp put dup 94 /slurbelow put dup 95 /slurabove put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndProcSet %%BeginProcSet: 0ef0afca.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmr5 % /TeX0ef0afcaEncoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /arrowup /arrowdown /quotesingle /exclamdown /questiondown /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /exclam /quotedblright /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /quotedblleft /bracketright /circumflex /dotaccent /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /endash /emdash /hungarumlaut /tilde /dieresis /suppress /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /arrowup /arrowdown /quotesingle /exclamdown /questiondown /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 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TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def end %%EndProcSet %%BeginProcSet: special.pro %! 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 7DB9DFB4372D3935C286E39EE7AC9FB6A9B5CE4D2FAE1BC0E55AE02BFC464378 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All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 66 /B put dup 83 /S put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI9 %!PS-AdobeFont-1.1: CMTI9 1.0 %%CreationDate: 1991 Aug 18 21:08:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-35 -250 1148 750}readonly def /UniqueID 5000827 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E3948FFB3DF7BFF10C9BDA4EFE5F68A8CB1526990D1357AE6D2F7C2D2EF8496 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 69 /E put dup 76 /L put dup 84 /T put dup 115 /s put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 40 /braceleftBigg put dup 80 /summationtext put dup 84 /intersectiontext put dup 86 /logicalandtext put dup 88 /summationdisplay put dup 94 /logicalanddisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D77639DF1232A4D6233A9CAF69B151DFD33F C0962EAC6E3EBFB8AD256A3C654EAAF9A50C51BC6FA90B61B60401C235AFAB7B 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6CC3F1E9AE32F234EB60FE7D E34995B1ACFF52428EA20C8ED4FD73E3935CEBD40E0EAD70C0887A451E1B1AC8 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A 27D1663E0B62F461F6E40A5D6676D1D12B51E641C1D4E8E2771864FC104F8CBF 5B78EC1D88228725F1C453A678F58A7E1B7BD7CA700717D288EB8DA1F57C4F09 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F0364CD5660F74BEE96790DE35AFA90CCF712 B1805DA88AE375A04D99598EADFC625BDC1F9C315B6CF28C9BD427F32C745C99 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-58 -250 1195 750}readonly def /UniqueID 5000767 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F0364CD5660F74BEE96790DE35AFA90CCF712 B1805DA88AE375A04D99598EADFC625BDC1F9C315B6CF28C9BD427F32C745C99 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%!PS-AdobeFont-1.1: CMTT9 1.0 %%CreationDate: 1991 Aug 20 16:46:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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/exclamdown /equal /questiondown /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /quotedblleft /bracketright /circumflex /dotaccent /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /endash /emdash /hungarumlaut /tilde /dieresis /suppress /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: texps.pro %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def end %%EndProcSet %%BeginProcSet: special.pro %! TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/setpagedevice{pop}N/copypage{}N/p 3 def @MacSetUp}N/doclip{psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N} N/@setspecial{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N /setpagedevice{pop}N/copypage{}N newpath}N/@endspecial{count ocount sub{ pop}repeat countdictstack dcount sub{end}repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N/@fedspecial{end}B/li{lineto}B /rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X/yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet %%BeginProcSet: color.pro %! TeXDict begin/setcmykcolor where{pop}{/setcmykcolor{dup 10 eq{pop setrgbcolor}{1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll }repeat setrgbcolor pop}ifelse}B}ifelse/TeXcolorcmyk{setcmykcolor}def /TeXcolorrgb{setrgbcolor}def/TeXcolorgrey{setgray}def/TeXcolorgray{ setgray}def/TeXcolorhsb{sethsbcolor}def/currentcmykcolor where{pop}{ /currentcmykcolor{currentrgbcolor 10}B}ifelse/DC{exch dup userdict exch known{pop pop}{X}ifelse}B/GreenYellow{0.15 0 0.69 0 setcmykcolor}DC /Yellow{0 0 1 0 setcmykcolor}DC/Goldenrod{0 0.10 0.84 0 setcmykcolor}DC /Dandelion{0 0.29 0.84 0 setcmykcolor}DC/Apricot{0 0.32 0.52 0 setcmykcolor}DC/Peach{0 0.50 0.70 0 setcmykcolor}DC/Melon{0 0.46 0.50 0 setcmykcolor}DC/YellowOrange{0 0.42 1 0 setcmykcolor}DC/Orange{0 0.61 0.87 0 setcmykcolor}DC/BurntOrange{0 0.51 1 0 setcmykcolor}DC /Bittersweet{0 0.75 1 0.24 setcmykcolor}DC/RedOrange{0 0.77 0.87 0 setcmykcolor}DC/Mahogany{0 0.85 0.87 0.35 setcmykcolor}DC/Maroon{0 0.87 0.68 0.32 setcmykcolor}DC/BrickRed{0 0.89 0.94 0.28 setcmykcolor}DC/Red{ 0 1 1 0 setcmykcolor}DC/OrangeRed{0 1 0.50 0 setcmykcolor}DC/RubineRed{ 0 1 0.13 0 setcmykcolor}DC/WildStrawberry{0 0.96 0.39 0 setcmykcolor}DC /Salmon{0 0.53 0.38 0 setcmykcolor}DC/CarnationPink{0 0.63 0 0 setcmykcolor}DC/Magenta{0 1 0 0 setcmykcolor}DC/VioletRed{0 0.81 0 0 setcmykcolor}DC/Rhodamine{0 0.82 0 0 setcmykcolor}DC/Mulberry{0.34 0.90 0 0.02 setcmykcolor}DC/RedViolet{0.07 0.90 0 0.34 setcmykcolor}DC /Fuchsia{0.47 0.91 0 0.08 setcmykcolor}DC/Lavender{0 0.48 0 0 setcmykcolor}DC/Thistle{0.12 0.59 0 0 setcmykcolor}DC/Orchid{0.32 0.64 0 0 setcmykcolor}DC/DarkOrchid{0.40 0.80 0.20 0 setcmykcolor}DC/Purple{ 0.45 0.86 0 0 setcmykcolor}DC/Plum{0.50 1 0 0 setcmykcolor}DC/Violet{ 0.79 0.88 0 0 setcmykcolor}DC/RoyalPurple{0.75 0.90 0 0 setcmykcolor}DC /BlueViolet{0.86 0.91 0 0.04 setcmykcolor}DC/Periwinkle{0.57 0.55 0 0 setcmykcolor}DC/CadetBlue{0.62 0.57 0.23 0 setcmykcolor}DC /CornflowerBlue{0.65 0.13 0 0 setcmykcolor}DC/MidnightBlue{0.98 0.13 0 0.43 setcmykcolor}DC/NavyBlue{0.94 0.54 0 0 setcmykcolor}DC/RoyalBlue{1 0.50 0 0 setcmykcolor}DC/Blue{1 1 0 0 setcmykcolor}DC/Cerulean{0.94 0.11 0 0 setcmykcolor}DC/Cyan{1 0 0 0 setcmykcolor}DC/ProcessBlue{0.96 0 0 0 setcmykcolor}DC/SkyBlue{0.62 0 0.12 0 setcmykcolor}DC/Turquoise{0.85 0 0.20 0 setcmykcolor}DC/TealBlue{0.86 0 0.34 0.02 setcmykcolor}DC /Aquamarine{0.82 0 0.30 0 setcmykcolor}DC/BlueGreen{0.85 0 0.33 0 setcmykcolor}DC/Emerald{1 0 0.50 0 setcmykcolor}DC/JungleGreen{0.99 0 0.52 0 setcmykcolor}DC/SeaGreen{0.69 0 0.50 0 setcmykcolor}DC/Green{1 0 1 0 setcmykcolor}DC/ForestGreen{0.91 0 0.88 0.12 setcmykcolor}DC /PineGreen{0.92 0 0.59 0.25 setcmykcolor}DC/LimeGreen{0.50 0 1 0 setcmykcolor}DC/YellowGreen{0.44 0 0.74 0 setcmykcolor}DC/SpringGreen{ 0.26 0 0.76 0 setcmykcolor}DC/OliveGreen{0.64 0 0.95 0.40 setcmykcolor} DC/RawSienna{0 0.72 1 0.45 setcmykcolor}DC/Sepia{0 0.83 1 0.70 setcmykcolor}DC/Brown{0 0.81 1 0.60 setcmykcolor}DC/Tan{0.14 0.42 0.56 0 setcmykcolor}DC/Gray{0 0 0 0.50 setcmykcolor}DC/Black{0 0 0 1 setcmykcolor}DC/White{0 0 0 0 setcmykcolor}DC end %%EndProcSet %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 3645B82392D5CAE11A7CB49D7E2E82DCD485CBA04C77322EB2E6A79D73DC194E 59C120A2DABB9BF72E2CF256DD6EB54EECBA588101ABD933B57CE8A3A0D16B28 51D7494F73096DF53BDC66BBF896B587DF9643317D5F610CD9088F9849126F23 DDE030F7B277DD99055C8B119CAE9C99158AC4E150CDFC2C66ED92EBB4CC092A AA078CE16247A1335AD332DAA950D20395A7384C33FF72EAA31A5B89766E635F 45C4C068AD7EE867398F0381B07CB94D29FF097D59FF9961D195A948E3D87C31 821E9295A56D21875B41988F7A16A1587050C3C71B4E4355BB37F255D6B237CE 96F25467F70FA19E0F85785FF49068949CCC79F2F8AE57D5F79BB9C5CF5EED5D 9857B9967D9B96CDCF73D5D65FF75AFABB66734018BAE264597220C89FD17379 26764A9302D078B4EB0E29178C878FD61007EEA2DDB119AE88C57ECFEF4B71E4 140A34951DDC3568A84CC92371A789021A103A1A347050FDA6ECF7903F67D213 1D0C7C474A9053866E9C88E65E6932BA87A73686EAB0019389F84D159809C498 1E7A30ED942EB211B00DBFF5BCC720F4E276C3339B31B6EABBB078430E6A09BB 377D3061A20B1EB98796B8607EECBC699445EAA866C38E02DF59F5EDD378303A 0733B90E7835C0AAF32BA04F1566D8161EA89CD4D14DDB953F8B910BFC8A7F03 5020F55EF8FC2640ADADA156F6CF8F2EB6610F7EE8874A26CBE7CD154469B9F4 ED76886B3FB679FFDEB59BB6C55AF7087BA48B75EE2FB374B19BCC421A963E15 FE05ECAAF9EECDF4B2715010A320102E6F8CCAA342FA11532671CA6F6725B014 B7DBEFFC14878F0304D673AB4F0D598206CCF6A7BBAFD35C6DE5087AD4BD2BE6 54AB965EB033BAE2B99C3C5AFD229782E04AEA350A73198621F06332C518B257 88CB1E4E520D31DA136F8227F933FC55D8A2E2980D9873F4FDCB43C175EBC4BB 026E0ECD2A83BD25369DF177332B93D079D9338EE9DB9F58A7D8A9A5946B2324 1D6EA35ED938D86072FB793E8CE762415BA78CC1894D0A27D4B1AE0EA088E355 3159FCDBEADC035AE747E15DA9A9DD750F0B9990962569AF0BC08BCDEDE02AA1 F7E7EEB626A6CCD5EEFCA4DC27885DAF6F0F9C56F7D2829AE86B705D161D7743 FBA7F5559E21DF3B34D09041560C94F13E61817EFB64347589F7B34FCB5DE7F9 42093006872E10D2EF65F6E262362CF27C7C97258B199214D67562F3D87AB0E2 BEF23E9042FE214B9BD4690EB788A5F2913C049026AC496CA4C13072479BB22B D67D9E4B0FA2535DA0F7356184FA63BD53EAF8AF69C2BE1CF035ED0A1157DBF3 02B93782C9379DDC6327BEFC0B19145F19B0357EA53DB928997A8CAE1AC753EA 4A0576E86981141DFC86D318DBC6890BC4473D60783DFF1578E04BA1CAB77F87 42B18053DEAE520401F5A77112B8393ADBA3882F6105141D7DFC3B936457BAE1 0D24B835C2F4D13D79C2CA5CA1E9C80EA7B3583159E04D37051F6154D1DDFEE6 65465BF8B5D1DF5E2179CD78588973B756CAC5F393D423DB0CC4C2AC6DAAC63D 9EBD35B4F5092D24C0C9B2ABF52784C45252834C36DF07CC8B1A548ED1D693CC 781D75FD2A9FF69695AD5E135CE5AFB2DB5D9F8B6891FEFCA3292DE0FA8A8EC7 616F28038657AAE373D8F45F5FDFBB0A205AE19E0D94D1E6ABEBBA0A7CAF3C55 E80779170748535F5AC89C650708DC7540D12AC20C7F9AF5BAF5A4A53B72AFA0 DAF6C0DB64D7E81C034AA8B746091C415EFA2C2A5223B11393C22112B771031D 7D5BCAFE47B0FB93DDB1A4FDFD1E533E3FEC4352156055077EC06EB6B32733BE 55AFF8B0A825EA279764C2EA67CA637A 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: aae443f0.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmmi10 cmmi12 cmmi5 cmmi6 cmmi7 cmmi8 cmmi9 cmmib10 % /TeXaae443f0Encoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /alpha /beta /gamma /delta /epsilon1 /zeta /eta /theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi /omega /epsilon /theta1 /pi1 /rho1 /sigma1 /phi1 /arrowlefttophalf /arrowleftbothalf /arrowrighttophalf /arrowrightbothalf /arrowhookleft /arrowhookright /triangleright /triangleleft /zerooldstyle /oneoldstyle /twooldstyle /threeoldstyle /fouroldstyle /fiveoldstyle /sixoldstyle /sevenoldstyle /eightoldstyle /nineoldstyle /period /comma /less /slash /greater /star /partialdiff /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /flat /natural /sharp /slurbelow /slurabove /lscript /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /dotlessi /dotlessj /weierstrass /vector /tie /psi /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /alpha /beta /gamma /delta /epsilon1 /zeta /eta /theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi /tie /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: f7b6d320.enc % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmb10 cmbx10 cmbx12 cmbx5 cmbx6 cmbx7 cmbx8 cmbx9 cmbxsl10 % cmdunh10 cmr10 cmr12 cmr17cmr6 cmr7 cmr8 cmr9 cmsl10 cmsl12 cmsl8 % cmsl9 cmss10cmss12 cmss17 cmss8 cmss9 cmssbx10 cmssdc10 cmssi10 % cmssi12 cmssi17 cmssi8cmssi9 cmssq8 cmssqi8 cmvtt10 % /TeXf7b6d320Encoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /exclam /quotedblright /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /exclamdown /equal /questiondown /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /quotedblleft /bracketright /circumflex /dotaccent /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /endash /emdash /hungarumlaut /tilde /dieresis /suppress /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %%EndProcSet %%BeginProcSet: texps.pro %! 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TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/setpagedevice{pop}N/copypage{}N/p 3 def @MacSetUp}N/doclip{psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N} N/@setspecial{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N /setpagedevice{pop}N/copypage{}N newpath}N/@endspecial{count ocount sub{ pop}repeat countdictstack dcount sub{end}repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N/@fedspecial{end}B/li{lineto}B /rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X/yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet %%BeginProcSet: color.pro %! 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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b(goal)g(states)g(are)g(those)g(states)g(where)g(all)h (tasks)f(ha)n(v)n(e)f(\014nished,)i(i.e.)g Fr(S)3177 4319 y Fo(g)r(oal)3340 4307 y Fz(=)523 4344 y Fm(V)592 4431 y Fo(i)634 4406 y Fr(t)664 4418 y Fc(\014nishe)m(d)6 b Fo(;i)942 4406 y Fz(.)523 4657 y Fd(4.2)95 b(Start)32 b(transition)g(relation)523 4825 y Fz(The)19 b(\014rst)f(phase)g(in)h (eac)n(h)e(iteration)h(is)h(the)f(image)g(computations)g(of)h(a)f (series)f(of)i(transition)523 4924 y(relations)26 b Fr(T)909 4936 y Fo(b)938 4944 y Ff(i)968 4924 y Fz(\()m Fr(~)-39 b(s;)11 b(~)-39 b(s)1115 4894 y Fs(0)1138 4924 y Fz(\))28 b(to)f(generate)e(the)j(states)e(with)i(all)e(p)r(ossible)h(com)n (binations)f(of)h(tasks)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 3367 232 a Fy(7)523 448 y Fz(started,)24 b(i.e.)g(the)h(state)f(where)f(no)h(tasks)g(are)f(attempted)i(started)e (and)h(the)h(states)f(where)523 548 y(task)29 b(one)h(is)g(started)f (if)h(p)r(ossible,)g(where)f(task)h(t)n(w)n(o)f(is)h(started)f(if)i(p)r (ossible,)e(where)h(task)523 648 y(one)d(and)h(t)n(w)n(o)e(are)h (started)g(if)h(p)r(ossible,)g(etc.)648 747 y(The)35 b(reason)f(w)n(e)h(are)f(using)h(a)g(n)n(um)n(b)r(er)h(of)f(transition) g(relations)f(for)h(this)h(purp)r(ose)523 847 y(instead)h(of)g(a)g (single)g(large)f(one)h(is)g(that)h(it)f(is)g(more)g(\015exible)g(and)g (mak)n(es)g(it)g(p)r(ossible)523 946 y(to)d(a)n(v)n(oid)f(a)h(large)e (n)n(um)n(b)r(er)i(of)h(constrain)n(ts)d(for)i(k)n(eeping)f(v)-5 b(ariables)33 b(unc)n(hanged.)h(Since)523 1046 y(a)d(large)g(relation)f (w)n(ould)i(b)r(e)g(a)f(disjunction)h(of)g(the)g(individual)g Fr(T)2669 1058 y Fo(b)2698 1066 y Ff(i)2728 1046 y Fz(\()m Fr(~)-39 b(s;)11 b(~)-39 b(s)2875 1016 y Fs(0)2898 1046 y Fz(\),)32 b(the)g(splitted)523 1146 y(relation)27 b(is)g(what)h(is)f (referred)g(to)g(as)g(a)g(disjunctiv)n(e)h(partioning)f([5].)648 1245 y(Eac)n(h)h Fr(T)905 1257 y Fo(b)934 1265 y Ff(i)964 1245 y Fz(\()m Fr(~)-39 b(s;)11 b(~)-39 b(s)1111 1215 y Fs(0)1134 1245 y Fz(\))30 b(allo)n(ws)e(a)h(transition)g(that)h(sets) f Fr(t)2273 1257 y Fc(starte)m(d)5 b Fo(;i)2557 1245 y Fz(to)29 b(true)g(and)h(decremen)n(ts)523 1345 y(the)f(n)n(um)n(b)r (er)f(of)h(free)f(pro)r(cessors)e(if)j(task)f Fr(t)1909 1357 y Fo(i)1965 1345 y Fz(has)g(not)h(b)r(een)g(started)f(y)n(et,)g (there)h(is)f(a)g(free)523 1445 y(pro)r(cessor)d(and)j(the)g(immediate) g(dep)r(endencies)g(of)f(task)g Fr(t)2400 1457 y Fo(i)2456 1445 y Fz(ha)n(v)n(e)f(\014nished:)606 1727 y Fr(T)655 1739 y Fo(b)684 1747 y Ff(i)714 1727 y Fz(\()m Fr(~)-39 b(s;)11 b(~)-39 b(s)861 1693 y Fs(0)885 1727 y Fz(\))23 b(=)g(\()p Fr(m)1133 1739 y Fc(fr)m(e)m(e)1271 1727 y Fq(6)p Fz(=)g(0\))18 b Fq(^)p 1525 1666 285 4 v 19 w Fr(t)1555 1739 y Fc(starte)m(d)5 b Fo(;i)1828 1727 y Fq(^)1903 1648 y Fm(^)1831 1830 y Fo(d)p Fs(2D)r Fp(\()p Fo(t)2016 1838 y Ff(i)2041 1830 y Fp(\))2010 1727 y Fr(t)2040 1739 y Fc(\014nishe)m(d)h Fo(;d)1239 1964 y Fq(^)18 b Fr(t)1342 1929 y Fs(0)1342 1984 y Fc(starte)m(d)6 b Fo(;i)1615 1964 y Fq(^)p 1689 1893 309 4 v 19 w Fr(t)1719 1935 y Fs(0)1719 1989 y Fc(\014nishe)m(d)g Fo(;i)2016 1964 y Fq(^)19 b Fz(\()p Fr(t)2152 1929 y Fs(0)2152 1984 y Fc(clo)m(ck)5 b Fo(;i)2374 1964 y Fz(=)22 b(0\))c Fq(^)h Fz(\()p Fr(m)2732 1929 y Fs(0)2732 1984 y Fc(fr)m(e)m(e)2871 1964 y Fz(=)j Fr(m)3031 1976 y Fc(fr)m(e)m(e)3165 1964 y Fq(\000)c Fz(1\))523 2122 y(As)28 b(an)f(example,)g(the)h(transition)f(relation)g (for)g(task)g(3)g(in)h(the)g(graph)e(in)i(Figure)f(1)g(is:)606 2305 y Fr(T)655 2317 y Fo(b)684 2325 y Fb(3)720 2305 y Fz(\()m Fr(~)-39 b(s)q(;)11 b(~)-39 b(s)868 2270 y Fs(0)891 2305 y Fz(\))23 b(=)g(\()p Fr(m)1139 2317 y Fc(fr)m(e)m(e)1250 2333 y Fb(1)1305 2305 y Fq(_)c Fr(m)1452 2317 y Fc(fr)m(e)m(e)1563 2333 y Fb(2)1599 2305 y Fz(\))g Fq(^)p 1724 2244 295 4 v 19 w Fr(t)1754 2317 y Fc(starte)m(d)5 b Fo(;)p Fp(3)2037 2305 y Fq(^)18 b Fr(t)2140 2317 y Fc(\014nishe)m(d)7 b Fo(;)p Fp(1)2447 2305 y Fq(^)19 b Fr(t)2551 2317 y Fc(\014nishe)m(d)6 b Fo(;)p Fp(2)1459 2445 y Fq(^)19 b Fr(t)1563 2411 y Fs(0)1563 2466 y Fc(starte)m(d)5 b Fo(;)p Fp(3)1845 2445 y Fq(^)p 1919 2374 319 4 v 19 w Fr(t)1949 2417 y Fs(0)1949 2470 y Fc(\014nishe)m(d)h Fo(;)p Fp(3)2256 2445 y Fq(^)p 2330 2374 238 4 v 19 w Fr(t)2360 2417 y Fs(0)2360 2470 y Fc(clo)m(ck)t Fo(;)p Fp(1)2586 2445 y Fq(^)19 b Fr(m)2733 2411 y Fs(0)2733 2466 y Fc(fr)m(e)m(e)2871 2445 y Fz(=)j Fr(m)3031 2457 y Fc(fr)m(e)m(e)3165 2445 y Fq(\000)c Fz(1\))523 2603 y(W)-7 b(e)27 b(a)n(v)n(oid)d(constrain)n(ts)h(for)h(k)n(eeping)g(the)g (v)-5 b(ariables)25 b(for)h(the)h(other)e(tasks)h(unc)n(hanged)f(b)n(y) 523 2703 y(letting)j(the)g(image)e(computation)h(quan)n(tify)h(and)f (rename)g(only)g(the)h(v)-5 b(ariables)26 b(that)h(can)523 2802 y(c)n(hange)33 b(from)g(the)h(curren)n(t)f(state)h(to)g(the)g (next)g(state)g(in)g(the)g(ab)r(o)n(v)n(e)f(expression.)f(This)523 2902 y(implies)21 b(that)h(the)g(BDD)f(no)r(des)g(for)g(curren)n(t)g (state)g(v)-5 b(ariables)19 b(for)i(the)h(other)e(tasks)h(are)f(not)523 3002 y(a\013ected)25 b(and)g(hence)g(sta)n(y)g(the)g(same.)g(The)g (omitted)h(constrain)n(ts)d(reduce)i(the)h(transition)523 3101 y(relations)g(and)i(thereb)n(y)f(the)h(computation)f(time)h (greatly)-7 b(.)648 3201 y(The)24 b(relations)g Fr(T)1199 3213 y Fo(b)1228 3221 y Ff(i)1258 3201 y Fz(\()m Fr(~)-39 b(s;)11 b(~)-39 b(s)1405 3171 y Fs(0)1429 3201 y Fz(\))25 b(can)f(b)r(e)h(used)g(to)g(compute)g(all)f(the)i(p)r(ossible)e(com)n (binations)523 3301 y(of)j(started)g(tasks)f(with)i(a)f(\014xed)g(p)r (oin)n(t)h(algorithm)e(that)h(at)h(eac)n(h)e(iteration)h(starts)f (either)523 3400 y(of)40 b(the)f(tasks)g(or)g(none.)g(But)h(in)g (practice)e(a)i(more)e(e\016cien)n(t)i(w)n(a)n(y)e(of)i(generating)e (the)523 3500 y(com)n(binations)27 b(is)g(the)h(recurrence:)1354 3658 y Fr(S)1405 3670 y Fo(i)p Fp(+1)1540 3658 y Fz(=)22 b Fq(I)1672 3670 y Fo(T)1711 3679 y Ff(b)1737 3692 y(i)1773 3658 y Fz(\()p Fr(S)1856 3670 y Fo(i)1884 3658 y Fz(\))c Fq([)h Fr(S)2059 3670 y Fo(i)2110 3658 y Fr(;)38 b Fz(1)22 b Fq(\024)h Fr(i)g Fq(\024)f(j)p Fr(G)p Fq(j)523 3816 y Fz(A)n(t)32 b(the)f(\014rst)g(step)h(of)f(the)h(recurrence)e(the)h (curren)n(t)g(state)g(set)g(consists)g(of)g(the)h(original)523 3915 y(state)37 b(set)g(and)f(the)i(states)e(where)h(task)f(1)g(is)h (allo)n(w)n(ed)f(to)h(start.)f(A)n(t)h(the)h(second)e(step)523 4015 y(the)28 b(curren)n(t)e(state)h(set)h(consists)e(of)h(the)h (original)e(state)h(set,)g(the)h(states)f(where)f(task)h(1)g(is)523 4115 y(allo)n(w)n(ed)c(to)h(start,)g(where)g(task)f(2)h(ma)n(y)g(start) g(and)g(where)g(b)r(oth)g(task)g(1)g(and)g(2)g(ma)n(y)g(start.)523 4214 y(And)k(so)f(forth.)523 4444 y Fd(4.3)95 b(Run)31 b(transition)g(relation)523 4590 y Fz(The)f(next)g(phase)g(in)g(eac)n (h)f(iteration)g(ensures)g(for)g(eac)n(h)h(task)f(that)h(either)g(the)g (task)g(has)523 4690 y(not)e(started,)f(it)h(has)f(\014nished)h(or)e (the)i(clo)r(c)n(k)f(of)h(the)g(task)f(is)g(incremen)n(ted)h(b)n(y)f (one:)968 4860 y Fr(T)1017 4872 y Fo(r)1053 4860 y Fz(\()m Fr(~)-39 b(s;)11 b(~)-39 b(s)1200 4825 y Fs(0)1223 4860 y Fz(\))24 b(=)1366 4781 y Fm(^)1401 4958 y Fo(i)p 1473 4799 285 4 v 1473 4860 a Fr(t)1503 4872 y Fc(starte)m(d)5 b Fo(;i)1776 4860 y Fq(_)18 b Fr(t)1879 4872 y Fc(\014nishe)m(d)7 b Fo(;i)2176 4860 y Fq(_)19 b Fr(t)2280 4825 y Fs(0)2280 4880 y Fc(clo)m(ck)5 b Fo(;i)2501 4860 y Fz(=)23 b Fr(t)2619 4872 y Fc(clo)m(ck)5 b Fo(;i)2836 4860 y Fz(+)18 b(1)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 523 232 a Fy(8)523 448 y Fz(Again,)35 b(the)g(image)f(computation)h(only)f(quan)n(ti\014es)h(and)f(renames)g (v)-5 b(ariables)34 b(that)h(are)523 548 y(supp)r(osed)28 b(to)h(c)n(hange,)f(here)g(the)h(clo)r(c)n(k)f(v)-5 b(ariables,)27 b(so)h(that)h(w)n(e)f(a)n(v)n(oid)g(ha)n(ving)f(to)i(set)f(all)523 648 y(other)d(v)-5 b(ariables)24 b(equal)h(to)g(their)h(successor.)d(F) -7 b(or)25 b(the)h(graph)e(in)i(Figure)f(1)g(the)h(transition)523 747 y(relation)h(will)h(b)r(e:)956 910 y Fr(T)1005 922 y Fo(r)1041 910 y Fz(\()m Fr(~)-39 b(s;)11 b(~)-39 b(s)1188 876 y Fs(0)1212 910 y Fz(\))23 b(=)p 1355 849 295 4 v 23 w Fr(t)1385 922 y Fc(starte)m(d)5 b Fo(;)p Fp(1)1667 910 y Fq(_)19 b Fr(t)1771 922 y Fc(\014nishe)m(d)6 b Fo(;)p Fp(1)2078 910 y Fq(_)18 b Fr(t)2181 876 y Fs(0)2181 931 y Fc(clo)m(ck)5 b Fo(;i)2403 910 y Fz(=)22 b Fr(t)2520 922 y Fc(clo)m(ck)5 b Fo(;i)2737 910 y Fz(+)18 b(1)c Fq(\001)g(\001)g(\001)523 1144 y Fd(4.4)95 b(End)31 b(transition)g (relation)523 1294 y Fz(The)26 b(last)g(phase)f(in)h(eac)n(h)f (iteration)h(ensures)f(that)h(all)g(tasks)f(that)h(ha)n(v)n(e)f(run)h (their)f(en)n(tire)523 1394 y(time)35 b(are)f(mark)n(ed)g(as)g (\014nished.)h(This)g(is)f(done)h(b)n(y)f(computing)h(the)g(image)f (where)h(the)523 1494 y(\014rst)i(task)g(is)h(stopp)r(ed)g(if)g(p)r (ossible,)f(then)h(computing)g(the)g(image)f(where)g(the)h(second)523 1593 y(task)33 b(is)g(stopp)r(ed)g(if)h(p)r(ossible,)f(etc.)h(The)f (relation)f(for)h(task)g Fr(t)2532 1605 y Fo(i)2593 1593 y Fz(is)g(a)g(disjunction)g(of)h(the)523 1693 y(t)n(w)n(o)25 b(p)r(ossibilities)h(that)h(either)f(the)h(task)e(is)h(not)g(running)g (and)g(the)h(state)f(should)g(remain)523 1792 y(unc)n(hanged,)31 b(or)g(the)i(task)e(is)h(running,)g(in)g(whic)n(h)g(case)f(it)h(has)g (\014nished)g(if)g(the)h(clo)r(c)n(k)e(of)523 1892 y(the)d(task)f(is)h Fn(T)p Fz(\()p Fr(t)1046 1904 y Fo(i)1073 1892 y Fz(\))g(or)f(else)g (remain)g(unc)n(hanged:)592 2055 y Fr(T)641 2067 y Fo(e)672 2075 y Ff(i)702 2055 y Fz(\()m Fr(~)-39 b(s)q(;)11 b(~)-39 b(s)850 2021 y Fs(0)873 2055 y Fz(\))23 b(=)1006 1988 y Fm(\000)1045 2055 y Fz(\()p 1077 1994 285 4 v Fr(t)1107 2067 y Fc(starte)m(d)5 b Fo(;i)1380 2055 y Fq(_)19 b Fr(t)1484 2067 y Fc(\014nishe)m(d)6 b Fo(;i)1762 2055 y Fz(\))1108 2189 y Fq(^)19 b Fr(t)1212 2155 y Fs(0)1212 2209 y Fc(starte)m(d)5 b Fo(;i)1489 2189 y Fq($)23 b Fr(t)1625 2201 y Fc(starte)m(d)6 b Fo(;i)1899 2189 y Fq(^)18 b Fr(t)2002 2155 y Fs(0)2002 2209 y Fc(\014nishe)m(d)7 b Fo(;i)2304 2189 y Fq($)23 b Fr(t)2440 2201 y Fc(\014nishe)m(d)6 b Fo(;i)2737 2189 y Fq(^)19 b Fr(m)2884 2155 y Fs(0)2884 2209 y Fc(fr)m(e)m(e)3022 2189 y Fz(=)k Fr(m)3183 2201 y Fc(fr)m(e)m(e)3298 2122 y Fm(\001)1025 2324 y Fq(_)1099 2257 y Fm(\000)1137 2324 y Fr(t)1167 2336 y Fc(starte)m(d)5 b Fo(;i)1440 2324 y Fq(^)p 1514 2263 309 4 v 19 w Fr(t)1544 2336 y Fc(\014nishe)m(d)h Fo(;i)1108 2449 y Fq(^)19 b Fz(\()p Fr(t)1244 2461 y Fc(clo)m(ck)5 b Fo(;i)1465 2449 y Fz(=)23 b Fn(T)p Fz(\()p Fr(t)1671 2461 y Fo(i)1698 2449 y Fz(\))c Fq(^)g Fr(t)1853 2415 y Fs(0)1853 2470 y Fc(starte)m(d)5 b Fo(;i)2126 2449 y Fq(^)19 b Fr(t)2230 2415 y Fs(0)2230 2470 y Fc(\014nishe)m(d)6 b Fo(;i)2527 2449 y Fq(^)18 b Fr(m)2673 2415 y Fs(0)2673 2470 y Fc(fr)m(e)m(e)2812 2449 y Fz(=)k Fr(m)2972 2461 y Fc(fr)m(e)m(e)3106 2449 y Fz(+)c(1)1191 2591 y Fq(_)h Fr(t)1295 2603 y Fc(clo)m(ck)5 b Fo(;i)1516 2591 y Fq(6)p Fz(=)23 b Fn(T)p Fz(\()p Fr(t)1722 2603 y Fo(i)1749 2591 y Fz(\))c Fq(^)g Fr(t)1904 2557 y Fs(0)1904 2612 y Fc(starte)m(d)5 b Fo(;i)2177 2591 y Fq(^)p 2250 2521 V 18 w Fr(t)2280 2563 y Fs(0)2280 2617 y Fc(\014nishe)m(d)i Fo(;i)2577 2591 y Fq(^)19 b Fr(t)2681 2557 y Fs(0)2681 2612 y Fc(clo)m(ck)5 b Fo(;i)2902 2591 y Fz(=)23 b Fr(t)3020 2603 y Fc(clo)m(ck)5 b Fo(;i)2678 2732 y Fq(^)19 b Fr(m)2825 2697 y Fs(0)2825 2752 y Fc(fr)m(e)m(e)2964 2732 y Fz(=)j Fr(m)3124 2744 y Fc(fr)m(e)m(e)3239 2732 y Fz(\))3271 2664 y Fm(\001)523 2895 y Fz(The)36 b(image)e(computation) i(then)g(only)f(quan)n(ti\014es)g(and)g(renames)g(all)g(the)h(v)-5 b(ariables)34 b(of)523 2994 y(task)27 b Fr(t)731 3006 y Fo(i)787 2994 y Fz(and)h Fr(m)1022 3006 y Fc(fr)m(e)m(e)1137 2994 y Fz(.)g(Note)g(that)g(there)g(are)f(no)g(requiremen)n(ts)g(for)g (the)i(clo)r(c)n(k)e(v)-5 b(ariables)26 b(if)523 3094 y(task)h Fr(t)731 3106 y Fo(i)786 3094 y Fz(has)g(not)h(started)f(or)g (has)g(\014nished.)523 3327 y Fd(4.5)95 b(Bounding)30 b(the)i(p)s(ossible)e(task)i(running)g(times)523 3478 y Fz(Because)g(of)h(the)g(dep)r(endencies)g(in)g(the)g(task)g(graphs,)e (most)i(tasks)f(cannot)g(start)h(un)n(til)523 3578 y(after)27 b(a)h(certain)f(p)r(oin)n(t)h(in)g(time,)g(the)g(earliest)f(p)r (ossible)h(start)f(time)h Fn(E)p Fz(\()p Fr(t)2840 3590 y Fo(i)2868 3578 y Fz(\).)g(F)-7 b(urthermore)523 3677 y(most)36 b(tasks)f(m)n(ust)h(ha)n(v)n(e)f(started)g(b)r(efore)h(a)f (certain)h(other)f(p)r(oin)n(t)h(in)g(time,)h(the)f(latest)523 3777 y(p)r(ossible)27 b(start)g(time)h Fn(L)p Fz(\()p Fr(t)1340 3789 y Fo(i)1369 3777 y Fz(\),)g(in)g(order)e(to)h(yield)h (an)f(optimal)h(sc)n(hedule.)648 3876 y(A)c(task)f Fr(t)938 3888 y Fo(i)989 3876 y Fz(cannot)g(p)r(ossibly)h(start)f(b)r(efore)g (its)h(complete)g(set)f(of)h(dep)r(endencies)g Fq(D)3249 3888 y Fo(c)3283 3876 y Fz(\()p Fr(t)3345 3888 y Fo(i)3373 3876 y Fz(\))523 3976 y(ha)n(v)n(e)h(run)g(so)g Fn(E)p Fz(\()p Fr(t)1080 3988 y Fo(i)1108 3976 y Fz(\))h(is)g(the)g(length)g (of)f(an)h(optimal)f(sc)n(hedule)h(for)f(the)h(subgraph)e(with)j(the) 523 4076 y(tasks)d(in)g Fq(D)888 4088 y Fo(c)922 4076 y Fz(\()p Fr(t)984 4088 y Fo(i)1012 4076 y Fz(\).)h(This)g(is)f(of)h (course)e(the)i(v)n(ery)e(problem)h(w)n(e)g(are)g(trying)g(to)g(solv)n (e,)f(but)i(w)n(e)523 4175 y(can)k(obtain)h(a)f(safe)g(estimate)h(quic) n(kly)f(b)n(y)g(observing)f(that)i(the)g(start)f(time)h(is)g(b)r (ounded)523 4275 y(from)c(b)r(elo)n(w)g(b)n(y)g(the)h(length)f(of)h (the)f(critical)g(path)h(to)f(the)h(task)e(and)h(also)g(b)n(y)g(the)h (sum)f(of)523 4375 y(the)i(execution)f(times)h(of)g(the)g(tasks)e(in)i Fq(D)1868 4387 y Fo(c)1902 4375 y Fz(\()p Fr(t)1964 4387 y Fo(i)1992 4375 y Fz(\))g(divided)g(b)n(y)f(the)h(n)n(um)n(b)r(er)g (of)f(pro)r(cessors:)1306 4554 y Fn(E)p Fz(\()p Fr(t)1423 4566 y Fo(i)1450 4554 y Fz(\))d Fq(\025)e Fz(max\()p Fl(CP)9 b Fz(\()p Fr(t)1966 4566 y Fo(i)1994 4554 y Fz(\))p Fr(;)14 b Fq(d)2102 4475 y Fm(X)2029 4657 y Fo(d)p Fs(2D)2161 4665 y Ff(c)2192 4657 y Fp(\()p Fo(t)2243 4665 y Ff(i)2269 4657 y Fp(\))2238 4554 y Fn(T)p Fz(\()p Fr(t)2356 4566 y Fo(d)2394 4554 y Fz(\))p Fr(=)-5 b(M)9 b Fq(e)p Fz(\))648 4814 y(F)-7 b(or)27 b(the)h(example)g(in)g(Figure)g(1,)f Fl(CP)9 b Fz(\()p Fr(t)1904 4826 y Fp(3)1942 4814 y Fz(\))24 b(=)f(3)28 b(and)2318 4752 y Fm(P)2405 4839 y Fo(d)p Fs(2D)2537 4847 y Ff(c)2567 4839 y Fp(\()p Fo(t)2618 4847 y Fb(3)2651 4839 y Fp(\))2681 4814 y Fz(\(2)18 b(+)h(3\))p Fr(=)p Fz(2)j(=)h(3)28 b(whic)n(h)523 4924 y(giv)n(es)e Fn(E)p Fz(\()p Fr(t)843 4936 y Fp(3)881 4924 y Fz(\))d(=)g(3.)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 3367 232 a Fy(9)648 448 y Fz(No)n(w)33 b(consider)g(a)h(task)f Fr(t)1463 460 y Fo(i)1525 448 y Fz(and)h(the)h(subgraph)d Fr(G)j Fz(with)g Fr(t)2535 460 y Fo(i)2596 448 y Fz(and)f(all)g(the)g(tasks)g(that)523 548 y(dep)r(end)c(on)g Fr(t)961 560 y Fo(i)1018 548 y Fz(\(i.e.)g Fq(f)p Fr(t)1258 560 y Fo(j)1292 548 y Fq(j)p Fr(t)1345 560 y Fo(i)1399 548 y Fq(2)d(D)1545 560 y Fo(c)1579 548 y Fz(\()p Fr(t)1641 560 y Fo(j)1676 548 y Fz(\))p Fq(g)p Fz(\).)j(T)-7 b(ask)29 b Fr(t)2066 560 y Fo(i)2123 548 y Fz(m)n(ust)h(ha)n(v)n(e)e(started)h(and)h(run)f(to)h(com-)523 648 y(pletion)k(some)g(time)h(b)r(efore)f(the)g(sc)n(hedule)g(is)g (complete)g({)g(else)g(the)g(other)g(tasks)f(in)i Fr(G)523 747 y Fz(w)n(ould)24 b(not)g(ha)n(v)n(e)f(time)i(to)f(run.)g(More)f (precisely)-7 b(,)23 b Fr(t)2143 759 y Fo(i)2195 747 y Fz(m)n(ust)h(ha)n(v)n(e)f(\014nished)i(in)f(time)h(for)f(the)523 847 y(longest)e(path)h(from)f Fr(t)1210 859 y Fo(i)1260 847 y Fz(to)h(some)f(leaf)h(no)r(de)f(in)h Fr(G)g Fz(to)g(execute.)f (Hence,)h Fn(L)p Fz(\()p Fr(t)2879 859 y Fo(i)2907 847 y Fz(\))g(is)g(the)g(length)523 946 y(of)31 b(the)g(critical)g(path)g (in)g Fr(G)g Fz(subtracted)g(from)f(the)h(length)g Fr(O)j Fz(of)d(an)g(optimal)f(sc)n(hedule,)523 1046 y Fn(L)p Fz(\()p Fr(t)640 1058 y Fo(i)668 1046 y Fz(\))25 b(=)g Fr(O)c Fq(\000)e Fl(CP)9 b Fz(\()p Fr(G)p Fz(\).)30 b(F)-7 b(or)28 b(the)h(example)f(in)h(Figure)f(1,)g Fr(G)2428 1058 y Fo(t)2453 1066 y Fb(3)2515 1046 y Fz(=)c Fq(f)p Fr(t)2676 1058 y Fp(4)2713 1046 y Fr(;)14 b(t)2780 1058 y Fp(5)2817 1046 y Fq(g)p Fz(,)29 b Fl(CP)9 b Fz(\()p Fr(G)3132 1058 y Fo(t)3157 1066 y Fb(3)3194 1046 y Fz(\))25 b(=)f(3,)523 1146 y Fr(O)i Fz(=)c(3)c(+)g(1)g(+)g(3)23 b(=)g(7)k(resulting)g(in)h Fn(L)p Fz(\()p Fr(t)1763 1158 y Fp(3)1800 1146 y Fz(\))c(=)e(6)c Fq(\000)g Fz(3)23 b(=)g(3.)648 1245 y Fr(O)31 b Fz(is)d(of)g(course)f(not)h(kno)n(wn,)g (but)h(an)f(o)n(v)n(erestimate)e(can)i(b)r(e)h(computed)f(with)h(a)f (fast)523 1345 y(heuristic)34 b(algorithm)f(that)i(generates)e(go)r(o)r (d)g(sc)n(hedules.)h(One)g(suc)n(h)g(simple)g(algorithm)523 1445 y(is)d(a)g(sligh)n(t)g(mo)r(di\014cation)g(of)g(this)h (breadth-\014rst)e(searc)n(h)g(algorithm)g(that)h(at)g(eac)n(h)g(step) 523 1544 y(only)c(pic)n(ks)g(the)h(sc)n(hedules)f(that)h(starts)f(the)h (most)f(tasks.)648 1644 y(A)n(t)35 b(eac)n(h)f(step)h(in)g(the)g (algorithm,)f(the)h(b)r(ounds)g Fn(E)p Fz(\()p Fr(t)2378 1656 y Fo(i)2406 1644 y Fz(\))g(and)g Fn(L)p Fz(\()p Fr(t)2759 1656 y Fo(i)2787 1644 y Fz(\))h(can)e(b)r(e)h(used)g(to)523 1743 y(a)n(v)n(oid)30 b(computing)i(images)f(o)n(v)n(er)f(transition)h (relations)f(that)i(cannot)f(p)r(ossibly)h(start)f(or)523 1843 y(end)e(a)f(task)g(at)h(that)f(step)h({)f(in)h(practice,)f(this)h (w)n(as)f(found)g(to)h(sa)n(v)n(e)e(at)h(least)h(50{60\045)d(of)523 1943 y(the)f(start)g(and)f(end)i(image)e(computations.)g(F)-7 b(urthermore,)24 b Fn(L)p Fz(\()p Fr(t)2566 1955 y Fo(i)2594 1943 y Fz(\))i(can)e(b)r(e)h(used)g(to)g(prune)523 2042 y(the)e(state)f(sets)g(from)g(states)g(where)f(a)h(task)g(that)h(w)n (ould)f(certainly)f(ha)n(v)n(e)g(b)r(een)i(completed)523 2142 y(in)28 b(an)f(optimal)g(sc)n(hedule)g(is)h(not)f(\014nished)h(y)n (et)f({)g(refer)g(to)h(Section)f(4.6)g(for)g(details.)g(This)523 2242 y(w)n(as)g(also)f(found)i(to)g(sp)r(eed)f(up)h(the)g(algorithm)f (\(b)n(y)g(more)g(than)h(10\045\).)648 2341 y(A)23 b(further)h(p)r (ossible)f(optimisation)f(is)i(to)f(k)n(eep)g(trac)n(k)f(of)h(already)f (reac)n(hed)g(states)h(and)523 2441 y(prune)35 b(the)h(curren)n(t)e (state)h(set)h(from)f(these)g(at)g(eac)n(h)f(iteration.)h(This)g(will)h (reduce)f(the)523 2540 y(n)n(um)n(b)r(er)29 b(of)g(states,)g(but)h (empirically)e(it)h(turns)g(out)h(that)f(the)h(BDDs)f(for)g(the)g (state)g(sets)523 2640 y(actually)k(end)h(up)f(larger)f(so)h(that)h (the)f(algorithm)g(b)r(ecomes)g(slo)n(w)n(er)f(\(ab)r(out)h(t)n(wice)h (the)523 2740 y(running)28 b(time\).)g(An)h(explanation)e(for)g(this)h (could)g(b)r(e)g(that)h(the)f(extra)f(states)g(can)h(mak)n(e)523 2839 y(v)-5 b(ariables)29 b(in)h(some)f(paths)h(irrelev)-5 b(an)n(t)29 b(whic)n(h)h(mak)n(es)f(it)h(p)r(ossible)g(to)g(reduce)f (the)i(BDDs)523 2939 y(more.)523 3176 y Fd(4.6)95 b(The)32 b(complete)e(algorithm)523 3330 y Fz(The)h(complete)h(algorithm)e(app)r (ears)g(in)i(Listing)f(1.1.)g(Line)g(3{5)f(put)i(the)g(curren)n(t)e (state)523 3430 y(set)i(through)g(the)g(start)g(transition)f(relations) g(of)h(the)h(tasks)e(that)i(are)e(able)h(to)g(start)f(at)523 3530 y(this)h(p)r(oin)n(t,)g(line)f(6)h(adv)-5 b(ances)30 b(the)i(clo)r(c)n(ks)f(one)g(step)h(with)g(the)g(run)f(transition)g (relation)523 3629 y(and)f(line)g(7{9)e(\014nishes)i(tasks)f(with)h (the)h(end)f(transition)f(relations)f(of)i(those)g(tasks)f(that)523 3729 y(can)23 b(p)r(ossibly)f(\014nish.)i(Line)f(10{11)d(cuts)k(of)f (states)f(with)i(un\014nished)f(tasks)f(that)i(certainly)523 3828 y(w)n(ould)j(ha)n(v)n(e)g(had)g(\014nished)h(in)g(an)f(optimal)g (sc)n(hedule.)648 3928 y(When)g(the)h(algorithm)e(terminates,)h(the)h (length)f(of)h(an)f(optimal)g(sc)n(hedule)g(is)g(simply)523 4028 y(the)37 b(n)n(um)n(b)r(er)f(of)h(iterations)f(in)h(the)g(lo)r (op.)f(T)-7 b(o)36 b(actually)g(obtain)h(an)f(optimal)g(sc)n(hedule)523 4127 y(the)e(algorithm)f(m)n(ust)i(b)r(e)f(mo)r(di\014ed)h(to)e(sa)n(v) n(e)g(the)h(curren)n(t)g(state)g Fr(S)k Fz(at)c(eac)n(h)g(iteration.) 523 4227 y(Denote)k(these)f(b)n(y)g Fr(S)1216 4239 y Fp(1)1253 4227 y Fr(;)14 b(:)g(:)g(:)g(;)g(S)1489 4239 y Fo(n)1571 4227 y Fz(and)37 b(let)h(for)f(the)g(sak)n(e)f(of)i(the)f (explanation)g Fr(T)48 b Fz(denote)523 4327 y(a)36 b(com)n(bined)g (abstract)f(transition)h(relation)f(that)i(encompasses)e(b)r(oth)i(the) f(start,)g(run)523 4426 y(and)29 b(end)g(transition)f(relations.)g (Then)h(a)g(sequence)f(of)h(states)f Fr(s)2592 4438 y Fp(1)2630 4426 y Fr(;)14 b(:)g(:)g(:)f(;)h(s)2853 4438 y Fo(n)2927 4426 y Fz(leading)28 b(to)h(an)523 4526 y(optimal)j(sc)n (hedule,)f(i.e.)h Fr(s)1366 4538 y Fp(1)1433 4526 y Fz(=)e Fr(s)1567 4538 y Fc(init)1681 4526 y Fz(,)i Fr(s)1775 4538 y Fo(n)1850 4526 y Fq(2)f Fr(S)1987 4538 y Fc(go)m(al)2144 4526 y Fz(and)h(\()p Fr(s)2381 4538 y Fo(\034)2423 4526 y Fr(;)14 b(s)2499 4538 y Fo(\034)7 b Fp(+1)2624 4526 y Fz(\))30 b Fq(2)h Fr(T)43 b Fz(for)31 b(1)f Fq(\024)g Fr(\034)39 b(<)30 b(n)p Fz(,)523 4625 y(can)c(b)r(e)h(obtained)g(b)n(y) f(setting)h Fr(s)1552 4637 y Fo(n)1623 4625 y Fz(to)g(a)f(solution)g (in)h Fr(S)2252 4637 y Fo(n)2324 4625 y Fz(and)f(pic)n(king)g Fr(s)2808 4637 y Fo(n)p Fs(\000)p Fp(1)2938 4625 y Fr(;)14 b(s)3014 4637 y Fo(n)p Fs(\000)p Fp(2)3144 4625 y Fr(;)g(:)g(:)g(:)g(;) g(s)3368 4637 y Fp(1)523 4725 y Fz(with)28 b(the)g(recurrence)e Fr(s)1296 4737 y Fo(\034)1361 4725 y Fq(2)d(I)1490 4690 y Fs(\000)p Fp(1)1484 4749 y Fo(T)1580 4725 y Fz(\()p Fr(s)1651 4737 y Fo(\034)7 b Fp(+1)1776 4725 y Fz(\))19 b Fq(\\)g Fr(S)1952 4737 y Fo(\034)1993 4725 y Fz(.)28 b(With)g(the)g(states)g Fr(s)2677 4737 y Fp(1)2714 4725 y Fr(;)14 b(:)g(:)g(:)f(;)h(s)2937 4737 y Fo(n)2982 4725 y Fz(,)28 b(a)f(sc)n(hedule)523 4825 y(can)i(b)r(e)h(constructed)e(b)n (y)h(examining)g(the)h(states)e(in)i(turn)f(to)g(disco)n(v)n(er)f(when) h(eac)n(h)g(task)523 4924 y(is)f(started)f(\(b)n(y)g(conjugation)g (with)h(the)g(constrain)n(t)e Fr(t)2238 4936 y Fc(starte)m(d)6 b Fo(;i)2493 4924 y Fz(\).)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 523 232 a Fy(10)412 429 y Fb(1)83 b Fj(S)30 b Fy(=)25 b Fx(f)p Fj(s)759 437 y Fh(init)870 429 y Fx(g)412 521 y Fb(2)83 b Fj(\034)30 b Fy(=)21 b(0)412 612 y Fb(3)83 b Fv(while)24 b Fy(\()p Fj(S)d Fx(\\)d Fj(S)965 621 y Fh(g)r(oal)1085 612 y Fy(\))j(=)g Fx(;)412 703 y Fb(4)152 b Fv(for)26 b(eac)n(h)g Fy(task)f Fj(t)1113 711 y Fh(i)412 795 y Fb(5)221 b Fv(if)25 b Fk(E)p Fy(\()p Fj(t)847 803 y Fh(i)873 795 y Fy(\))c Fx(\024)g Fj(\034)30 b Fx(\024)21 b Fk(L)p Fy(\()p Fj(t)1259 803 y Fh(i)1285 795 y Fy(\))412 886 y Fb(6)290 b Fj(S)30 b Fy(=)25 b Fx(I)934 894 y Fh(T)970 906 y Ff(b)996 919 y(i)1032 886 y Fy(\()p Fj(S)t Fy(\))17 b Fx([)g Fj(S)412 991 y Fb(7)152 b Fj(S)30 b Fy(=)25 b Fx(I)796 999 y Fh(T)832 1007 y Ff(r)870 991 y Fy(\()p Fj(S)t Fy(\))412 1083 y Fb(8)152 b Fv(for)26 b(eac)n(h)g Fy(task)f Fj(t)1113 1091 y Fh(i)412 1174 y Fb(9)221 b Fv(if)25 b Fk(E)p Fy(\()p Fj(t)847 1182 y Fh(i)873 1174 y Fy(\))17 b(+)g Fk(T)p Fy(\()p Fj(t)1106 1182 y Fh(i)1132 1174 y Fy(\))k Fx(\024)g Fj(\034)30 b Fx(\024)21 b Fk(L)p Fy(\()p Fj(t)1518 1182 y Fh(i)1544 1174 y Fy(\))c(+)f Fk(T)p Fy(\()p Fj(t)1776 1182 y Fh(i)1803 1174 y Fy(\))384 1265 y Fb(10)290 b Fj(S)30 b Fy(=)25 b Fx(I)934 1273 y Fh(T)970 1281 y Ff(e)998 1294 y(i)1034 1265 y Fy(\()p Fj(S)t Fy(\))384 1368 y Fb(11)221 b Fv(if)25 b Fy(=)h Fk(L)p Fy(\()p Fj(t)933 1376 y Fh(i)959 1368 y Fy(\))17 b(+)f Fk(T)p Fy(\()p Fj(t)1191 1376 y Fh(i)1217 1368 y Fy(\))384 1459 y Fb(12)290 b Fj(S)30 b Fy(=)25 b Fj(S)d Fx(\\)17 b Fj(t)1057 1468 y Fa(\014nishe)m(d)5 b Fh(;i)384 1550 y Fb(13)152 b Fj(\034)30 b Fy(=)21 b Fj(\034)k Fy(+)17 b(1)612 1675 y Fd(Listing)30 b(1.1.)d Fz(The)g(breadth-\014rst)g(based)g(algorithm;)f Fr(S)33 b Fz(is)27 b(the)h(curren)n(t)f(state)g(set.)523 1996 y Ft(5)112 b(A)37 b(guided)i(BDD)d(algorithm)523 2217 y Fz(Instead)23 b(of)f(searc)n(hing)f(exhaustiv)n(ely)h(for)g(an)h (optimal)g(sc)n(hedule)f(in)h(a)g(breadth-\014rst)e(man-)523 2317 y(ner,)28 b(it)h(is)f(p)r(ossible)g(to)g(guide)g(the)h(searc)n(h)d (with)j(an)f(adaptation)f(of)i(A)2775 2287 y Fs(\003)2841 2317 y Fz(for)f(BDDs)h([7,)13 b(8].)523 2416 y(The)27 b(cost)g(of)h(reac)n(hing)d(eac)n(h)i(set)g(of)h(states)f(is)g(main)n 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y(Instead)h(w)n(e)g(let)h(the)f(cost)g(b)r(e)h(the)g(n)n(um)n (b)r(er)f(of)h(free)f(time)h(slots)f(in)g(the)h(sc)n(hedule)f(and)523 3421 y(use)38 b(the)g(n)n(um)n(b)r(er)f(of)h(free)f(slots)h(at)f(the)h (next)g(step)g(of)g(the)g(sc)n(hedule)g(as)f(an)g(estimate)523 3521 y(of)32 b(the)g(remaining)f(free)h(slots.)f(These)g(c)n(hoices)g (will)h(giv)n(e)f(optimal)h(sc)n(hedules)f(since)h(the)523 3621 y(n)n(um)n(b)r(er)h(of)g(free)h(slots)e(at)i(the)f(next)h(step)f (is)h(a)f(lo)n(w)n(er)e(b)r(ound)j(on)f(the)h(total)f(remaining)523 3720 y(free)i(slots,)g(and)g(a)g(sc)n(hedule)h(with)g(the)f(few)n(est)h (p)r(ossible)f(free)g(time)h(slots)f(m)n(ust)g(b)r(e)h(an)523 3820 y(optimal)28 b(sc)n(hedule.)h(T)-7 b(o)28 b(see)g(this,)h (consider)f(the)h(sc)n(hedules)f(in)h(Figure)f(1)g(where)g(the)h(t)n(w) n(o)523 3919 y(optimal)23 b(sc)n(hedules)g(b)r(oth)h(ha)n(v)n(e)f (three)g(free)h(time)g(slots,)f(whereas)f(the)i(sc)n(hedule)f(with)h (\014v)n(e)523 4019 y(free)34 b(slots)g(is)h(one)f(step)h(longer)e({)h (more)g(free)g(time)h(slots)f(necessarily)f(means)h(a)g(longer)523 4119 y(sc)n(hedule.)648 4223 y(The)29 b(algorithm,)e(giv)n(en)h(in)i (Listings)e(1.2,)g(w)n(orks)f(b)n(y)i(main)n(taining)f(a)h(priorit)n(y) e(queue)523 4322 y(of)33 b(visited)h(states)f(and)g(their)h(costs.)e (When)i(a)g(state)f(set)g(has)g(b)r(een)h(through)f(the)h(start)523 4422 y(transition)d(\(line)h(7-8\),)f(it)h(is)f(\(on)h(line)g(9-15\))e (split)i(according)e(to)h(ho)n(w)g(man)n(y)g(free)h(slots)523 4522 y(the)c(transition)e(creates,)g(and)h(the)g(new)h(state)e(sets)h (are)f(then)i(enqueued)f(with)h(their)f(new)523 4621 y(cost.)g(States)h(with)g(equal)f(cost)g(are)g(merged.)648 4725 y(When)34 b(the)h(algorithm)e(has)g(\014nished,)i(the)f(length)g (of)g(an)g(optimal)g(sc)n(hedule)g(is)g(the)523 4825 y(sum)26 b(of)f(the)h(execution)f(times)h(for)f(all)g(tasks)g(and)g (the)h(n)n(um)n(b)r(er)f(of)g(free)h(slots,)f(divided)g(b)n(y)523 4924 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add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall currentdict end definefont pop end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 630 700 M 63 0 V 6269 0 R -63 0 V 546 700 M ( 0) Rshow 630 1048 M 63 0 V 6269 0 R -63 0 V -6353 0 R ( 10) Rshow 630 1395 M 63 0 V 6269 0 R -63 0 V -6353 0 R ( 20) Rshow 630 1743 M 63 0 V 6269 0 R -63 0 V -6353 0 R ( 30) Rshow 630 2091 M 63 0 V 6269 0 R -63 0 V -6353 0 R ( 40) Rshow 630 2439 M 63 0 V 6269 0 R -63 0 V -6353 0 R ( 50) Rshow 630 2786 M 63 0 V 6269 0 R -63 0 V -6353 0 R ( 60) Rshow 682 700 M 0 63 V 0 2093 R 0 -63 V 682 560 M ( 0) Cshow 1729 700 M 0 63 V 0 2093 R 0 -63 V 0 -2233 R ( 20) Cshow 2776 700 M 0 63 V 0 2093 R 0 -63 V 0 -2233 R ( 40) Cshow 3822 700 M 0 63 V 0 2093 R 0 -63 V 0 -2233 R ( 60) Cshow 4869 700 M 0 63 V 0 2093 R 0 -63 V 0 -2233 R ( 80) Cshow 5915 700 M 0 63 V 0 2093 R 0 -63 V 0 -2233 R ( 100) Cshow 6962 700 M 0 63 V 0 2093 R 0 -63 V 0 -2233 R ( 120) Cshow 1.000 UL LTb 630 700 M 6332 0 V 0 2156 V -6332 0 V 630 700 L 140 1778 M currentpoint gsave translate 90 rotate 0 0 M (Running time \(minutes\)) Cshow grestore 3796 350 M (Task graph no.) Cshow 1.000 UP 1.000 UL LT0 1670 70 M (2 processors) Rshow 682 2786 Pls 735 763 Pls 787 862 Pls 839 2786 Pls 892 717 Pls 944 1270 Pls 996 2786 Pls 1049 2786 Pls 1101 860 Pls 1153 2786 Pls 1206 2786 Pls 1258 748 Pls 1310 1185 Pls 1363 2786 Pls 1415 2786 Pls 1467 740 Pls 1520 2786 Pls 1572 2786 Pls 1624 760 Pls 1677 1049 Pls 1729 2786 Pls 1781 730 Pls 1834 774 Pls 1886 2498 Pls 1938 2786 Pls 1991 884 Pls 2043 2786 Pls 2095 2786 Pls 2148 796 Pls 2200 1529 Pls 2252 2786 Pls 2305 2786 Pls 2357 2786 Pls 2409 2786 Pls 2462 2786 Pls 2514 2786 Pls 2566 2786 Pls 2619 2786 Pls 2671 2786 Pls 2723 2786 Pls 2776 2786 Pls 2828 2786 Pls 2880 2786 Pls 2933 2786 Pls 2985 2786 Pls 3037 2786 Pls 3090 2786 Pls 3142 2786 Pls 3194 2786 Pls 3247 2786 Pls 3299 2786 Pls 3351 2786 Pls 3404 2786 Pls 3456 2786 Pls 3508 2786 Pls 3561 2786 Pls 3613 2786 Pls 3665 2786 Pls 3718 2786 Pls 3770 2786 Pls 3822 705 Pls 3874 708 Pls 3927 2786 Pls 3979 705 Pls 4031 708 Pls 4084 2786 Pls 4136 705 Pls 4188 707 Pls 4241 783 Pls 4293 705 Pls 4345 705 Pls 4398 706 Pls 4450 2786 Pls 4502 705 Pls 4555 706 Pls 4607 786 Pls 4659 705 Pls 4712 706 Pls 4764 714 Pls 4816 2786 Pls 4869 705 Pls 4921 706 Pls 4973 794 Pls 5026 704 Pls 5078 705 Pls 5130 723 Pls 5183 2786 Pls 5235 706 Pls 5287 708 Pls 5340 2786 Pls 5392 714 Pls 5444 755 Pls 5497 1370 Pls 5549 716 Pls 5601 725 Pls 5654 982 Pls 5706 2786 Pls 5758 738 Pls 5811 926 Pls 5863 2786 Pls 5915 716 Pls 5968 719 Pls 6020 826 Pls 6072 2786 Pls 6125 712 Pls 6177 854 Pls 6229 2786 Pls 6282 755 Pls 6334 796 Pls 6386 2786 Pls 6439 712 Pls 6491 717 Pls 6543 761 Pls 6596 2786 Pls 6648 713 Pls 6700 762 Pls 6753 2786 Pls 6805 739 Pls 6857 757 Pls 6910 843 Pls 1953 70 Pls 1.000 UP 1.000 UL LT5 3958 70 M (4 processors) Rshow 682 2786 Circle 735 837 Circle 787 765 Circle 839 2786 Circle 892 710 Circle 944 1673 Circle 996 2786 Circle 1049 2786 Circle 1101 729 Circle 1153 2786 Circle 1206 2786 Circle 1258 714 Circle 1310 793 Circle 1363 1638 Circle 1415 2786 Circle 1467 716 Circle 1520 1577 Circle 1572 2786 Circle 1624 708 Circle 1677 1262 Circle 1729 2786 Circle 1781 745 Circle 1834 823 Circle 1886 2786 Circle 1938 2786 Circle 1991 846 Circle 2043 2786 Circle 2095 2786 Circle 2148 725 Circle 2200 1834 Circle 2252 2786 Circle 2305 2786 Circle 2357 2786 Circle 2409 2786 Circle 2462 2786 Circle 2514 2786 Circle 2566 2786 Circle 2619 2786 Circle 2671 2786 Circle 2723 2786 Circle 2776 2786 Circle 2828 2786 Circle 2880 2786 Circle 2933 2786 Circle 2985 2786 Circle 3037 2786 Circle 3090 2786 Circle 3142 2786 Circle 3194 2786 Circle 3247 2786 Circle 3299 2786 Circle 3351 2786 Circle 3404 2786 Circle 3456 2786 Circle 3508 2786 Circle 3561 2786 Circle 3613 2786 Circle 3665 2786 Circle 3718 2786 Circle 3770 2786 Circle 3822 705 Circle 3874 705 Circle 3927 2786 Circle 3979 705 Circle 4031 705 Circle 4084 2786 Circle 4136 705 Circle 4188 705 Circle 4241 720 Circle 4293 705 Circle 4345 705 Circle 4398 705 Circle 4450 2786 Circle 4502 705 Circle 4555 705 Circle 4607 723 Circle 4659 705 Circle 4712 706 Circle 4764 707 Circle 4816 2786 Circle 4869 705 Circle 4921 705 Circle 4973 761 Circle 5026 705 Circle 5078 705 Circle 5130 706 Circle 5183 2786 Circle 5235 706 Circle 5287 706 Circle 5340 2786 Circle 5392 746 Circle 5444 1000 Circle 5497 2786 Circle 5549 759 Circle 5601 2786 Circle 5654 2786 Circle 5706 2786 Circle 5758 1200 Circle 5811 2786 Circle 5863 2786 Circle 5915 727 Circle 5968 756 Circle 6020 987 Circle 6072 2786 Circle 6125 712 Circle 6177 2786 Circle 6229 2786 Circle 6282 912 Circle 6334 2786 Circle 6386 1524 Circle 6439 726 Circle 6491 775 Circle 6543 1034 Circle 6596 2786 Circle 6648 763 Circle 6700 1420 Circle 6753 2786 Circle 6805 1386 Circle 6857 2786 Circle 6910 2318 Circle 4241 70 Circle 1.000 UP 1.000 UL LT6 6246 70 M (8 processors) Rshow 682 961 CircleF 735 708 CircleF 787 717 CircleF 839 2786 CircleF 892 706 CircleF 944 719 CircleF 996 739 CircleF 1049 2786 CircleF 1101 716 CircleF 1153 758 CircleF 1206 1294 CircleF 1258 713 CircleF 1310 711 CircleF 1363 891 CircleF 1415 2786 CircleF 1467 712 CircleF 1520 763 CircleF 1572 2786 CircleF 1624 709 CircleF 1677 716 CircleF 1729 723 CircleF 1781 708 CircleF 1834 708 CircleF 1886 772 CircleF 1938 2786 CircleF 1991 713 CircleF 2043 781 CircleF 2095 2786 CircleF 2148 708 CircleF 2200 726 CircleF 2252 2786 CircleF 2305 2786 CircleF 2357 990 CircleF 2409 2786 CircleF 2462 2786 CircleF 2514 733 CircleF 2566 2786 CircleF 2619 2786 CircleF 2671 2786 CircleF 2723 2786 CircleF 2776 2786 CircleF 2828 2786 CircleF 2880 1035 CircleF 2933 901 CircleF 2985 2786 CircleF 3037 898 CircleF 3090 1418 CircleF 3142 2786 CircleF 3194 2786 CircleF 3247 1743 CircleF 3299 2786 CircleF 3351 2786 CircleF 3404 2786 CircleF 3456 2786 CircleF 3508 2786 CircleF 3561 2786 CircleF 3613 2786 CircleF 3665 2786 CircleF 3718 2786 CircleF 3770 2786 CircleF 3822 705 CircleF 3874 705 CircleF 3927 2786 CircleF 3979 705 CircleF 4031 705 CircleF 4084 870 CircleF 4136 705 CircleF 4188 705 CircleF 4241 710 CircleF 4293 705 CircleF 4345 705 CircleF 4398 705 CircleF 4450 877 CircleF 4502 705 CircleF 4555 705 CircleF 4607 712 CircleF 4659 705 CircleF 4712 705 CircleF 4764 707 CircleF 4816 2786 CircleF 4869 704 CircleF 4921 705 CircleF 4973 710 CircleF 5026 704 CircleF 5078 705 CircleF 5130 706 CircleF 5183 2786 CircleF 5235 705 CircleF 5287 705 CircleF 5340 2786 CircleF 5392 709 CircleF 5444 775 CircleF 5497 714 CircleF 5549 712 CircleF 5601 2786 CircleF 5654 725 CircleF 5706 2786 CircleF 5758 709 CircleF 5811 813 CircleF 5863 2786 CircleF 5915 713 CircleF 5968 707 CircleF 6020 714 CircleF 6072 2786 CircleF 6125 708 CircleF 6177 710 CircleF 6229 901 CircleF 6282 721 CircleF 6334 716 CircleF 6386 805 CircleF 6439 719 CircleF 6491 739 CircleF 6543 748 CircleF 6596 2786 CircleF 6648 724 CircleF 6700 781 CircleF 6753 2786 CircleF 6805 842 CircleF 6857 2786 CircleF 6910 758 CircleF 6529 70 CircleF 1.000 UL LT2 682 700 M 0 2086 V 735 700 M 0 63 V 52 -63 R 0 162 V 839 700 M 0 2086 V 892 700 M 0 17 V 52 -17 R 0 570 V 996 700 M 0 2086 V 1049 700 M 0 2086 V 1101 700 M 0 160 V 52 -160 R 0 2086 V 1206 700 M 0 2086 V 1258 700 M 0 48 V 52 -48 R 0 485 V 53 -485 R 0 2086 V 1415 700 M 0 2086 V 1467 700 M 0 40 V 53 -40 R 0 2086 V 1572 700 M 0 2086 V 1624 700 M 0 60 V 53 -60 R 0 349 V 52 -349 R 0 2086 V 1781 700 M 0 30 V 53 -30 R 0 74 V 52 -74 R 0 1798 V 1938 700 M 0 2086 V 1991 700 M 0 184 V 52 -184 R 0 2086 V 2095 700 M 0 2086 V 2148 700 M 0 96 V 52 -96 R 0 829 V 52 -829 R 0 2086 V 2305 700 M 0 2086 V 2357 700 M 0 2086 V 2409 700 M 0 2086 V 2462 700 M 0 2086 V 2514 700 M 0 2086 V 2566 700 M 0 2086 V 2619 700 M 0 2086 V 2671 700 M 0 2086 V 2723 700 M 0 2086 V 2776 700 M 0 2086 V 2828 700 M 0 2086 V 2880 700 M 0 2086 V 2933 700 M 0 2086 V 2985 700 M 0 2086 V 3037 700 M 0 2086 V 3090 700 M 0 2086 V 3142 700 M 0 2086 V 3194 700 M 0 2086 V 3247 700 M 0 2086 V 3299 700 M 0 2086 V 3351 700 M 0 2086 V 3404 700 M 0 2086 V 3456 700 M 0 2086 V 3508 700 M 0 2086 V 3561 700 M 0 2086 V 3613 700 M 0 2086 V 3665 700 M 0 2086 V 3718 700 M 0 2086 V 3770 700 M 0 2086 V 3822 700 M 0 5 V 52 -5 R 0 8 V 53 -8 R 0 2086 V 3979 700 M 0 5 V 52 -5 R 0 8 V 53 -8 R 0 2086 V 4136 700 M 0 5 V 52 -5 R 0 7 V 53 -7 R 0 83 V 52 -83 R 0 5 V 52 -5 R 0 5 V 53 -5 R 0 6 V 52 -6 R 0 2086 V 4502 700 M 0 5 V 53 -5 R 0 6 V 52 -6 R 0 86 V 52 -86 R 0 5 V 53 -5 R 0 6 V 52 -6 R 0 14 V 52 -14 R 0 2086 V 4869 700 M 0 5 V 52 -5 R 0 6 V 52 -6 R 0 94 V 53 -94 R 0 4 V 52 -4 R 0 5 V 52 -5 R 0 23 V 53 -23 R 0 2086 V 5235 700 M 0 6 V 52 -6 R 0 8 V 53 -8 R 0 2086 V 5392 700 M 0 14 V 52 -14 R 0 55 V 53 -55 R 0 670 V 52 -670 R 0 16 V 52 -16 R 0 25 V 53 -25 R 0 282 V 52 -282 R 0 2086 V 5758 700 M 0 38 V 53 -38 R 0 226 V 52 -226 R 0 2086 V 5915 700 M 0 16 V 53 -16 R 0 19 V 52 -19 R 0 126 V 52 -126 R 0 2086 V 6125 700 M 0 12 V 52 -12 R 0 154 V 52 -154 R 0 2086 V 6282 700 M 0 55 V 52 -55 R 0 96 V 52 -96 R 0 2086 V 6439 700 M 0 12 V 52 -12 R 0 17 V 52 -17 R 0 61 V 53 -61 R 0 2086 V 6648 700 M 0 13 V 52 -13 R 0 62 V 53 -62 R 0 2086 V 6805 700 M 0 39 V 52 -39 R 0 57 V 53 -57 R 0 143 V stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument @endspecial 174 x Fv(Fig.)14 b(4.)26 b Fy(Breadth-\014rst)g(searc)n(h) i(results)f(with)g(2,)g(4)g(and)g(8)g(pro)r(cessors)h(for)g(the)e(120)i (\014rst)e(graphs)523 1840 y(with)d(50)h(tasks.)f(52.5\045)h(w)n(ere)g (solv)n(ed)f(with)g(t)n(w)n(o)g(pro)r(cessors,)i(48.3\045)f(with)g (four)f(and)f(66.7\045)j(with)523 1931 y(eigh)n(t.)523 3399 y @beginspecial 50 @llx 50 @lly 410 @urx 201 @ury 3458 @rwi @setspecial %%BeginDocument: a-star-50-tasks.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: a-star-50-tasks.eps %%Creator: gnuplot 3.7 patchlevel 3 %%CreationDate: Tue May 18 09:12:53 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 201 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall currentdict end definefont pop end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 630 700 M 63 0 V 6269 0 R -63 0 V 546 700 M ( 0) Rshow 630 1048 M 63 0 V 6269 0 R -63 0 V -6353 0 R ( 10) Rshow 630 1395 M 63 0 V 6269 0 R -63 0 V -6353 0 R ( 20) Rshow 630 1743 M 63 0 V 6269 0 R -63 0 V -6353 0 R ( 30) Rshow 630 2091 M 63 0 V 6269 0 R -63 0 V -6353 0 R ( 40) Rshow 630 2439 M 63 0 V 6269 0 R -63 0 V -6353 0 R ( 50) Rshow 630 2786 M 63 0 V 6269 0 R -63 0 V -6353 0 R ( 60) Rshow 682 700 M 0 63 V 0 2093 R 0 -63 V 682 560 M ( 0) Cshow 1729 700 M 0 63 V 0 2093 R 0 -63 V 0 -2233 R ( 20) Cshow 2776 700 M 0 63 V 0 2093 R 0 -63 V 0 -2233 R ( 40) Cshow 3822 700 M 0 63 V 0 2093 R 0 -63 V 0 -2233 R ( 60) Cshow 4869 700 M 0 63 V 0 2093 R 0 -63 V 0 -2233 R ( 80) Cshow 5915 700 M 0 63 V 0 2093 R 0 -63 V 0 -2233 R ( 100) Cshow 6962 700 M 0 63 V 0 2093 R 0 -63 V 0 -2233 R ( 120) Cshow 1.000 UL LTb 630 700 M 6332 0 V 0 2156 V -6332 0 V 630 700 L 140 1778 M currentpoint gsave translate 90 rotate 0 0 M (Running time \(minutes\)) Cshow grestore 3796 350 M (Task graph no.) Cshow 1.000 UP 1.000 UL LT0 1670 70 M (2 processors) Rshow 682 2786 Pls 735 798 Pls 787 960 Pls 839 2786 Pls 892 717 Pls 944 1212 Pls 996 2786 Pls 1049 2786 Pls 1101 839 Pls 1153 2786 Pls 1206 2786 Pls 1258 741 Pls 1310 1208 Pls 1363 2439 Pls 1415 2786 Pls 1467 2786 Pls 1520 2786 Pls 1572 2786 Pls 1624 2786 Pls 1677 1266 Pls 1729 2786 Pls 1781 732 Pls 1834 774 Pls 1886 2786 Pls 1938 2786 Pls 1991 954 Pls 2043 2786 Pls 2095 2786 Pls 2148 965 Pls 2200 1468 Pls 2252 2786 Pls 2305 2786 Pls 2357 2786 Pls 2409 2786 Pls 2462 2786 Pls 2514 2786 Pls 2566 2786 Pls 2619 2786 Pls 2671 2786 Pls 2723 2786 Pls 2776 2786 Pls 2828 2786 Pls 2880 2578 Pls 2933 2786 Pls 2985 2786 Pls 3037 2786 Pls 3090 2786 Pls 3142 2786 Pls 3194 2786 Pls 3247 2786 Pls 3299 2786 Pls 3351 2786 Pls 3404 2786 Pls 3456 2786 Pls 3508 2786 Pls 3561 2786 Pls 3613 2786 Pls 3665 2786 Pls 3718 2786 Pls 3770 2786 Pls 3822 730 Pls 3874 708 Pls 3927 2786 Pls 3979 715 Pls 4031 766 Pls 4084 2786 Pls 4136 723 Pls 4188 895 Pls 4241 830 Pls 4293 725 Pls 4345 718 Pls 4398 706 Pls 4450 2786 Pls 4502 709 Pls 4555 760 Pls 4607 2786 Pls 4659 724 Pls 4712 775 Pls 4764 2786 Pls 4816 2786 Pls 4869 711 Pls 4921 738 Pls 4973 792 Pls 5026 709 Pls 5078 730 Pls 5130 724 Pls 5183 2786 Pls 5235 783 Pls 5287 1034 Pls 5340 2786 Pls 5392 714 Pls 5444 766 Pls 5497 1170 Pls 5549 723 Pls 5601 724 Pls 5654 1279 Pls 5706 2786 Pls 5758 764 Pls 5811 1158 Pls 5863 2786 Pls 5915 717 Pls 5968 723 Pls 6020 837 Pls 6072 2786 Pls 6125 712 Pls 6177 819 Pls 6229 2786 Pls 6282 830 Pls 6334 897 Pls 6386 2786 Pls 6439 714 Pls 6491 719 Pls 6543 758 Pls 6596 2786 Pls 6648 712 Pls 6700 785 Pls 6753 2786 Pls 6805 742 Pls 6857 757 Pls 6910 882 Pls 1953 70 Pls 1.000 UP 1.000 UL LT5 3958 70 M (4 processors) Rshow 682 2786 Circle 735 2786 Circle 787 2786 Circle 839 2786 Circle 892 2786 Circle 944 2786 Circle 996 2786 Circle 1049 2786 Circle 1101 2786 Circle 1153 2786 Circle 1206 2786 Circle 1258 2786 Circle 1310 2786 Circle 1363 2786 Circle 1415 2786 Circle 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726 Circle 4607 2786 Circle 4659 716 Circle 4712 733 Circle 4764 2786 Circle 4816 2786 Circle 4869 712 Circle 4921 858 Circle 4973 2786 Circle 5026 710 Circle 5078 733 Circle 5130 2786 Circle 5183 2786 Circle 5235 731 Circle 5287 779 Circle 5340 2786 Circle 5392 2104 Circle 5444 1205 Circle 5497 2786 Circle 5549 2786 Circle 5601 2786 Circle 5654 2786 Circle 5706 2786 Circle 5758 2786 Circle 5811 2786 Circle 5863 2786 Circle 5915 2786 Circle 5968 2786 Circle 6020 2786 Circle 6072 2786 Circle 6125 2786 Circle 6177 2786 Circle 6229 2786 Circle 6282 2786 Circle 6334 2786 Circle 6386 2786 Circle 6439 2786 Circle 6491 2786 Circle 6543 1058 Circle 6596 2786 Circle 6648 1625 Circle 6700 2786 Circle 6753 2786 Circle 6805 2786 Circle 6857 2786 Circle 6910 2786 Circle 4241 70 Circle 1.000 UP 1.000 UL LT6 6246 70 M (8 processors) Rshow 682 2786 CircleF 735 2786 CircleF 787 2786 CircleF 839 2786 CircleF 892 2464 CircleF 944 2786 CircleF 996 2786 CircleF 1049 2786 CircleF 1101 2786 CircleF 1153 2786 CircleF 1206 2786 CircleF 1258 2786 CircleF 1310 2786 CircleF 1363 2786 CircleF 1415 2786 CircleF 1467 2786 CircleF 1520 2786 CircleF 1572 2786 CircleF 1624 2786 CircleF 1677 2786 CircleF 1729 2786 CircleF 1781 2786 CircleF 1834 2786 CircleF 1886 2786 CircleF 1938 2786 CircleF 1991 2786 CircleF 2043 2786 CircleF 2095 2786 CircleF 2148 2786 CircleF 2200 2786 CircleF 2252 2786 CircleF 2305 2786 CircleF 2357 2786 CircleF 2409 2786 CircleF 2462 2786 CircleF 2514 2786 CircleF 2566 2786 CircleF 2619 2786 CircleF 2671 2786 CircleF 2723 2786 CircleF 2776 2786 CircleF 2828 2786 CircleF 2880 2786 CircleF 2933 2786 CircleF 2985 2786 CircleF 3037 2786 CircleF 3090 2786 CircleF 3142 2786 CircleF 3194 2786 CircleF 3247 2786 CircleF 3299 2786 CircleF 3351 2786 CircleF 3404 2786 CircleF 3456 2786 CircleF 3508 2786 CircleF 3561 2786 CircleF 3613 2786 CircleF 3665 2786 CircleF 3718 2786 CircleF 3770 2786 CircleF 3822 739 CircleF 3874 925 CircleF 3927 2786 CircleF 3979 720 CircleF 4031 765 CircleF 4084 2786 CircleF 4136 731 CircleF 4188 781 CircleF 4241 2786 CircleF 4293 728 CircleF 4345 732 CircleF 4398 807 CircleF 4450 2786 CircleF 4502 713 CircleF 4555 750 CircleF 4607 2786 CircleF 4659 723 CircleF 4712 765 CircleF 4764 1939 CircleF 4816 2786 CircleF 4869 716 CircleF 4921 783 CircleF 4973 2786 CircleF 5026 712 CircleF 5078 738 CircleF 5130 2786 CircleF 5183 2786 CircleF 5235 764 CircleF 5287 812 CircleF 5340 2786 CircleF 5392 2786 CircleF 5444 2786 CircleF 5497 2786 CircleF 5549 2786 CircleF 5601 2786 CircleF 5654 2786 CircleF 5706 2786 CircleF 5758 2786 CircleF 5811 2786 CircleF 5863 2786 CircleF 5915 2786 CircleF 5968 2786 CircleF 6020 2786 CircleF 6072 2786 CircleF 6125 2786 CircleF 6177 2786 CircleF 6229 2786 CircleF 6282 2786 CircleF 6334 2786 CircleF 6386 2786 CircleF 6439 2786 CircleF 6491 2786 CircleF 6543 2786 CircleF 6596 2786 CircleF 6648 2786 CircleF 6700 2786 CircleF 6753 2786 CircleF 6805 2786 CircleF 6857 2786 CircleF 6910 2786 CircleF 6529 70 CircleF 1.000 UL LT2 682 700 M 0 2086 V 735 700 M 0 98 V 52 -98 R 0 260 V 839 700 M 0 2086 V 892 700 M 0 17 V 52 -17 R 0 512 V 996 700 M 0 2086 V 1049 700 M 0 2086 V 1101 700 M 0 139 V 52 -139 R 0 2086 V 1206 700 M 0 2086 V 1258 700 M 0 41 V 52 -41 R 0 508 V 53 -508 R 0 1739 V 1415 700 M 0 2086 V 1467 700 M 0 2086 V 1520 700 M 0 2086 V 1572 700 M 0 2086 V 1624 700 M 0 2086 V 1677 700 M 0 566 V 52 -566 R 0 2086 V 1781 700 M 0 32 V 53 -32 R 0 74 V 52 -74 R 0 2086 V 1938 700 M 0 2086 V 1991 700 M 0 254 V 52 -254 R 0 2086 V 2095 700 M 0 2086 V 2148 700 M 0 265 V 52 -265 R 0 768 V 52 -768 R 0 2086 V 2305 700 M 0 2086 V 2357 700 M 0 2086 V 2409 700 M 0 2086 V 2462 700 M 0 2086 V 2514 700 M 0 2086 V 2566 700 M 0 2086 V 2619 700 M 0 2086 V 2671 700 M 0 2086 V 2723 700 M 0 2086 V 2776 700 M 0 2086 V 2828 700 M 0 2086 V 2880 700 M 0 1878 V 2933 700 M 0 2086 V 2985 700 M 0 2086 V 3037 700 M 0 2086 V 3090 700 M 0 2086 V 3142 700 M 0 2086 V 3194 700 M 0 2086 V 3247 700 M 0 2086 V 3299 700 M 0 2086 V 3351 700 M 0 2086 V 3404 700 M 0 2086 V 3456 700 M 0 2086 V 3508 700 M 0 2086 V 3561 700 M 0 2086 V 3613 700 M 0 2086 V 3665 700 M 0 2086 V 3718 700 M 0 2086 V 3770 700 M 0 2086 V 3822 700 M 0 30 V 52 -30 R 0 8 V 53 -8 R 0 2086 V 3979 700 M 0 15 V 52 -15 R 0 66 V 53 -66 R 0 2086 V 4136 700 M 0 23 V 52 -23 R 0 195 V 53 -195 R 0 130 V 52 -130 R 0 25 V 52 -25 R 0 18 V 53 -18 R 0 6 V 52 -6 R 0 2086 V 4502 700 M 0 9 V 53 -9 R 0 60 V 52 -60 R 0 2086 V 4659 700 M 0 24 V 53 -24 R 0 75 V 52 -75 R 0 2086 V 4816 700 M 0 2086 V 4869 700 M 0 11 V 52 -11 R 0 38 V 52 -38 R 0 92 V 53 -92 R 0 9 V 52 -9 R 0 30 V 52 -30 R 0 24 V 53 -24 R 0 2086 V 5235 700 M 0 83 V 52 -83 R 0 334 V 53 -334 R 0 2086 V 5392 700 M 0 14 V 52 -14 R 0 66 V 53 -66 R 0 470 V 52 -470 R 0 23 V 52 -23 R 0 24 V 53 -24 R 0 579 V 52 -579 R 0 2086 V 5758 700 M 0 64 V 53 -64 R 0 458 V 52 -458 R 0 2086 V 5915 700 M 0 17 V 53 -17 R 0 23 V 52 -23 R 0 137 V 52 -137 R 0 2086 V 6125 700 M 0 12 V 52 -12 R 0 119 V 52 -119 R 0 2086 V 6282 700 M 0 130 V 52 -130 R 0 197 V 52 -197 R 0 2086 V 6439 700 M 0 14 V 52 -14 R 0 19 V 52 -19 R 0 58 V 53 -58 R 0 2086 V 6648 700 M 0 12 V 52 -12 R 0 85 V 53 -85 R 0 2086 V 6805 700 M 0 42 V 52 -42 R 0 57 V 53 -57 R 0 182 V stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument @endspecial 174 x Fv(Fig.)14 b(5.)30 b Fy(A)844 3542 y Fu(\003)910 3573 y Fy(searc)n(h)g(results)h(with)g(2,)g(4)f(and)g(8)h (pro)r(cessors)h(for)f(the)f(120)i(\014rst)e(graphs)h(with)f(50)523 3665 y(tasks.)c(50\045)h(w)n(ere)f(solv)n(ed)g(with)g(t)n(w)n(o)g(pro)r (cessors,)i(18.3\045)f(with)f(four)g(and)f(17.5\045)i(with)f(eigh)n(t.) 523 5028 y @beginspecial 50 @llx 50 @lly 410 @urx 188 @ury 3458 @rwi @setspecial %%BeginDocument: all-dep-50.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: all-dep-50.eps %%Creator: gnuplot 3.8k patchlevel 1 %%CreationDate: Mon May 17 13:45:27 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 188 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10.0 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /Rounded false def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /N {newpath moveto} bind def /C {setrgbcolor} bind def /f {rlineto fill} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop 0 setgray Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth Rounded { 1 setlinejoin 1 setlinecap } if } def /AL { stroke userlinewidth 2 div setlinewidth Rounded { 1 setlinejoin 1 setlinecap } if } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth Rounded { 1 setlinejoin 1 setlinecap } if } def /LTw { PL [] 1 setgray } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /BoxColFill { gsave Rec /Fillden exch def currentrgbcolor /ColB exch def /ColG exch def /ColR exch def /ColR ColR Fillden mul Fillden sub 1 add def /ColG ColG Fillden mul Fillden sub 1 add def /ColB ColB Fillden mul Fillden sub 1 add def ColR ColG ColB setrgbcolor fill grestore } def % % PostScript Level 1 Pattern Fill routine % Usage: x y w h s a XX PatternFill % x,y = lower left corner of box to be filled % w,h = width and height of box % a = angle in degrees between lines and x-axis % XX = 0/1 for no/yes cross-hatch % /PatternFill { gsave /PFa [ 9 2 roll ] def PFa 0 get PFa 2 get 2 div add PFa 1 get PFa 3 get 2 div add translate PFa 2 get -2 div PFa 3 get -2 div PFa 2 get PFa 3 get Rec gsave 1 setgray fill grestore clip currentlinewidth 0.5 mul setlinewidth /PFs PFa 2 get dup mul PFa 3 get dup mul add sqrt def 0 0 M PFa 5 get rotate PFs -2 div dup translate 0 1 PFs PFa 4 get div 1 add floor cvi { PFa 4 get mul 0 M 0 PFs V } for 0 PFa 6 get ne { 0 1 PFs PFa 4 get div 1 add floor cvi { PFa 4 get mul 0 2 1 roll M PFs 0 V } for } if stroke grestore } def % /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall currentdict end definefont pop end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 630 2604 M 63 0 V 6269 0 R -63 0 V -6353 0 R gsave 0 setgray (1) Rshow grestore 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M (Dependency fraction) Cshow grestore grestore LTb 3796 70 M gsave 0 setgray (Task graph no.) 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1.000 UL LTb 630 420 M 6332 0 V 0 2184 V -6332 0 V 630 420 L 1.000 UP stroke grestore end showpage %%EndDocument @endspecial 175 x Fv(Fig.)14 b(6.)24 b Fy(The)h(dep)r(endency)f (fraction)i(for)g(the)e(\014rst)h(120)h(graphs.)g(The)f(dep)r (endencies)f(are)i(gener-)523 5294 y(ated)g(using)g(di\013eren)n(t)f (algorithms)i(in)f(in)n(terv)l(als)g(of)g(30.)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 3328 232 a Fy(13)523 448 y Fz(are)27 b(solv)n(ed)h(fastest)g(with)h(t)n(w)n(o)e(pro)r(cessors)f(and)i(some)g (with)h(four,)f(although)f(clearly)h(the)523 548 y(algorithm)d(is)h(b)r (est)g(with)h(eigh)n(t)e(pro)r(cessors.)f(The)i(di\013erences)g(seem)f (to)h(ha)n(v)n(e)f(to)h(do)f(with)523 648 y(the)37 b(n)n(um)n(b)r(er)g (of)g(states)g(in)g(the)g(state)g(sets.)g(More)f(pro)r(cessors)e (result)j(in)g(more)f(states)523 747 y(since)c(there)g(are)f(more)g(p)r (ossible)h(sc)n(hedules.)f(The)h(reason)f(that)h(eigh)n(t)g(pro)r (cessors)d(then)523 847 y(w)n(as)34 b(faster)h(seems)f(to)h(b)r(e)h (that)f(the)g(state)g(set)g(BDDs)h(exhibited)f(more)g(sharing.)e(Also) 523 946 y(the)j Fn(E)g Fz(and)f Fn(L)h Fz(estimates)g(are)f(more)f (accurate)h(with)h(eigh)n(t)f(pro)r(cessors,)f(whic)n(h)h(allo)n(ws)523 1046 y(the)28 b(algorithm)e(to)h(cut)g(o\013)h(a)e(lot)i(of)f(states.)g (And)g(\014nally)g(the)h(algorithm)e(runs)h(for)g(few)n(er)523 1146 y(iterations)g(since)g(the)h(length)g(of)f(the)h(sc)n(hedule)g(is) f(shorter.)648 1245 y(In)34 b(comparison,)e(the)j(A)1438 1215 y Fs(\003)1511 1245 y Fz(algorithm)e(p)r(erformed)g(w)n(orse)g(on) h(almost)g(all)f(instances.)523 1345 y(There)22 b(are)f(t)n(w)n(o)g (explanations)g(for)h(this.)g(First,)g(the)h(A)2240 1315 y Fs(\003)2300 1345 y Fz(algorithm)e(will)h(searc)n(h)f(all)h(sc)n (hed-)523 1445 y(ules)34 b(with)g(no)g(free)g(slots)f(\014rst,)h(then)g (those)g(with)g(one)g(free)f(slot,)h(etc.)g(If)h(there)e(are)g(no)523 1544 y(free)e(slots)f(in)h(the)g(optimal)g(sc)n(hedules)f(this)h (reduces)g(the)g(n)n(um)n(b)r(er)f(of)h(states)g(to)f(searc)n(h,)523 1644 y(but)38 b(if)f(the)g(optimal)g(sc)n(hedules)f(con)n(tain)h Fr(x)g Fz(free)g(slots)f(then)i(the)f(searc)n(h)e(has)i(to)f(fully)523 1743 y(explore)31 b Fr(x)i Fz(non-optimal)f(branc)n(hes,)f(whereas)g (the)i(breadth-\014rst)e(algorithm)h(examines)523 1843 y(all)23 b(of)f(these)h(at)g(the)g(same)f(time.)i(Second,)f(since)f (the)h(A)2275 1813 y Fs(\003)2337 1843 y Fz(algorithm)e(do)r(es)i(not)g (k)n(eep)f(trac)n(k)523 1943 y(of)29 b(the)g(elapsed)g(time)h(it)f(is)g (not)g(p)r(ossible)g(to)g(use)f(the)i Fn(E)f Fz(and)g Fn(L)g Fz(optimisations.)g(Indeed,)523 2042 y(exp)r(erimen)n(ts)k(with) h(earlier)d(v)n(ersions)h(of)h(the)g(breadth-\014rst)g(algorithm)f (without)h(the)h Fn(E)523 2142 y Fz(and)g Fn(L)g Fz(optimisations)f (sho)n(w)n(ed)g(that)h(the)g(A)1989 2112 y Fs(\003)2062 2142 y Fz(algorithm)e(then)j(w)n(as)e(faster)g(on)g(graphs)523 2242 y(where)27 b(the)h(optimal)g(sc)n(hedules)e(did)i(not)g(ha)n(v)n (e)e(an)n(y)h(free)h(slots.)555 3179 y @beginspecial 50 @llx 50 @lly 230 @urx 201 @ury 963 @rhi @setspecial %%BeginDocument: av-compl-dep.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: ../av-compl-dep.eps %%Creator: gnuplot 3.8k patchlevel 1 %%CreationDate: Tue May 18 09:57:34 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 230 201 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10.0 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /Rounded false def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /N {newpath moveto} bind def /C {setrgbcolor} bind def /f {rlineto fill} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop 0 setgray Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth Rounded { 1 setlinejoin 1 setlinecap } if } def /AL { stroke userlinewidth 2 div setlinewidth Rounded { 1 setlinejoin 1 setlinecap } if } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth Rounded { 1 setlinejoin 1 setlinecap } if } def /LTw { PL [] 1 setgray } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /BoxColFill { gsave Rec /Fillden exch def currentrgbcolor /ColB exch def /ColG exch def /ColR exch def /ColR ColR Fillden mul Fillden sub 1 add def /ColG ColG Fillden mul Fillden sub 1 add def /ColB ColB Fillden mul Fillden sub 1 add def ColR ColG ColB setrgbcolor fill grestore } def % % PostScript Level 1 Pattern Fill routine % Usage: x y w h s a XX PatternFill % x,y = lower left corner of box to be filled % w,h = width and height of box % a = angle in degrees between lines and x-axis % XX = 0/1 for no/yes cross-hatch % /PatternFill { gsave /PFa [ 9 2 roll ] def PFa 0 get PFa 2 get 2 div add PFa 1 get PFa 3 get 2 div add translate PFa 2 get -2 div PFa 3 get -2 div PFa 2 get PFa 3 get Rec gsave 1 setgray fill grestore clip currentlinewidth 0.5 mul setlinewidth /PFs PFa 2 get dup mul PFa 3 get dup mul add sqrt def 0 0 M PFa 5 get rotate PFs -2 div dup translate 0 1 PFs PFa 4 get div 1 add floor cvi { PFa 4 get mul 0 M 0 PFs V } for 0 PFa 6 get ne { 0 1 PFs PFa 4 get div 1 add floor cvi { PFa 4 get mul 0 2 1 roll M PFs 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b(ha)n(v)n(e)f(represen)n(ted)g(the)h(task)g (graph)f(sc)n(heduling)g(problem)g(as)h(a)f(BDD)i(state)f(space)523 3828 y(searc)n(h)e(problem)h(and)h(explored)e(a)h(n)n(um)n(b)r(er)h(of) f(design)g(alternativ)n(es.)f(The)i(state)f(repre-)523 3928 y(sen)n(tation)j(w)n(as)f(c)n(hosen)h(so)f(that)i(an)n(y)f(temp)r (orarily)f(un)n(used)h(BDD)h(v)-5 b(ariables)32 b(could)i(b)r(e)523 4028 y(ignored,)21 b(and)i(the)f(transition)g(relation)g(w)n(as)f (split)i(in)n(to)f(three)g(seman)n(tic)g(phases)g(of)g(whic)n(h)523 4127 y(the)29 b(\014rst)g(and)f(last)h(phase)f(w)n(ere)g(further)h (partitioned)f(in)n(to)h(a)f(relation)g(p)r(er)h(task.)f(These)523 4227 y(t)n(w)n(o)k(ideas)g(made)g(the)h(algorithms)e(tractable)g(in)i (practice.)f(F)-7 b(urthermore,)31 b(the)i(b)r(ounds)523 4327 y(for)j(the)h(time)g(frame)f(in)g(whic)n(h)h(a)f(giv)n(en)f(task)h (could)h(run)f(w)n(ere)f(estimated)i(and)f(used)523 4426 y(to)29 b(sa)n(v)n(e)f(more)g(than)h(half)h(of)f(the)g(image)g (computations,)f(and)h(also)f(allo)n(w)n(ed)g(pruning)h(of)523 4526 y(some)e(non-optimal)g(sc)n(hedules.)648 4625 y(Some)h(other)h (tec)n(hniques)f(from)h(the)g(literature)f([4])h(turned)g(out)g(to)g (ha)n(v)n(e)f(a)g(negativ)n(e)523 4725 y(impact)35 b(on)g(the)h(p)r (erformance.)e(Dynamic)h(v)-5 b(ariable)35 b(ordering)e(w)n(as)h(tried) i(but)f(did)h(not)523 4825 y(reduce)d(the)g(state)g(set)g(BDDs,)h(and)f (pruning)f(of)h(already)f(visited)h(states)g(enlarged)f(the)523 4924 y(state)27 b(set)h(BDDs)g(instead)g(of)f(reducing)g(their)h(size.) p eop end %%Page: 15 15 TeXDict begin 15 14 bop 3328 232 a Fy(15)648 448 y Fz(But)19 b(o)n(v)n(erall,)d(a)i(breadth-\014rst)g(searc)n(h)f(algorithm)h(w)n (as)g(found)h(to)f(b)r(e)h(able)g(to)f(solv)n(e)g(50\045)523 548 y(of)30 b(the)g(task)f(graphs)f(with)i(50)e(tasks)h(and)h(at)f (least)g(40\045)g(of)h(the)f(p)r(ossible)h(dep)r(endencies)523 648 y(within)35 b(60)f(min)n(utes.)h(Larger)e(task)h(graphs)f(with)i (man)n(y)g(dep)r(endencies)g(could)f(also)g(b)r(e)523 747 y(solv)n(ed,)26 b(although)h(it)g(then)h(tak)n(es)e(more)h(than)g (ten)g(hours)g(to)g(\014nd)g(an)g(optimal)g(sc)n(hedule.)523 847 y(A)i(guided)g(searc)n(h)f(strategy)f(with)j(a)e(BDD)i(adaptation)e (of)h(the)g(A)2658 817 y Fs(\003)2725 847 y Fz(algorithm)f(w)n(as)g (also)523 946 y(tried,)40 b(but)h(w)n(as)e(found)h(to)g(b)r(e)h (ine\016cien)n(t.)f(W)-7 b(e)41 b(attribute)f(this)g(to)g(the)h (fundamen)n(tal)523 1046 y(problem)23 b(of)g(\014nding)g(a)g(heuristic) g(for)f(the)i(sc)n(heduling)e(problem)h(that)g(is)g(able)g(to)g(comp)r (ete)523 1146 y(with)28 b(the)g(brute-force)f(breadth-\014rst)f(searc)n (h.)648 1245 y(F)-7 b(uture)41 b(w)n(ork)f(could)i(b)r(e)f(to)h(extend) f(the)h(BDD)g(represen)n(tation)e(to)h(mo)r(del)h(more)523 1345 y(complex)31 b(sc)n(heduling)g(problems,)g(suc)n(h)g(as)g(sc)n (heduling)g(with)g(comm)n(unication)g(costs)g(or)523 1445 y(job-shop)c(problems.)523 1710 y Ft(References)561 1901 y Fy(1.)43 b(Y.)32 b(Ab)r(dedda)-9 b(\177)-30 b(\020m,)34 b(E.)f(Asarin,)g(and)g(O.)f(Maler.)57 b(Sc)n(heduling)32 b(with)h(timed)g(automata.)57 b(In)663 1993 y Fg(The)l(or)l(etic)l(al) 29 b(Computer)f(Scienc)l(e)h(\(to)f(app)l(e)l(ar\))p Fy(,)g(2004.)561 2084 y(2.)43 b(Henrik)32 b(Reif)h(Andersen.)55 b(An)32 b(in)n(tro)r(duction)h(to)g(binary)g(decision)h(diagrams.)57 b(T)-6 b(ec)n(hnical)663 2175 y(rep)r(ort,)26 b(IT)g(Univ)n(ersit)n(y)e (of)j(Cop)r(enhagen,)f(1999.)561 2267 y(3.)43 b(Randal)33 b(E.)h(Bry)n(an)n(t.)57 b(Graph-based)34 b(algorithms)h(for)f(Bo)r (olean)h(function)f(manipulation.)663 2358 y Fg(IEEE)27 b(T)-6 b(r)l(ansactions)30 b(on)d(Computers)p Fy(,)h (C-35\(8\):677{691,)i(August)25 b(1986.)561 2449 y(4.)43 b(Randal)19 b(E.)h(Bry)n(an)n(t.)k(Binary)c(decision)g(diagrams)h(and)e (b)r(ey)n(ond:)g(Enabling)h(tec)n(hnologies)h(for)663 2540 y(formal)30 b(v)n(eri\014cation.)44 b(In)28 b Fg(IEEE/A)n(CM)h (International)j(Confer)l(enc)l(e)g(on)e(Computer)i(A)n(ide)l(d)663 2632 y(Design,)27 b(ICCAD,)f(San)i(Jose/CA)p Fy(,)f(pages)g(236{243,)i (1995.)561 2723 y(5.)43 b(Edm)n(und)24 b(M.)j(Clark)n(e,)g(Orna)f(Grum) n(b)r(erg,)h(and)f(Doron)g(A.)g(P)n(eled.)36 b Fg(Mo)l(del)28 b(Che)l(cking)p Fy(.)36 b(MIT)663 2814 y(Press,)27 b(2000.)561 2906 y(6.)43 b(Thomas)32 b(H.)f(Cormen,)h(Charles)g(E.)g(Leiserson,)g (Ronald)f(L.)g(Riv)n(est,)g(and)g(Cli\013ord)h(Stein.)663 2997 y Fg(Intr)l(o)l(duction)d(to)f(A)n(lgorithms)p Fy(.)35 b(The)26 b(MIT)g(Press,)h(2nd)f(edition,)g(2001.)561 3088 y(7.)43 b(Stefan)22 b(Edelk)l(amp)h(and)f(F)-6 b(rank)22 b(Re\013el.)29 b(OBDDs)22 b(in)h(heuristic)g(searc)n(h.)30 b(In)22 b Fg(KI)i(-)h(Kunstliche)663 3180 y(Intel)t(ligenz)p Fy(,)h(pages)g(81{92,)i(1998.)561 3271 y(8.)43 b(R.)30 b(M.)h(Jensen,)h(R.)e(E.)h(Bry)n(an)n(t,)g(and)f(M.)h(M.)h(V)-6 b(eloso.)50 b(SetA)2513 3239 y Fu(\003)2548 3271 y Fy(:)31 b(An)f(e\016cien)n(t)g(BDD-based)663 3362 y(heuristic)19 b(searc)n(h)h(algorithm.)25 b(In)18 b Fg(Pr)l(o)l(c)l(e)l(e)l(dings)24 b(of)d(18th)h(National)g(Confer)l(enc)l(e)h(on)f(A)n(rti\014cial)663 3454 y(Intel)t(ligenc)l(e)28 b(\(AAAI'02\))p Fy(,)e(pages)g(668{673,)j (2002.)561 3545 y(9.)43 b(Y)-6 b(u-Kw)n(ong)28 b(Kw)n(ok)i(and)f (Ishfaq)h(Ahmad.)45 b(Benc)n(hmarking)30 b(and)f(comparison)i(of)g(the) e(task)663 3636 y(graph)c(sc)n(heduling)h(algorithms.)36 b Fg(J.)28 b(Par)l(al)t(lel)f(Distrib.)g(Comput.)p Fy(,)f (59\(3\):381{422,)k(1999.)523 3728 y(10.)43 b(Kasahara)34 b(Lab)r(oratory)-6 b(.)57 b(Standard)33 b(T)-6 b(ask)33 b(Graph)h(Set.)56 b Fw(http://www.kasahara.elec.)663 3819 y(waseda.ac.jp/schedule/index.htm)q(l)p Fy(.)523 3910 y(11.)43 b(E.)24 b(L.)f(La)n(wler.)32 b(Sequencing)23 b(jobs)h(to)f(minimize)i(total)f(w)n(eigh)n(ted)g(completion)g(time)f (sub)t(ject)663 4002 y(to)j(precedence)f(constrain)n(ts.)35 b Fg(A)n(nnals)28 b(of)f(Discr)l(ete)i(Mathematics)p Fy(,)f(1978.)523 4093 y(12.)43 b(J\034rn)22 b(Lind-Nielsen.)27 b Fg(BuDDy:)d(Binary)g(De)l(cision)g(Diagr)l(am)g(p)l(ackage)p Fy(,)g(2002.)29 b Fw(http://www.)663 4184 y(itu.dk/research/buddy/)p Fy(.)523 4276 y(13.)43 b(J.)32 b(Rasm)n(ussen,)f(K.)h(G.)g(Larsen,)g (and)f(K.)g(Subramani.)51 b(Sc)n(heduling)31 b(using)h(priced)f(timed) 663 4367 y(automata.)40 b(In)26 b Fg(Pr)l(o)l(c.)k(10th)g(Int.)e(Conf.) h(of)f(T)-6 b(o)l(ols)30 b(and)f(A)n(lgorithms)h(for)f(the)h (Construction)663 4458 y(and)e(A)n(nalysis)g(of)f(Systems)i(\(T)-6 b(A)n(CAS'2004\))p Fy(,)27 b(pages)f(220{235,)j(2004.)523 4550 y(14.)43 b(Theo)20 b(Ruys.)25 b(Optimal)20 b(Sc)n(heduling)g (Using)g(Branc)n(h)g(and)g(Bound)f(with)i(SPIN)e(4.0.)26 b(In)20 b Fg(Mo)l(del)663 4641 y(Che)l(cking)27 b(Softwar)l(e)h({)f(Pr) l(o)l(c)l(e)l(e)l(dings)i(of)d(the)i(10th)g(International)f(SPIN)g (Workshop)h(\(SPIN)663 4732 y(2003\))p Fy(,)f(pages)f(1{17,)i(P)n (ortland,)e(OR,)f(USA,)g(Ma)n(y)h(2003.)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF