ࡱ>  M bjbj==WWUl lllzjzjzj8jlb>Vn F|"h|h||)<e9^^^^^^^$Sf sh&^-l%9)%%&^ h||CSaV %j >h|:l|^ %^ , 9R>.lD]|Jn ]֨8SzjTV`D]albViIiD]   8653>@>4A:89 3>AC40@AB25==K9 C=825@A8B5B 8<. .. >10G52A:>3> $0:C;LB5B 2KG8A;8B5;L=>9 <0B5<0B8:8 8 :815@=5B8:8 # _________________________________________________________________ #G51=>-8AA;54>20B5;LA:0O ;01>@0B>@8O "=D>@<0F8>==K5 B5E=>;>388" 040G0 > @07<5I5=88 M;5<5=B>2 =0 AE5<5 07@01>BG8:8: >@=O:>2 8@8;; 0:>25520 ;L30 030@8=>20 !25B;0=0 8:>;052 8:8B0 82>45@>2 @B5< !C11>B8=0 :0B5@8=0 @5DL52 <8B@89 C@0B>@: 55@>2 >A8D >@8A>28G 86=89 >23>@>4 2006 !>45@60=85  TOC \o "1-2" \h \z  HYPERLINK \l "_Toc128494243" 2545=85  PAGEREF _Toc128494243 \h 3  HYPERLINK \l "_Toc128494244" 040G0 > @07<5I5=88  PAGEREF _Toc128494244 \h 3  HYPERLINK \l "_Toc128494245" "8?K VLSI placement  PAGEREF _Toc128494245 \h 3  HYPERLINK \l "_Toc128494246" @8B5@88 >?B8<0;L=>AB8 @07<5I5=8O  PAGEREF _Toc128494246 \h 6  HYPERLINK \l "_Toc128494247" >4E>4K : @5H5=8N 7040G8  PAGEREF _Toc128494247 \h 8  HYPERLINK \l "_Toc128494248" Partition-based placement (Min-cut placement)  PAGEREF _Toc128494248 \h 8  HYPERLINK \l "_Toc128494249" Analytical Placement  PAGEREF _Toc128494249 \h 9  HYPERLINK \l "_Toc128494250" Simulated annealing  PAGEREF _Toc128494250 \h 9  HYPERLINK \l "_Toc128494251" =AB@C<5=BK 4;O @5H5=8O 7040G8  PAGEREF _Toc128494251 \h 11  HYPERLINK \l "_Toc128494252" Capo  PAGEREF _Toc128494252 \h 11  HYPERLINK \l "_Toc128494253" Feng Shui  PAGEREF _Toc128494253 \h 17  HYPERLINK \l "_Toc128494254" FastPlace  PAGEREF _Toc128494254 \h 21  HYPERLINK \l "_Toc128494255" mPL  PAGEREF _Toc128494255 \h 25  HYPERLINK \l "_Toc128494256" Dragon  PAGEREF _Toc128494256 \h 32  HYPERLINK \l "_Toc128494257" 0:;NG5=85  PAGEREF _Toc128494257 \h 35  HYPERLINK \l "_Toc128494258" !?8A>: ;8B5@0BC@K  PAGEREF _Toc128494258 \h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ircuit placement problem), :>B>@0O O2;O5BAO 8725AB=>9 7040G59 48A:@5B=>9 >?B8<870F88. 5 0:BC0;L=>ABL 2KB5:05B 87 =C64 A>2@5<5==>9 ?@><KH;5==>AB8, 4;O :>B>@>9 206=>ABL =0E>645=8O >?B8<0;L=KE ?> =5:>B>@K< :@8B5@8O< @07<5I5=89 M;5<5=B>2 =0 AE5<0E =5 2K7K205B A><=5=89. "@51>20=8O : M;5:B@>==K< AE5<0< AB0=>2OBAO 2A5 1>;55 AB@>38<8, @07@01>BG8:8 AB@5<OBAO A45;0BL 8E 2A5 1>;55 :><?0:B=K<8, 1KAB@K<8 8 <0:A8<0;L=> M:>=><=K<8 2 A<KA;5 M=5@3>?>B@51;5=8O. ;O B>3> GB>1K >15A?5G8BL E>@>H85 E0@0:B5@8AB8:8 @07@010BK205<KE AE5<, =5>1E>48<> ?@545;L=> MDD5:B82=> @01>B0BL =0 :064>9 AB0488 ?@>872>4AB20. ">, :0:8< >1@07>< 1C4CB @07<5I5=K M;5<5=BK =0 ?;0B5, =0?@O<CN ?>2;8O5B =0 2@5<5==K5 E0@0:B5@8AB8:8 AE5<K, GB> AB0=>28BAO >A>15==> 206=K< 2 =0H5 2@5<O 2KA>:8E A:>@>AB59. >?KB:8 @5H5=8O 7040G8 > @07<5I5=88 2 @07=KE ?>AB0=>2:0E ?@54?@8=8<0NBAO 4>2>;L=> 402=>. 5@2K5 ?C1;8:0F88 >B=>AOBAO : =0G0;C 70-E 3>4>2.  A>60;5=8N, =0 A53>4=OH=89 45=L 4;O 1>;LH8=AB20 ?>AB0=>2>: =5 ACI5AB2C5B 0;3>@8B<>2, A?>A>1=KE =09B8 >?B8<0;L=>5 @5H5=85 70 ?@85<;5<>5 2@5<O. ">G=K5 0;3>@8B<K G0I5 2A53> O2;ONBAO <>48D8:0F8O<8 ?>;=>3> ?5@51>@0 8 ?>B><C 8<5NB =5C4>2;5B2>@8B5;L=CN 2KG8A;8B5;L=CN A;>6=>ABL, 8, A>>B25BAB25==>, =5?@8<5=8<K =0 ?@0:B8:5. 5;> 2 B><, GB> @07<5@=>AB8 @5H05<KE 2 =0AB>OI55 2@5<O 7040G B0:>2K, GB> =0E>645=85 >?B8<0;L=KE @5H5=89 ?>B@51CNB 3>4K 2KG8A;5=89 =0 ;CGH8E AC?5@:><?LNB5@0E <8@0.  B> 65 2@5<O ?>=OB=>, GB> ?@>872>48B5;8 =5 <>3CB ?>72>;8BL A515 B@0B8BL AB>;L:> 2@5<5=8 =0 @5H5=85 B0:8E 7040G, 2 >A>15==>AB8, 5A;8 CG5ABL, GB> @07<5I5=85 O2;O5BAO ;8HL >4=>9 87 AB0489 ?@>5:B8@>20=8O.  MB8E CA;>28OE 1>;LH>5 7=0G5=85 ?@8>1@5B05B @07@01>B:0 8 0=0;87 ?@81;865==KE 0;3>@8B<>2, =0E>4OI8E 1;87:85 : >?B8<0;L=K< @5H5=8O A C4>2;5B2>@8B5;L=K<8 2@5<5==K<8 70B@0B0<8 =0 8E ?>;CG5=85. 0==K9 :>=A?5:B ?>A2OI5= @0AA<>B@5=8N 8 0=0;87C =5:>B>@KE ?>?C;O@=KE 0;3>@8B<>2 @07<5I5=8O 8 8E ?@>3@0<<=KE @50;870F89. 040G0 > @07<5I5=88 "8?K VLSI placement N10O 8=B53@0;L=0O AE5<0 2 ?@>F5AA5 A2>53> A>740=8O ?@>E>48B MB0?, =07K205<K9 @07<5I5=85< (placement), 2> 2@5<O :>B>@>3> :064><C M;5<5=BC ?@8A208205BAO 53> <5AB> =0 AE5<5. 07<5I5=85 ?@>872>48BAO ?>A;5 B>3>, :0: >?@545;5=0 ;>38:0 AE5<K, >?@545;5=>, :0:85 M;5<5=BK 1C4CB 2 =55 2E>48BL, 8 CAB0=>2;5=K A2O78 <564C =8<8. E>4=K<8 40==K<8 4;O 7040G8 @07<5I5=8O O2;ONBAO: 01>@ <>4C;59 A C:070==K<8 D>@<>9, @07<5@0<8 8 ?>;>65=85< :>=B0:B>2 !2O78 <564C <>4C;O<8  :0G5AB25 :@8B5@852 >?B8<0;L=>AB8 @07<5I5=8O @0AA<0B@820NB 4;8=C ?@>2>4>2, ?;>I04L AE5<K, ?>:070B5;8 ?@>872>48B5;L=>AB8 (7045@6:0 A8AB5<K, ?>B@51;5=85 M=5@388, B0:B>20O G0AB>B0), 2>7<>6=>ABL ?@>2545=8O A>548=5=89 ?>A;5 @07<5I5=8O. >A;54=89 ?0@0<5B@ =5<0;>2065=, ?>A:>;L:C =5@54:0 A8BC0F8O, :>340 2 @57C;LB0B5 @07<5I5=8O A>7405BAO AE5<0, 2 :>B>@>9 ?@>AB> =52>7<>6=> ?@>25AB8 =5>1E>48<K5 A>548=5=8O. <5NBAO @07;8G=K5 :;0AA8D8:0F88 7040G @07<5I5=8O. '0I5 2A53> 8E @07;8G0NB ?> B8?C @07<5I05<KE M;5<5=B>2. ;0AA8D8:0F8O, ?@82545==0O 2 [14], ?@54AB02;5=0 =0 A;54CNI8E AE5<0E:  @8A.1 ;0AA8D8:0F8O 7040G @07<5I5=8O, ?@8254Q==0O 2 [14]  gate-array 48709=5 =0 AE5<5 @0A?>;0305BAO <0AA82 845=B8G=KE ?> @07<5@C M;5<5=B>2. Standard-cell 48709= ?@54?>;0305B C:;04:C 2 @O4K >48=0:>2KE ?> >4=><C 87<5@5=8N M;5<5=B>2. 0: 284=> 87 @8A.1, C 2A5E M;5<5=B>2 >48=0:>20O 2KA>B0, => @07=0O 4;8=0.  A;CG05 general cell layout D>@<0 M;5<5=B>2 O2;O5BAO ?@>872>;L=>9. B;8G8B5;L=>9 >A>15==>ABLN G5B25@B>3> A;CG0O O2;O5BAO B>, GB> =0 >4=>9 ?;0B5 ?@54?>;0305BAO @07<5AB8BL CG0AB:8 A M;5<5=B0<8 @07=KE B8?>2.  B0:8E A;CG0OE @07<5I5=85 G0AB> ?@>872>48BAO @C:0<8, G5;>25: =5?>A@54AB25==> CG0AB2C5B 2 ?@>F5AA5. >E>60O :;0AA8D8:0F8O ?@82545=0 2 [7]:  @8A.2 ;0AA8D8:0F8O 7040G @07<5I5=8O, ?@8254Q==0O 2 [7] Gate-array AE5<K 87=0G0;L=> =5 8<5NB >?@545;5==>9 DC=:F88, =0 =8E ;8HL @0A?>;>65=K 2 2845 ?@O<>C3>;L=>9 B01;8FK B@0=78AB>@K, AB0=40@B=K5 ;>38G5A:85 M;5<5=BK 8 4@. 0:B82=K5 CAB@>9AB20. >=5G=0O AE5<0 A >?@545;5==>9 DC=:F8>=0;L=>ABLN A>7405BAO ?CB5< 4>102;5=8O A>548=5=89 <564C M;5<5=B0<8. "> 5ABL @07@01>BG8: A0< ?@>2>48B =C6=K5 A>548=5=8O, 2 =5:>B>@>< A<KA;5 ?@>3@0<<8@CO AE5<C. @>F5AA A>740=8O ?>4>1=KE AE5< 3>@074> 1KAB@55, G5< standard-cell 8 full-custom 48709=. @><5 B>3>, C?@>I5= ?@>F5AA B5AB8@>20=8O AE5<K 2 A8;C B>3>, GB> AE5<K 8<5NB AB0=40@B=K9 @07<5@ 8 <>3CB 1KBL 8A?>;L7>20=K >4=8 8 B5 65 8=AB@C<5=BK. B=>H5=85 AC<<0@=>9 ?;>I048 @07<5I05<KE M;5<5=B>2 : ?;>I048 ?;0BK =07K205BAO placement densities.  =0AB>OI55 2@5<O 2 ?@><KH;5==>AB8 8A?>;L7C5BAO fixed-die <>45;L, 2 :>B>@>9 4;O AE5<K 8725AB=0 25@E=OO 3@0=8F0 ?> ?;>I048.  MB>9 A8BC0F88 B@51C5BAO ;8HL C;>68BLAO 2 MBC ?;>I04L. -B0 <>45;L :>=B@0AB8@C5B A @0=55 8A?>;L7>202H59AO variable-die <>45;LN, 2 :>B>@>9 1K;> 206=> <8=8<878@>20BL ?;>I04L AE5<K. 040G8 ?5@2>3> B8?0, 0 2 =8E :0: ?>4B8? standard cell placement, =081>;55 G0AB> 2AB@5G0NBAO 2 ?@><KH;5==>AB8. 040G8 65 2B>@>3> 8 B@5BL53> B8?0 3>@074> A;>6=55 8 2AB@5G0NBAO @565, 2 ?@54K4CI59 :;0AA8D8:0F88 >=8 1K;8 =0720=K full-custom layout design. @8B5@88 >?B8<0;L=>AB8 @07<5I5=8O  40==>< @0745;5 2 >A=>2=>< ?@82545=0 8=D>@<0F8O 87 [11]. 8=8<870F8O 4;8=K ?@>2>4>2 8 70=8<05<>9 ?;>I048 (chip area) O2;ONBAO B@048F8>==K<8 F5;O<8 >?B8<870F88. > A0<8 ?> A515 >=8 =5 ?@54AB02;ONB 8=B5@5A0 4;O @07@01>BG8:0.  A0<>< 45;5, 5A;8 AE5<0 M:>=><=> ?>B@51;O5B M=5@38N 8 8<55B E>@>HCN B0:B>2CN G0AB>BC, B> <0;> :>3> 8=B5@5ACNB B0:85 E0@0:B5@8AB8:8, :0: 4;8=0 ?@>2>4>2 8 ?;>I04L AE5<K. 865 ?@82545=K 4@C385 @0AA<0B@8205<K5 :@8B5@88. >7<>6=>ABL ?@>:;04:8 A>548=5=89 (Routability). 8=8<870F8O 4;8=K A>548=5=89 8 ?;>I048 <>65B 2 :>=5G=>< 8B>35 ?@825AB8 : B><C, GB> ?@>:;04:0 A>548=5=89 1C45B =52>7<>6=0. ;O B>3> GB>1K CG5ABL MB>B <><5=B, 2 :0G5AB25 :@8B5@8O @0AA<0B@820NB ?5@53@C65==>ABL (congestion). Placer K, CG8BK20NI85 MB>B :@8B5@89, AB@5<OBAO @02=><5@=> @0A?@545;8BL M;5<5=BK ?> AE5<5 (2 @0<:0E fixed-die <>45;8), GB>1K 871560BL A;8H:>< 1;87:>3> @0A?>;>65=8O A>548=5=89. >:070B5;8 ?@>872>48B5;L=>AB8 (Performance issues): @0AE>4 M=5@388 (power consumption), 7045@6:0 A8AB5<K (system delay), F5;>AB=>ABL A83=0;>2 (signal integrity). -B8 ?>:070B5;8 =5 <>3CB 1KBL >F5=5=K ?@8 ?><>I8 ?@>ABKE 35><5B@8G5A:8E ?@028;, 8E CG5B =5B@8280;5=. 48= >?0A=K9 MDD5:B, :>B>@K9 B@C4=> 2KO28BL 70@0=55  MB> coupling capacitance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irelength) ?>-?@56=5<C @0AA<0B@8205BAO 2 :0G5AB25 >A=>2=>9 E0@0:B5@8AB8:8 ?;0BK. -B> >1JOA=O5BAO A;54CNI8<8 ?@8G8=0<8: 4;8=0 ?@>2>40 A8;L=> A2O70=0 A >AB0;L=K<8 E0@0:B5@8AB8:0<8. 0G0ABCN <>6=> 4>18BLAO 2KA>:>3> :0G5AB20 AE5<K 8<5==> 70 AG5B A>:@0I5=8O 4;8=K ?@>2>4>2. G5284=>, GB> G5< >=8 :>@>G5, B5< 1KAB@55 4>;6=0 @01>B0BL AE5<0 8 B5< <5=LH5 70=8<0BL <5AB0 8 ?>B@51;OBL M=5@388. %>BO A 4@C38<8 ?>:070B5;O<8 ?@>B82>?>;>6=0O A8BC0F8O: ?@8 <0;>9 4;8=5 A>548=5=89 2KH5 8E ?;>B=>ABL 8 B5< A;>6=55 8E ?@>:;04K20BL, B0:65 A8;L=55 1C4CB ?5@5:@5AB=K5 ?><5E8 <564C A>548=5=8O<8. 0 @8A.3 ?@54AB02;5=K =5A:>;L:> 2>7<>6=KE A?>A>1>2 ?@81;865=8O 4;8=K A>548=5=89.  @8A.3 KG8A;5=85 4;8=K A>548=5=89 5@2K9 87 =8E (A0<K9 B>G=K9, => 8 =081>;55 A;>6=K9) A2O70= A ?>AB@>5=85< Steiner Tree  :@0BG09H8< ?CB5<, A>548=ONI89 :>=B0:BK M;5<5=B>2. @8 MB>< ?@>2>4 <>65B 25B28BLAO 2 ;N1>9 B>G:5. B>@>9  ?>AB@>5=85 Minimal spanning tree, >B;8G85 >B ?@54K4CI53> 2 B><, GB> 25B2;5=85 2>7<>6=> =5 2 ?@>872>;L=>< <5AB5, 0 ;8HL =0 :>=B0:B0E. Chain connections =5 4>?CA:05B 25B2;5=8O 2>>1I5, :>=B0:BK A>548=ONBAO ?>A;54>20B5;L=>. 5B>4 ?@>I5, G5< ?@54K4CI85 420, => 405B A;53:0 1>;LHCN >F5=:C. Source-to-Sink <5B>4 ?@54?>;0305B A>548=5=85 8AB>G=8:>2 A> AB>:0<8 >B45;L=K<8 ?@>2>40<8. 5B>4 =081>;55 ?@>AB 2 @50;870F88, => 405B 702KH5==CN >F5=:C 8 A;8H:>< 1>;LH>9 ?>:070B5;L ?5@53@C65==>AB8 (congestion), ?>MB><C 8A?>;L7C5BAO :@09=5 @54:>. -DD5:B82=K9 8 =081>;55 H8@>:> 8A?>;L7C5<K9 <5B>4  half-perimeter method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teiner tree, G5B25@BK9  ;52K9 =86=89 C3>; M;5<5=B>2.  =0AB>OI55 2@5<O G0I5 2A53> 8A?>;L7CNB 2B>@>9 8 B@5B89 <5B>4K. "@5B89 ?@54AB02;O5BAO =081>;55 045:20B=K<, 0 2B>@>9 O2;O5BAO A0<K< ?@>ABK<.  @8A.4 K1>@ >:09<;ONI53> ?@O<>C3>;L=8:0 >4E>4K : @5H5=8N 7040G8 081>;55 ?>;=K9 >17>@ ACI5AB2CNI8E ?>4E>4>2 8 ?;59A5@>2 ?@82545= 2 ;5:F8OE ?@>D5AA>@0 Pan a: " Simulated Annealing  Timberwolf package  Dragon " Partitioning-Based Placement  Capo  FengShui " Analytical Placement  Gordian  Kraftwerk  FastPlace " Hall s Quadratic Placement " Genetic Algorithm 5=5B8G5A:85 0;3>@8B<K (Genetic algorithms) ?>:070;8 A51O <0;>MDD5:B82=K<8.  :>=F5 80-E  =0G0;5 90-E ?>O28;>AL =5A:>;L:> ?C1;8:0F89, ?>A2OI5==KE 40==>9 B5<5, => >=8 =5 1K;8 ?@>4>;65=K. #65 B>340 : 35=5B8G5A:8< 0;3>@8B<0< 1K;0 ?@8<5=5=0 ?0@0;;5;870F8O, 8 2A5 65 0;3>@8B<K @01>B0;8 :@09=5 <54;5==>.  2000-E 3>40E A=>20 =01;N405BAO =5:89 2A?;5A: 8=B5@5A0 : 40==>9 B5<0B8:5, 2=8<0=85 C45;O5BAO A>25@H5=AB2>20=8N 0;3>@8B<0 8 B5E=8: 53> ?0@0;;5;870F88, A<. =0?@8<5@ [10]. >, :0: CB25@640NB A0<8 02B>@K, 2@5<O @01>BK 2A5 5I5 >G5=L 2KA>:>. 5@2K5 65 B@8 ?>4E>40 4>AB0B>G=> MDD5:B82=K 8 ACI5AB2CNB placer K, 8A?>;L7CNI85 8E. %>@>H89 >17>@ 8E E0@0:B5@8AB8:, 2>7<>6=>AB59 8 @57C;LB0B>2 A@02=5=8O <>65B 1KBL =0945= 2 [1]. Partition-based placement (Min-cut placement) >4E>4 2?5@2K5 C?><8=05BAO 2 [6]. ;02=0O 845O A>AB>8B 2 ?>AB5?5==>< ?>=865=88 @07<5@=>AB8 7040G8. 0 :064>9 8B5@0F88 <=>65AB2> M;5<5=B>2 @0718205BAO =0 =5A:>;L:> ?>4<=>65AB2, :064>5 87 :>B>@KE A=>20 4@>18BAO, 8 B0: 40;55. 064K9 @07 A>>B25BAB2CNI8< >1@07>< ?;0B0 45;8BAO =0 G0AB8, 8 ?>4<=>65AB20< M;5<5=B>2 ?@8A20820NBAO A>1AB25==K5 CG0AB:8 ?;0BK. @>F5AA ?@>4>;605BAO 4> B5E ?>@, ?>:0 ?>4<=>65AB20 =5 AB0=CB 4>AB0B>G=> <0;K<8 (=0?@8<5@, 2 Capo - ?> 8 M;5<5=B>2 2 :064><). >A;5 B>3>, :0: :064><C 87 <=>65AB2 1C45B ?@8A2>5=> <5AB> =0 ?;0B5, ?@>8AE>48B @07<5I5=85 M;5<5=B>2 2=CB@8 2K45;5==>3> CG0AB:0. B0:, 0;3>@8B< <>6=> 70?8A0BL 2 A;54CNI5< 2845: 0718BL <=>65AB2> M;5<5=B>2 =0 ?>4<=>65AB20 B0:8< >1@07><, GB>1K G8A;> A2O759 <564C ?>4<=>65AB20<8 1K;> <8=8<0;L=K<. !>>B25BAB2CNI8< >1@07>< @0718BL ?;>I04L, =0 :>B>@>9 ?@>8AE>48B @07<5I5=85. ?@545;8BL <5AB> 4;O :064>3> ?>4<=>65AB20 M;5<5=B>2. A;8 G8A;> M;5<5=B>2 2 :064>< ?>4<=>65AB25 1>;LH5 =5:>B>@>3> k, 25@=CBLAO =0 H03 (1), 2 ?@>B82=>< A;CG05 ?5@5E>4 =0 H03 (5). 07<5AB8BL M;5<5=BK :064>3> ?>4<=>65AB20 2=CB@8 2K45;5==>3> CG0AB:0 A>3;0A=> =5:>B>@><C 0;3>@8B<C @07<5I5=8O. !CI5AB2CNI85 @50;870F88 ?>4E>40 @07;8G0NBAO ?> A?>A>10< @07185=8O =0 ?>4<=>65AB20, A?>A>10< @07185=8O =0 CG0AB:8, 0;3>@8B<0< :>=5G=>3> @07<5I5=8O. 0<8 1K;8 @0AA<>B@5=K 420 0:045<8G5A:8E placer a, A;54CNI85 40==><C ?>4E>4C: Capo 8 FengShui. 5@2K9 @07@010BK205BAO 2 University of Michigan, 2B>@>9 - 2 Binghamton University. Analytical Placement  @0<:0E 40==>3> ?>4E>40 7040G0 @07<5I5=8O D>@<C;8@C5BAO :0: <0B5<0B8G5A:0O 7040G0. 5 <>6=> ?@54AB028BL :0: 7040GC <0B5<0B8G5A:>3> ?@>3@0<<8@>20=8O A :204@0B8G=>9 DC=:F859 AB>8<>AB8. Quadratic placement A;8 <8=8<878@C5BAO 4;8=0 ?@>2>4>2, B> 55 :204@0B  =5:0O :204@0B8G=0O DC=:F8O: Force-directed placement  MB>< ?>4E>45 7040G0 @07<5I5=8O ?@5>1@07C5BAO 2 7040GC <5E0=8:8. @8<5=ONBAO A;54CNI85 0=0;>388: Module (Block/Cell/Gate) = Object Net = Spring Net weight = Spring constant. Optimal placement = Equilibrium configuration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an a, AB@.19). 0 >A=>25 40==>3> ?>4E>40 @50;87>20= placer Kraftwerk [6], => >= ?>:07K205B ?>A@54AB25==K5 @57C;LB0BK [1]. @5645 2A53>, =C6=> C:070BL =0 B>, 2 @57C;LB0B5 53> @01>BK A>7405BAO @07<5I5=85, 2 :>B>@>< 8<5NBAO =0;>65=8O M;5<5=B>2 4@C3 =0 4@C30. >MB><C 8<55BAO =5>1E>48<>ABL 2 >B45;L=>< MB0?5 ;530;870F88 ?>;CG5==>3> @07<5I5=8O, :>B>@>5 >ACI5AB2;O5BAO ?@8 ?><>I8 4@C3>3> 8=AB@C<5=B0. Simulated annealing -B>B ?>4E>4 >4=8< 87 ?5@2KE 1K; ?@8<5=5= : 7040G5 @07<5I5=8O. 45O 53> 8A?>;L7>20=8O ?@>AB0: 15@5BAO =5:>B>@>5 =0G0;L=>5 @07<5I5=85 M;5<5=B>2 (?CABL 4065 ?@>872>;L=>5), 70B5< 53> ?KB0NBAO C;CGH8BL, A>25@H0O A;CG09=K5 ?5@5AB0=>2:8 M;5<5=B>2 8 ?@8=8<0O 8;8 >B25@30O ?@5>1@07>20=85 2 A>>B25BAB288 A =5:>B>@K< :@8B5@85< . 5B>4 A@07C 70@5:><5=4>20; A51O :0: 25AL<0 MDD5:B82=K9, => B@51>20; >B=>A8B5;L=> 1>;LH>3> 2@5<5=8 4;O @01>BK. =AB@C<5=BK 4;O @5H5=8O 7040G8 Capo A=>2=K< ?@58<CI5AB2>< Capo ?> >B=>H5=8N : 4@C38< placer 0< O2;O5BAO A:>@>ABL 53> @01>BK, 2 B> 2@5<O :0: :0G5AB2> ?>;CG05<>3> A ?><>ILN =53> @07<5I5=8O CABC?05B :>=:C@5=B0<. ;>10;L=>5 @07<5I5=85 2 Capo 2K?>;=5=> A ?><>ILN @5:C@A82=>3> 45;5=8O ?>?>;0< A 8A?>;L7>20=85< 8=AB@C<5=B0 4;O <=>3>C@>2=52>3> 45;5=8O A C?@02;O5<>9 ;8=859-@0745;8B5;5<. 5B0;L=>5 @07<5I5=85 2K?>;=5=> A ?><>ILN ?@>F54C@K branch-and-bound (<5B>4 25B259 8 3@0=8F). Capo G8B05B AB0=40@B=K9 D>@<0B LEF/DEF (8 B0:65 C?@>I5==K9 D>@<0B 4;O 0:045<8G5A:8E 8AA;54>20=89) 8 @01>B05B 2 fixed-die @568<5. Capo 405B @57C;LB0BK, A@02=8<K5 A :><<5@G5A:8<8 8=AB@C<5=B0<8; 4>ABC?5= (A> 2A5<8 181;8>B5:0<8 ?>445@6:8) A>3;0A=> open-source ;8F5=788. 0<8 A:><?8;8@>20=0 8 1C45B ?@>B5AB8@>20=0 25@A8O Capo 9.1, 4>ABC?=0O =0  HYPERLINK "http://vlsicad.eecs.umich.edu/BK/Pdtools" http://vlsicad.eecs.umich.edu/BK/Pdtools. 725AB=>, GB> ACI5AB2C5B 1>;55 ?>74=OO 25@A8O  Capo 9.3, 55 <>6=> A:0G0BL 745AL:  HYPERLINK "http://vlsicad.eecs.umich.edu/BK/PDtools/tar.gz/LATEST/UMpack-43-051014.tar.gz" http://vlsicad.eecs.umich.edu/BK/PDtools/tar.gz/LATEST/UMpack-43-051014.tar.gz. -B0 25@A8O, ?>O282H0OAO 2 A5=BO1@5 2005 3>40, 4>ABC?=0 ;8HL 70@538AB@8@>202H8<AO ?>;L7>20B5;O< Capo. A5 ?>A;54CNI85 25@A88 B0: 65 =5 1C4CB 2K;>65=K =0 homepage, 0 1C4CB 4>ABC?=K ;8HL 70@538AB@8@>202H8<AO ?>;L7>20B5;O<. @8=F8?K @01>BK !E5<0 @01>BK  ;>10;L=>5 @07<5I5=85 ;3>@8B< 3;>10;L=>3> @07<5I5=8O Capo AB@5<8BAO @0745;8BL ?;0BC A M;5<5=B0<8 =0 G0AB8 (@538>=K), A2O7K20O ?@8 MB>< >B45;L=K5 3@C??K M;5<5=B>2 (<>4C;8) A ?>;CG5==K<8 @538>=0<8 8 C<5=LH0O A?8A>: A2O759 <564C <>4C;O<8 87 @07=KE @538>=>2 B0:8< >1@07><, GB>1K A=0G0;0 =09B8 >?B8<0;L=>5 @07<5I5=85 2 :064>< @538>=5, 0 70B5< ?>;CG8BL >?B8<0;L=>5 @07<5I5=85 2A5E M;5<5=B>2. 0 ?@0:B8:5 B0:>5 @07185=85 @50;87C5BAO =0 >A=>25 @5H5=8O 7040G8 > <=>3>C@>2=52>< >?B8<0;L=>< @07185=88 38?5@3@0D0 (A<. @01>BK 0@8?8A0, 2 G0AB=>AB8, AB0BL8 [21], [22]). 8?5@3@0D  EMBED Equation.3 ?@54AB02;O5B A>1>9 <=>65AB2> 25@H8=  EMBED Equation.3 8 <=>65AB2> @515@  EMBED Equation.3 , 345 :064>5 @51@> 7040=> ?>4<=>65AB2>< <=>65AB20 25@H8=  EMBED Equation.3 . ?B8<0;L=>5 @07185=85 38?5@3@0D0 AB@5<8BAO <8=8<878@>20BL G8A;> A2O759, 8=F845=B=KE M;5<5=B0< 2 @07;8G=KE @07185=8OE. 064>5 @07185=85 38?5@3@0D0 ?@>872>48BAO 4;O ?@O<>C3>;L=>9 >1;0AB8, 8;8 1;>:0, 8 2:;NG05BAO 2 :>=5G=>5 @0A?>;>65=85 M;5<5=B>2. ;>: A>>B25BAB2C5B: @538>=C A 4>?CAB8<K< @0A?>;>65=85< M;5<5=B>2; =01>@C <>4C;59, :>B>@K5 4>;6=K 1KBL @07<5I5=K 2 40==>< @538>=5; 2A5< A2O7O<, 8=F845=B=K< 2A5< 2:;NG5==K< <>4C;O< 8 @0A?>;>65=8N 2A5E <>4C;59 2=5 40==>3> @538>=0, A<56=KE A <>4C;O<8, @0A?>;>65==K<8 2 40==>< @538>=5. -;5<5=BK, @0A?>;>65==K5 2=CB@8 1;>:0, ?@54AB02;ONB A>1>9 25@H8=K 38?5@3@0D0, 0 @51@0<8 38?5@3@0D0 AG8B0NBAO A2O78, 8=F845=B=K5 M;5<5=B0< 2=CB@8 1;>:0. 5A>< 25@H8=K AG8B05BAO ?;>I04L, 70=8<05<0O 0AA>F88@>20==>9 A =59 OG59:>9. 5H5=85-@07185=85 4>;6=> 8<5BL ?@8<5@=> >48=0:>2K9 AC<<0@=K9 25A 2 :064>< @07185=88. =0G5, =5:>B>@K5 1;>:8 <>3CB 8<5BL 1>;LH5 M;5<5=B>2, G5< <>65B 1KBL @0A?>;>65=> 2=CB@8 ?;0BK 157 ?5@5:@KB89. @>F5AA @07<5I5=8O <>65B 1KBL ?@54AB02;5= :0: ?>A;54>20B5;L=>ABL H03>2, 345 :064K9 H03 ?@>25@O5B 2A5 1;>:8: 1;>:8 <>3CB 1KBL ?@>?CI5=K, =5 B@>=CBK5 >?B8<870F859 B5:CI53> C@>2=O, 8;8 ?>4@0745;5=K =0 420 <5=LH8E 1;>:0. >340 <K ?@8<5=O5< @5:C@A82=>5 45;5=85 ?>?>;0<, AB0=>28BAO 206=K< 2K1>@ =0?@02;5=8O 45;5=8O  3>@87>=B0;L=>3> 8;8 25@B8:0;L=>3>: MB>B 2K1>@ <>65B 2;8OBL =0 4;8=C ?@>2>4>2 8 A:>?;5=8O M;5<5=B>2 2 D8=0;L=>< @0A?>;>65=88 M;5<5=B>2.  Capo 8A?>;L7>20= ?@8=F8? 2K1>@0 =0?@02;5=8O 45;5=8O, >?8A0==K9 2 [19]. 5B0;L=>5 @07<5I5=85 5B0;L=>5 @07<5I5=85 2 ?;59A5@5 Capo 2K?>;=O5BAO A ?><>ILN 8=AB@C<5=B0 Row Ironing. 3> @01>B0 ?>AB@>5=0 ?> A;54CNI59 AE5<5: 2 :064>< @O4C 2K45;O5BAO 3@C??0 87 7-8 ?>A;54>20B5;L=> @0A?>;>65==KE M;5<5=B>2, 8 :0640O 3@C??0 =57028A8<> >B 4@C38E 3@C?? ?5@5<5I05BAO A ?><>ILN >?B8<878@CNI59 branch-and-bound ?@>F54C@K ([19]).  :064>< @O4C 2K45;5=85 B0:8E 3@C?? ?@>8AE>48B A 8=B5@20;>< 2 2-3 M;5<5=B0. 07<=>65=85 B5@<8=0;>2 07<=>65=85 B5@<8=0;>2 O2;O5BAO 206=>9 A>AB02;ONI59, 2;8ONI59 =0 :0G5AB2> @07<5I5=8O A ?><>ILN min-cut placer 0. > 2@5<O @07<5I5=8O, :>340 :0:>9-;81> 1;>: @0745;5= =0 =5A:>;L:> ?>41;>:>2, =5:>B>@K5 M;5<5=BK 2=CB@8 ?>41;>:0 <>3CB 1KBL A2O70=K A M;5<5=B0<8 2=5 MB>3> ?>41;>:0 (B0:85 25@H8=K =07K20NBAO B5@<8=0;0<8). 5 CG8BK20O =0;8G8O B5@<8=0;>2, <K @8A:C5< ?>;CG8BL @07<5I5=85 =0<=>3> EC65 >?B8<0;L=>3>. @>1;5<C @07<=>65=8O B5@<8=0;>2 @5H0NB 70?@5I5=85< A2>1>4=>3> A2O7K20=8O B5@<8=0;>2 A =5:>B>@K<8 G0ABO<8 @07185=8O 8;8 D8:A8@>20=85< B5@<8=0;>2, B.5. :064><C B5@<8=0;C 70405BAO >?@545;5==>5 =587<5==>5 <5AB> =0 ?;0B5. ! @07<=>65=85< B5@<8=0;>2 G0I5 2A53> 1>@NBAO A ?><>ILN 35><5B@8G5A:>3> ?@81;865=8O B5@<8=0;>2 : G0ABO< @07185=8O, c :>B>@K<8 >=8 A2O70=K. =>385 placer K ?@8<5=ONB B0: =07K205<K5 ad hoc B5E=8:8, =0?@8<5@, @0A?>;030NB B5@<8=0;K 2 F5=B@5 G0AB8 @07185=8O 8;8 15@CB ?@>872>;L=>5 4>?CAB8<>5 @07<5I5=85 B0:8E :;5B>:. 0?@8<5@, :0G5AB2> @07<5I5=8O Capo 1K;> CA>25@H5=AB2>20==> 1;03>40@O A;8O=8N Capo A >A=>20==K< =0 SOR-<5B>45 0=0;8B8G5A:8< placer ><. -B>B placer 2K7K205BAO ?>A;5 :064>3> H030 min-cut @07185=8O Capo 8 =0E>48B @07<5I5=85 :;5B>:, :>B>@>5 <8=8<878@C5B :204@0B 4;8=K ?@>2>4>2, @0A?>;>65==KE 2=CB@8 >3@0=8G5=89 @538>=>2, :>B>@K< A>>B25BAB2C5B >?@545;5==K9 M;5<5=B.  8450;5 2 :>=F5 @07185=8O MB>B M;5<5=B 4>;65= 1KBL 2=CB@8 1;>:0, :>B>@><C A>>B25BAB2C5B, 8 =0E>48BLAO 2 F5=B@5 MB>3> 1;>:0. SOR (Successive Over Relaxation)  MB> 8B5@0B82=K9 <5B>4 @5H5=8O A8AB5< ;8=59=KE C@02=5=89 2840 AX = B. ?8A0=85 40==>3> <5B>40 2> <=>3>< ?5@5A5:05BAO A >?8A0=85< 3;>10;L=>3> @07<5I5=8O, 8A?>;L7C5<>3> 2 0=0;8B8G5A:8E placer 0E.  Capo =0 40==K9 <><5=B 8A?>;L7C5BAO ?0@0;;5;L=0O 25@A8O SOR0: 45B0;L=0O 53> @50;870F8O >?8A0=0 2 [18]. >@L10 A ?5@5:@KB8O<8 >@L10 A ?5@5:@KB8O<8 2 Capo ?@>8AE>48B 2 2 MB0?0: 8A:;NG05BAO ?@52KH5=85 AC<< 4;8= M;5<5=B>2 2 @O4C =04 4;8=>9 @O40 (<5B>4 0=8?C;8@>20=85 OG59:0<8 (Cell Juggling)); 8A:;NG0NBAO ?5@5:@KB8O <564C 4;8=0<8 M;5<5=B>2 2 ?@545;0E @O40 (<5B>4 530;870F8O OG55: 2 :064>< @O4C (Row Legalization)). @><5 ?5@5G8A;5==KE <5B>4>2 A>2@5<5==K9 Capo 8A?>;L7C5B <5B>4, =0?@02;5==K9 =0 CA>25@H5=AB2>20=85 @0A?>;>65=8O M;5<5=B>2 A F5;LN <8=8<870F88 4;8=K ?@>2>4>2, A>548=ONI8E M;5<5=BK, =0E>4OI85AO 2 >4=>< @O4C ( 540:B8@>20=85 ?>@O4:0 OG55: 2 :064>< @O4C (Cell Order Polishing)). 5B>4 B5A=> A2O70= A @01>B>9 <5B>40 530;870F8O OG55: 2 :064>< @O4C, B0: GB> AE5<0 1>@L1K A ?5@5:@KB8O<8 <>65B 1KBL ?@54AB02;5=0 2 A;54CNI5< 2845:   0AA<>B@8< ?5@5G8A;5==K5 <5B>4K 1>;55 ?>4@>1=>. 0=8?C;8@>20=85 OG59:0<8 (Cell Juggling)  @57C;LB0B5 @01>BK 40==>3> <5B>40 <K C156405<AO, GB> 4;O :064>3> @O40 AC<<0 4;8= 2A5E M;5<5=B>2 =5 ?@52KH05B @07<5@ @O40. ;O MB>3> 8A?>;L7C5BAO ?@>AB0O M2@8AB8:0  0=8?C;8@>20=85 OG59:0<8 (Cell Juggling), :>B>@0O =5<=>3> 2;8O5B =0 ?>;C?5@8<5B@ ?@O<>C3>;L=8:0, >:09<;ONI53> A>548=5=85 - HPWL (Half Perimeter Wire Length). ?@545;8< <5B>4 0=8?C;8@>20=85 OG59:0<8 :0: ?5@5<5I5=85 M;5<5=B>2 B>;L:> 2 25@B8:0;L=>< =0?@02;5=88. >A:>;L:C HPWL =57028A8<0 >B 2K1>@0 3>@87>=B0;L=>3> 8;8 25@B8:0;L=>3> =0?@02;5=8O, <5B>4 A>E@0=O5B 7=0G5=85 >1I59 4;8=K ?@>2>4>2, :>B>@K5 @0A?>;030NBAO 3>@87>=B0;L=>. ;3>@8B< 0=>: @07<5I5=85 A ?5@5:@KB8O<8, 2 :>B>@>< =5 CGB5=K @07<5@K @O4>2 57C;LB0B: @07<5I5=85 A ?5@5:@KB8O<8, 2 :>B>@>< CGB5=K @07<5@K @O4>2 _________________________________________________________________________________ For i = 1 to r: 4;O :064>3> @O40 EMBED Equation.3 >?@545;8< @07=>ABL <564C AC<<>9 4;8= M;5<5=B>2 2 40==>< @O4C  EMBED Equation.3  8 4;8=>9 @O40  EMBED Equation.3  :0:  EMBED Equation.3 , B> 5ABL EMBED Equation.3 . #?>@O4>G8< 2A5 @O4K 2 ?>@O4:5 =5C1K20=8O ?0@0<5B@0 EMBED Equation.3  ;O :064>3> @O40  EMBED Equation.3 , 4;O :>B>@>3>  EMBED Equation.3 , 2K?>;=8<: While  EMBED Equation.3   EMBED Equation.3  For each cell  EMBED Equation.3 : For each k = 1 to r: If  EMBED Equation.3  and  EMBED Equation.3  then !G8B05<, =0A:>;L:> 2>7@>A;0 25;8G8=0  EMBED Equation.3  (HPWL) ?@8 ?5@5<5I5=88 M;5<5=B0  EMBED Equation.3 2 @O4  EMBED Equation.3  If  EMBED Equation.3  then  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3  ?5@5<5AB8BL  EMBED Equation.3  2 @O4  EMBED Equation.3  8 >1=>28BL 7=0G5=8O ?0@0<5B@0  EMBED Equation.3 4;O @O4>2  EMBED Equation.3  8  EMBED Equation.3  _________________________________________________________________________________ 0 40==>< MB0?5 87102;5=8O >B ?5@5:@KB89 <K ?@5=51@5305< ;N1K<8 ?5@5:@KB8O<8, 2>7=8:0NI8<8 2 @57C;LB0B5 ?5@5<5I5=8O M;5<5=B0  EMBED Equation.3  2 @O4  EMBED Equation.3 , ?>A:>;L:C 2B>@>9 MB0? 30@0=B8@C5B =0< 87102;5=85 >B MB8E ?5@5:@KB89. 530;870F8O OG55: 2 :064>< @O4C (Row Legalization) @54AB028< :064K9 @O4 :0: <=>65AB2> C?>@O4>G5==KE <5AB  EMBED Equation.3 . ;O :064>3> @O40  EMBED Equation.3 <K 4>;6=K @0A?>;>68BL <=>65AB2> 53> M;5<5=B>2  EMBED Equation.3  B0:, GB>1K 2 =5< =5 1K;> ?5@5:@KB89, B.5.  EMBED Equation.3 , 345  EMBED Equation.3 - :>>@48=0B0 ;52>3> :>=F0 M;5<5=B0. ;O ?>AB@>5=8O @07<5I5=8O 157 ?5@5:@KB89 ?>AB@>8< >@85=B8@>20==K9 0F8:;8G5A:89 3@0D  EMBED Equation.3 , :0: ?>:070=> =0 @8A.2, A <=>65AB2>< 25@H8=  EMBED Equation.3 8 <=>65AB2>< @515@:  EMBED Equation.3  =>65AB2> @515@ E A>AB02;5=> 87 3>@87>=B0;L=KE 8 4803>=0;L=KE @515@ 3@0D0 G. 064>5 @51@> E0@0:B5@87C5BAO 53> =0G0;>< 8 :>=F>< 8 ?@54AB028<> 2 2845 ?0@K 25@H8=. 040G0 > =0E>645=88 4>?CAB8<>3> @07<5I5=8O 157 ?5@5:@KB89 A>>B25BAB2C5B =0E>645=8N ?CB8 >B 25@H8=K (0,0) 4> (m,n).  @8A.5. !0<K9 1KAB@K9 A?>A>1 =0E>645=8O @07<5I5=8O 157 ?5@5:@KB89  E>45 ?>;CG5=8O 4>?CAB8<>3> @07<5I5=8O <K <>65< AB@5<8BLAO <8=8<878@>20BL @07;8G=K5 ?0@0<5B@K, =0?@8<5@: >;8G5AB2> ?5@5AB0=>2>: M;5<5=B>2 2> 2A5E @O40E (minPERB). ;8=0 ?@>2>4>2, >F5=8205<0O :0: ?>;C?5@8<5B@ ?@O<>C3>;L=8:0, >:09<;ONI53> A>548=5=85 (minHPWL) 0:A8<C< ?5@5AB0=>2>: 2 @O45(minMAX).  7028A8<>AB8 >B 2K1>@0 ?0@0<5B@0 0;3>@8B< =0E>645=8O @07<5I5=8O 157 ?5@5:@KB89 1C45B 8<5BL A;54CNI89 284. ;3>@8B< 0=>: @O4  EMBED Equation.3 , <=>65AB2> M;5<5=B>2  EMBED Equation.3 , ?0@0<5B@ 4;O <8=8<870F88 (objective) 57C;LB0B: @07<5I5=85 M;5<5=B>2  EMBED Equation.3 , ?@8 :>B>@>< 4>AB8305BAO <8=8<C< 2K1@0==>3> ?0@0<5B@0 4;O <8=8<870F88 _________________________________________________________________________________ cost (0, 0)=0 For j = 1 to m: cost (j, 0) = EMBED Equation.3  For k = 1 to n: cost (0, k) = 0 For j=1 to m For k=1 to n - w (cj) If objective is minPERB, minHPWL Then cost (j, k) = min (cost (j, k-1), cost (j-1, k- w (cj))) If objective is minMAX Then cost (j, k) = min (cost (j, k-1), max (cost (j-1, k- w (cj)), |k- w (cj)-xcj |) Return cost (m, n) _________________________________________________________________________________ 540:B8@>20=85 ?>@O4:0 OG55: 2 :064>< @O4C (Cell Order Polishing) -2@8AB8:0, :>B>@0O 70;>65=0 2 40==>< <5B>45, 8725AB=0 402=>: 2?5@2K5 (2 1961 3>4C) 1K;0 8A?>;L7>20=0 2 7040G5 @0A?>;>65=8O ?;0BK A ?5G0B=>9 AE5<>9.  @0<:0E 40==>3> ?>4E>40 4;O :064>3> @O40 2K18@05< =57028A8<>5 <=>65AB2> M;5<5=B>2, ?>4@07C<520O ?>4 MB8< B5@<8=>< <=>65AB2> M;5<5=B>2, =0E>4OI8EAO 2 >4=>< @O4C 8 =5 A<56=KE 4@C3 A 4@C3><. K 2K18@05< B0:>5 <=>65AB2>, A=0G0;0 35=5@8@CO ?@>872>;L=K9 ?>@O4>: M;5<5=B>2, 0 40;55 >1@010BK205< M;5<5=BK A>3;0A=> MB><C ?>@O4:C 8, 2K18@0O >G5@54=>9 M;5<5=B 2 :0G5AB25 M;5<5=B0 =57028A8<>3> <=>65AB20, 5A;8 MB> =57028A8<>5 <=>65AB2> 2 B5:CI5< A>AB>O=88 =5 A>45@68B =8 >4=>3> M;5<5=B0, A<56=>3> A 40==K<. (:07K205BAO, GB> 2 A@54=5< 30-40% M;5<5=B>2 2 @O4C O2;ONBAO =57028A8<K<8). >A;5 2K1>@0 =57028A8<>3> <=>65AB20 <K ?5@5<5I05< M;5<5=BK 2 A>>B25BAB288 A 0;3>@8B<>< =0E>645=8O <8=8<0;L=>9 4;8=K ?CB8, A>548=ONI53> 2A5 M;5<5=BK =57028A8<>3> <=>65AB20 (7040G0 =0 3@0D5). >A:>;L:C 4;8=K 2=>2L ?5@5AB02;5==KE 40==K< <5B>4>< M;5<5=B>2 <>3CB 1KBL @07;8G=K, 8E ?5@5<5I5=85 <>65B ?>@>48BL ?5@5:@KB8O.  MB>< A;CG05 <K 4>;6=K 25@=CBLAO : <5B>4C 530;870F8O OG55: 2 :064>< @O4C. Feng Shui Feng Shui  MB> standard-cell ?;59A5@. 3> @07@01>B:>9 70=8<0NBAO 2 8=35<B>=A:>< C=825@A8B5B5 !(. @8B5@85< O2;O5BAO <8=8<870F8O >1I59 4;8=K ?@>2>4>2. ! 4><0H=59 AB@0=8FK  HYPERLINK "http://vlsicad.cs.binghamton.edu/pubs.html" http://vlsicad.cs.binghamton.edu/pubs.html 1K;0 27OB0 25@A8O Feng Shui2.6, 2K?CI5==0O 2 8N=5 2004. !CI5AB2CNB B0:65 1>;55 =>2K5 25@A88: Feng Shui5.0 8 Feng Shui5.1. > >=8 =5 O2;ONBAO A2>1>4=> @0A?@>AB@0=O5<K<8. @8=F8?K @01>BK A=>2>9 Feng Shui O2;O5BAO <5B>4 @5:C@A82=>9 18A5:F88. !>3;0A=> MB><C <5B>4C, 8B5@0F8>==> >ACI5AB2;O5BAO 45;5=85 <=>65AB20 ;>38G5A:8E M;5<5=B>2 ?@8 ?><>I8 181;8>B5:8 hMetis, ?>72>;ONI59 @5H0BL 7040GC > @07185=88 3@0D0; B0:65 ?@>8AE>48B 45;5=85 >1;0AB8 @07<5I5=8O =0 ?>4@538>=K. 064>5 ?>4<=>65AB2> M;5<5=B>2 @07<5I05BAO 2 =5:>B>@>< ?>4@538>=5. >340 <K @0718205< @538>=, 2>7=8:05B A8BC0F8O, GB> M;5<5=BK 87 >4=>3> @538>=0 <>3CB 1KBL A>548=5=K A M;5<5=B0<8 87 4@C38E @538>=>2. G5284=>, GB> MB8  2=5H=85 A>548=5=8O B0:65 =5>1E>48<> CG8BK20BL, 8=0G5 =52>7<>6=> ?>;CG8BL >?B8<0;L=>3> @5H5=8O. ;O MB8E F5;59 2 Feng Shui 8A?>;L7C5BAO CA>25@H5=AB2>20==K9 <5B>4  Terminal propagation . 45O A>AB>8B 2 B><, GB> A>7405BAO D8:B82=K9 M;5<5=B, =0<5@5==> CBO65;5==K9, 4;O B>3>, GB>1K >1@0B8BL 2=8<0=85 =0 MB> A>548=5=85. A5 A2O78 M;5<5=B>2 87 ?>4@538>=0 A  2=5H=8< M;5<5=B>< 70<5=ONBAO =0 A2O78 A D8:B82=K< M;5<5=B><. > A;54C5B >B<5B8BL, GB> ?@8 8A?>;L7>20=88 40==>3> <5B>40 @07185=85 @538>=>2 AB0=>28BAO 2708<>7028A8<K<. ".5. >B B>3>, :0: <K @0A?>;>68< M;5<5=BK 2 >4=>< 87 A>A54=8E @538>=>2, 1C45B 7028A5BL @5H5=85 8 4;O 2B>@>3>. 0: ?@028;>, ?@8E>48BAO ?@>872>48BL =5A:>;L:> F8:;>2 ?5@5AB0=>2>:, 4;O B>3>, GB> 1K ?>;CG8BL E>@>H55 @5H5=85. ;O C<5=LH5=8O G8A;0 F8:;>2 ?5@54 Terminal propagation <>65B 8A?>;L7>20BLAO <5B>4 8B5@0F8>==>3> 2KG5@:820=8O (iterative deletion). 5B>4 8B5@0F8>==>3> 2KG5@:820=8O @8 @07185=88 @538>=0 53> M;5<5=BK @07<5I0NB 2 :064>< ?>4@538>=5 (B.5. 4C1;8@CNB). -;5<5=BK  A:;04K20NBAO 2 F5=B@ @538>=0. "0:8< >1@07><, :064><C M;5<5=BC <>6=> ?>AB028BL 2 A>>B25BAB285 :>>@48=0BK F5=B@0 B>3> @538>=0, 2 :>B>@>< >= @0A?>;0305BAO. 0B5< 4;O :064>3> M;5<5=B0 2KG8A;O5BAO G8A;5==0O E0@0:B5@8AB8:0: AC<<0 4;8= @0AAB>O=89 >B =53> 4> 4@C38E M;5<5=B>2, A :>B>@K<8 >= 4>;65= 1KBL A>548=5=, ?@8G5< CG8BK205BAO @0AAB>O=85 4> :064>3> 87 4C1;8:0B>2 2> 2A5E ?>4@538>=0E (MB> =5>1E>48<>, GB>1K >1J5:B82=> >F5=8BL, M;5<5=B 87 :0:>3> ?>4@538>=0 >AB028BL). 0B5< =0G8=05BAO ?@>F5AA C40;5=8O ;8H=8E M;5<5=B>2: 4;O @538>=0 A 1>;LH8< G8A;>< M;5<5=B>2 C40;O5BAO M;5<5=B A =082KAH59 E0@0:B5@8AB8:>9 (:>=5G=>, 5A;8 4;O =53> 8<55BAO 4C1;8:0B); ?@8 C40;5=88 ;8H=53> M;5<5=B0 E0@0:B5@8AB8:8 4;O >AB0;L=KE =5>1E>48<> ?5@5AG8B0BL. !;54C5B 70<5B8BL, GB> 45;5=85 <>6=> >ACI5AB2;OBL =0 1>;LH5, G5< 2* <G8A;> ?>4@538>=>2> @07. @8<5@: 0AA<>B@8< 2 M;5<5=B0: A1 8 A2. !G8B05< GB> A1 A>548=O5BAO A A2, 0 A2 - A B5@<8=0;L=K< (D8:B82=K<) M;5<5=B><. CABL =0 =5:>B>@>< MB0?5 ?>;CG8;0AL A;54CNI0O A8BC0F8O: A1 =0E>48BAO 2 @538>=5 n1, 0 A2 2 n2. 0;55 ?@>872>48BAO >G5@54=>9 H03 18A5:F88. !>3;0A=> <5B>4C 8B5@0F8>==>3> 2KG5@:820=8O, A1 @0A?>;0305BAO (4C1;8@C5BAO) 2 :064>< 87 ?>4@538=>2 n1; 0=0;>38G=> 8 4;O A2.  =0G0;L=>5 @0A?>;>65=85 M;5<5=B>2C40;5=85 >4=>3> 87 4C1;8:0B>2 A2C40;5=85 >4=>3> 87 4C1;8:0B>2 A1 @8A.6 5B>4 8B5@0F8>==>3> 2KGQ@:820=8O 5B>4 8B5@0F8>==>3> 2KG5@:820=8O A?>A>15= ?@54>AB028BL 1>;55 >?B8<0;L=>5 =0G0;L=>5 4;O Terminal propagation @0A?>;>65=85 M;5<5=B>2 ?> >1;0AB8, GB> 2 A2>N >G5@54L 4>;6=> C<5=LH8BL G8A;> F8:;>2, >ACI5AB2;O5<>5 2 Terminal propagation. 07185=85 >1;0AB8 07185=85 >1;0AB59 =0 ?>4@538>=K >ACI5AB2;O5BAO :0: 2 3>@87>=B0;L=><, B0: 8 2 25@B8:0;L=>< =0?@02;5=88. @8 @07185=88 7=0G8B5;L=>5 2=8<0=85 C45;O5BAO B><C, GB>1K <8=8<878@>20BL G8A;> ?5@5A5G5=89 3@0=8F @07185=8O A @51@0<8 38?5@3@0D0 (G5< <5=LH5 B0:8E ?5@5A5G5=89, B5< 1>;55 =57028A8<K ?>4>1;0AB8 8 ?@>I5 40;L=59H0O @01>B0). ACI5AB2;OO 18A5:F8N, <>6=> 2K18@0BL :0: 1C45B @071820BLAO @538>=: 25@B8:0;L=> 8;8 3>@87>=B0;L=>.  Feng Shui 4;O :064>3> @538>=0 @0AAG8BK205BAO :>MDD8F85=B ?@>?>@F8>=0;L=>AB8 @07<5@>2 @538>=0. 0 53> >A=>25 >?@545;O5BAO =0?@02;5=85 @07185=8O: 5A;8 @538>=, ?@54=07=0G5==K9 4;O @07185=8O, A>45@68B 1>;55 G5< 1 ?>;>AC M;5<5=B>2, B> <>65B 1KBL 2K1@0=> 3>@87>=B0;L=>5 @07185=85, 2 A;CG05, :>340 :>MDD8F85=B ?@>?>@F8>=0;L=>AB8 (2KA>B0 : 4;8=5) 1>;LH5 =5:>3> ?>@>3>2>3> 7=0G5=8O, 7040==>3> ?>;L7>20B5;5<. @>F5AA 18A5:F88 1C45B ?@>4>;60BLAO 4> B5E ?>@, ?>:0 2 :064>< @538>=5 G8A;> M;5<5=B>2 =5 1C45B <5=LH5 CAB0=>2;5==>3> (>1KG=> MB> 6 M;5<5=B>2). @8G5< 2 :064>< @538>=5 4>;6=0 1KBL B>;L:> >4=0 ?>;>A0. 0B5< :064><C M;5<5=BC CAB0=02;8205BAO 53> ?>;>65=85 2 AB@>G:5; 8E C?0:>2K20NB 2<5AB5 157 ?@>AB@0=AB20 8 ?5@5A5G5=89 (5A;8 ?>4>1=>5 2>7<>6=>). >4>1=>5 @07<5I5=85 CAB0=02;8205BAO A ?><>ILN AB0=40@B=>3> <5B>40 25B259 8 3@0=8F (branch-and-bound reordering). 0AA<>B@8< @07185=85 @538>=0 =0 ?@8<5@5: @8A.7a 07185=85 @538>=0, 8A?>;L7CNI55 <5B>4 @5:C@A82=>9 18A5:F88  @8A.7b 07;8G=K5 20@80=BK @07185=8O >1;0AB59, 8A?>;L7CNI53> :>MDD8F85=B ?@>?>@F8>=0;L=>AB8. 0 @8A.7b ?>:070=K @07;8G=K5 20@80=BK @07185=8O >1;0AB8. &8D@0< A>>B25BAB2C5B ?>@O4>: A;54>20=8O, 2 :>B>@>< ?@>E>48;> @07185=85. !;54C5B >B<5B8BL, GB> 2 ?>A;54=8E 25@A8OE FengShui 2 7028A8<>AB8 >B :>MDD8F85=B0 ?@>?>@F8>=0;L=>AB8 @538>= <>65B 45;8BLAO =0 =5@02=K5 ?>4>1;0AB8. 865 ?@54AB02;5=0 >1I0O AE5<0 @01>BK Feng Shui:  SHAPE \* MERGEFORMAT  FastPlace FastPlace " MB> standard-cell ?;59A5@.  :0G5AB25 :@8B5@8O >?B8<0;L=>AB8 A>740205<>3> 8< @07<5I5=8O @0AA<0B@8205BAO 4;8=0 ?@>2>4>2. FastPlace O2;O5BAO 4>AB0B>G=> 1KAB@K< ?;59A5@><.  AB0BL5 [29] ?@82>4OBAO A;54CNI85 F8D@K: >= 2 13 8 97,4 @070 1KAB@55, G5< Capo 8.8 8 Dragon 2.2.3 A>>B25BAB25==>, 0 4;8=0 ?@>2>4>2 @07<5I5=8O, A>740==>3> FastPlace, B>;L:> =0 1% 8 1,6% EC65. 0<8 1K;0 =0945=0 25@A8O FastPlace1.0 2004 3>40, >4=0:> A:>@> ?>O28BAO 25@A8O FastPlace2.0, => C65 2 :0G5AB25 ?;59A5@0 4;O mix-mode @07<5I5=8O, B.5. 4;O @07<5I5=8O 4>2>;L=> 1>;LH>3> G8A;0 :@C?=KE 1;>:>2 8 <8;;8>=>2 =51>;LH8E 1;>:>2 AB0=40@B=KE @07<5@>2. >:0 =0 4><0H=59 AB@0=8G:5 4>ABC?=0 B>;L:> 25@A8O 2004 3>40. @8=F8?K @01>BK FastPlace  1KAB@K9 8 4>AB0B>G=> MDD5:B82=K9 ?;59A5@.  40==>< @0745;5 @0AA<0B@8205BAO 0;3>@8B<, =0 :>B>@>< >A=>20=K ?@8=F8?K 53> @01>BK. 7;>65=85 ?@>872>48BAO =0 >A=>25 AB0BL8 [29]. ;3>@8B< >A=>20= =0 0=0;8B8G5A:>< ?>4E>45.  =Q< <>6=> 2K45;8BL 3 AB0488. 0 ?5@2>9 AB0488 ?@>8AE>48B <8=8<870F8O 4;8=K ?@>2>4>2 0=0;8B8G5A:8< <5B>4>< 8 3@C1>5 @07<5I5=85 M;5<5=B>2 ?> >1;0AB8. @8 MB>< 4>?CA:0NBAO ?5@5:@KB8O M;5<5=B>2, 4;8=0 ?@>2>4>2 >F5=8205BAO 8AE>4O 87 A5B52>9 <>45;8. !5B520O <>45;L  MB> 3@0D, ?>:07K20NI89, :0:85 87 M;5<5=B>2 A>548=5=K 8 :0:>20 4;8=0 MB8E A>548=5=89. 0 MB>< MB0?5 ?>2B>@ONBAO 2 H030: 3;>10;L=0O >?B8<870F8O (Global Optimization) 8 ?5@5<5I5=85 M;5<5=B>2 (Cell Shifting). >4@>1=55 >=8 1C4CB @0AA<>B@5=K =865. &5;LN 2B>@>9 AB0488 O2;O5BAO <8=8<870F8O 4;8=K ?@>2>4>2 8 C=8GB>65=85 ?5@5:@KB89 <564C M;5<5=B0<8. -B> 4>AB8305BAO G5@54>20=85< <5B>4>2 ;>:0;L=>3> 8B5@0B82=>3> CA>25@H5=AB2>20=8O (Iterative local refinement), 3;>10;L=>9 >?B8<870F88 8 ?5@5<5I5=8O M;5<5=B>2. 0 40==>< MB0?5 4;8=0 ?@>2>4>2 >F5=8205BAO half-perimeter <5B>4><. &5;LN B@5BL59 AB0488 O2;O5BAO >:>=G0B5;L=>5 @0A?@545;5=85 M;5<5=B>2 ?> @O40< 8 <8=8<870F8O 4;8=K ?@>2>4>2 A ?><>ILN 604=KE M2@8AB8G5A:8E <5B>4>2. -B8 AB0488 <>3CB 1KBL ?@54AB02;5=K 2 2845: I.@C1>5 3;>10;L=>5 @07<5I5=85 (Coarse Global Placement (CGP)) Repeat ;>10;L=0O >?B8<870F8O 5@5<5I5=85 M;5<5=B>2 8 4>102;5=85 : M;5<5=B0< @0AB0A:820NI8E A8; (spreading forces) Until =5 ?>;CG5=> @07<5I5=85, >?B8<0;L=>5 4;O 40==>9 AB0488 II. #;CGH5=85 4;8=K ?@>2>4>2 ?@8 3;>10;L=>< @07<5I5=88 (Wirelength Improved Global Placement (WIGP)) Repeat ;>10;L=0O >?B8<870F8O >:0;L=>5 8B5@0B82=>5 CA>25@H5=AB2>20=85 5@5<5I5=85 M;5<5=B>2 8 4>102;5=85 : M;5<5=B0< @0AB0A:820NI8E A8; Until =5 1C4CB C=8GB>65=K ?5@5:@KB8O 8 @07<5I5=85 =5 1C45B >?B8<0;L=K< III.5B0;L=>5 @07<5I5=85 (Detailed placement) Repeat 8=8<870F8O 4;8=K ?@>2>4>2 604=K<8 M2@8AB8G5A:8<8 <5B>40<8 530;870F8O B5:CI53> @07<5I5=8O Until 4;8=0 ?@>2>4>2 70<5B=> C<5=LH05BAO 0AA<>B@8< >B45;L=K5 AB0488 40==>3> 0;3>@8B<0. ;>10;L=0O >?B8<870F8O. CABL (Ei,Ci)  :>>@48=0BK F5=B@0 i->3> M;5<5=B0. >AB028< 2 A>>B25BAB285 A>548=5=8N i->3> M;5<5=B0 A j-K< 25A wij . ">340 AB>8<>ABL A>548=5=8O 1C45B @02=0: wij [(xi-xj)+(yi-yj)] A;8 i-K9 M;5<5=B A>548=O5BAO A f-K< D8:A8@>20==K< (=5?>4286=K<) M;5<5=B><, B> AB>8<>ABL @02=0 wij [(xi-xf)+(yi-yf)]. ">340 AC<<0@=0O AB>8<>ABL, :>B>@0O @0AA<0B@8205BAO :0: F5;520O DC=:F8O, @02=0: $(E,C)=E'Qx+d'xx+y'Qy+d'yy+const, 345 Q 5ABL <0B@8F0 A<56=>AB8 4;O A5B52>9 <>45;8, 2 Q 70?8AK20NBAO 25A0 A>548=5=89. ">340 @5H5=85 7040G8 A2>48BAO : @5H5=8N A8AB5<K ;8=59=KE C@02=5=89 Qx + dx = 0 !:>@>ABL 5Q @5H5=8O 7028A8B >B G8A;0 =5=C;52KE M;5<5=B>2 2 <0B@8F5 Q. -B0 ?@>1;5<0 @5H05BAO A ?><>ILN A?5F8D8G5A:>9 A5B52>9 <>45;8. !5B520O <>45;L.  40==>9 A5B52>9 <>45;8 8A?>;L7CNBAO 2 <>45;8: clique model 8 star model. =8 ?@54AB02;5=K =0 @8AC=:5:  @8A.8 >45;8 A>548=5=89. ;O A>548=5=89 2B>@>9 8 B@5BL59 AB5?5=8 8A?>;L7C5BAO <>45;L ?>;=>3> 3@0D0 (clique model), 0 4;O A>548=5=89 G5B25@B>9 AB5?5=8 8 2KH5  <>45;L 72574K (star model). 50;L=>5 A>548=5=85 70<5=O5BAO A5B52>9 <>45;LN. @8<5=5=85 MB>3> <5B>40 ?>72>;O5B <8=8<878@>20BL G8A;> =5=C;52KE M;5<5=B>2 2 <0B@8F5 A2O7=>AB8 Q, 0 7=0G8B, CA:>@8BL 3;>10;L=CN >?B8<870F8N. 4=0:> 8A?>;L7>20=85 MB>9 <>45;8 2<5AB> half-perimeter <5B>40 45;05B @07<5I5=85, ?>;CG5==>5 2 @57C;LB0B5 3;>10;L=>9 >?B8<870F88, 3@C1K<. "> 5ABL >F5=:0 AC<<0@=>9 4;8=K ?@>2>4>2 ?>;CG05BAO =5B>G=>9. 5@5:@KB8O, 2>7=8:0NI85 2 @57C;LB0B5 3;>10;L=>9 >?B8<870F88, CAB@0=ONBAO A ?><>ILN ?5@5<5I5=8O M;5<5=B>2. 5@5<5I5=85 M;5<5=B>2. AO >1;0ABL @07<5I5=8O @0718205BAO =0 >48=0:>2K5 ?> @07<5@C 18=K.  A@54=5< 2 :064>< 87 =8E <>6=> @07<5AB8BL ?> 4 M;5<5=B0.  @8A.9 07185=85 =0 18=K. ;O :064>3> 18=0 ?>4AG8BK205BAO ?;>I04L (Ui) , 70=OB0O M;5<5=B0<8. 5@5<5I5=85 ?@>8AE>48B 2 2 H030. !=0G0;0 @07<5@K 18=>2 ?5@5AG8BK20NBAO 2 A>>B25BAB288 A 25;8G8=>9 Ui 40==>3> 8 A>A54=8E 18=>2: NBi =(OBi"1(Ui+1 + ) + OBi+1(Ui + ))/(Ui + Ui+1 + 2), 345 OBi  :>>@48=0B0 ?> E 3@0=8FK i->3> 18=0 4> ?5@5AGQB0, NBi  :>>@48=0B0 ?> E 3@0=8FK i->3> 18=0 ?>A;5 ?5@5AGQB0,   :>=AB0=B0, 2 40==>9 @50;870F88 @02=0O 1.5. =0 =5>1E>48<0, GB>1K 3@0=8FK 18=>2 =5 ?5@5:@K20;8AL <564C A>1>9. @54?>;>68<, GB> =0 8 Ui+1=0, B>340 NBi = OBi+1 8 NBi+1=OBi. >MB><C 15@QBAO `"0. 0B5< 2A5 M;5<5=BK 2 AB0@>< 18=5 ;8=59=> >B>1@060NBAO =0 =>2K9 18=:  , 345 xj  :>>@48=0B0 M;5<5=B0 4> ?5@5<5I5=8O, x j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dx 8 dy. >:0;L=>5 8B5@0B82=>5 CA>25@H5=AB2>20=85. 0 40==>< H035 @0AA<0B@820NBAO 18=K ?@8<5@=> 2 5 @07 1>;LH85, G5< =0 ?@54K4CI5< MB0?5. -B> 45;05BAO 4;O B>3>, GB>1K <>6=> 1K;> ?5@5<5I0BL M;5<5=B =0 1>;LH85 @0AAB>O=8O 8 <8=8<878@>20BL 4;8=C ?@>2>4>2, A>548=ONI8E 40;Q:85 M;5<5=BK. ;O :064>3> M;5<5=B0 2KG8A;ONBAO 25;8G8=K, E0@0:B5@87CNI85 2>7<>6=>ABL 53> ?5@5<5I5=8O 2 >4=>< 87 G5BK@QE =0?@02;5=89 =0 @0AAB>O=85, @02=>5 H8@8=5 18=0. >4AG8B0==0O 25;8G8=0 5ABL A>:@0I5=85 4;8=K ?@>2>4>2 ?@8 ?5@5<5I5=88 M;5<5=B0 2 40==>< =0?@02;5=88. => AG8B05BAO 8AE>4O 87 4;8=K ?>;C?5@8<5B@0 ?@O<>C3>;L=8:0, >3@0=8G820NI53> 2A5 A>548=5=8O, 2 :>B>@KE CG0AB2C5B 40==K9 M;5<5=B. "0:65 ?>4AG8BK205BAO @07=>ABL Ui F5;52>3> 8 8AE>4=>3> 18=0. A;8 25;8G8=K >B@8F0B5;L=K5, B> M;5<5=B >AB0QBAO =0 <5AB5.  ?@>B82=>< A;CG05 >= ?5@5<5I05BAO 2 18=, =081>;55 2K3>4=K9 A B>G:8 7@5=8O ?>;CG5==KE 25;8G8=. 5B0;L=>5 @07<5I5=85. -;5<5=BK @07<5I0NBAO ?> 87=0G0;L=> >?@545;Q==K< @O40<. 0B5< : @07<5I5=8N ?@8<5=O5BAO M2@8AB8G5A:89 <5B>4, ?>72>;ONI89 C<5=LH8BL 4;8=C ?@>2>4>2. @8 MB>< 1>;LH>5 2=8<0=85 C45;O5BAO >BACBAB28N ?5@5:@KB89 8 A>>B25BAB28N @O40<. A=>2=K<8 A>AB02;ONI8<8 <5B>40 O2;ONBAO: Global Swap " ?0@0 M;5<5=B>2 <5=O5BAO <5AB0<8, >AB0;L=K5 ?@8 MB>< =5 ?5@5<5I0NBAO; Vertical Swap " M;5<5=B <5=O5BAO <5AB0<8 A A>A54=8< M;5<5=B><, @0A?>;>65==K< 2KH5 8;8 =865 40==>3>; Local Re-ordering " ?5@5C?>@O4>G820=85 @0A?>;>65==KE 2 >4=>9 AB@>:5 =5A:>;L:8E M;5<5=B>2; Single-Segment Clustering " >?B8<0;L=>5 @07<5I5=85 M;5<5=B>2 2 >4=>< 18=5. 564C A>1>9 >=8 A2O70=K A;54CNI8< >1@07><:  SHAPE \* MERGEFORMAT  mPL 0==K9 >17>@ @0AA<0B@8205B 0;3>@8B<, ?@54AB02;5==K9, 2 ?;59A5@5 mPL. ;3>@8B<, 8A?>;L7C5<K9 2 mPL5, 4>AB8305B ;CGH53> 7=0G5=8O ?> 4;8=5 ?@>2>4>2, A>E@0=OO ?@85<;5<>5 2@5<O @01>BK.  mPL5 @07@01>B0= =>2K9 0;3>@8B< @07<5I5=8O, 2 @57C;LB0B5 G53> 1K;8 4>AB83=CBK C4828B5;L=K5 @57C;LB0BK.  A@02=5=88 A Capo 9.0, mPL5 35=5@8@C5B @07<5I5=85 A 4;8=>9 ?@>2>4>2, =0 8% <5=LH59, 0 @01>B05B 242>5 1KAB@55.  ?> A@02=5=8N A Dragon 3.01 ?>;CG5==0O 4;8=0 ?@>2>4>2 =0 3% <5=LH5, 0 2@5<O @01>BK >B;8G05BAO 2 12 @07, B0:65 2 ?>;L7C mPL. !@02=820O 65 mPL5 A FengShui 5.0 8 FastPlace, ?>;CG5==K5 4;8=K ?@>2>4>2 1C4CB =0 5% 8 =0 8% ;CGH5, 0 2@5<O @01>BK  242>5 1KAB@55 8 2 6 @07 <54;5==55 A>>B25BAB25==>. !:0G0BL mPL <>6=> ?> A;54CNI59 AAK;:5:  HYPERLINK "http://ballade.cs.ucla.edu/cpmo/" http://ballade.cs.ucla.edu/cpmo/ 0 A09B5 4>ABC?=K ?5@2K5 B@8 87 ?OB8 ACI5AB2CNI8E 25@A89 ?;59A5@0. "@5BLO 25@A8O ?>O28;0AL 2 >:BO1@5 2003 3. @8=F8?K @01>BK >AB0=>2:0 7040G8 @8=F8? @01>BK mPL @0AA<0B@8205BAO =0 >A=>25 <0B5@80;0 AB0BL8  HYPERLINK \l "_!?8A>:_;8B5@0BC@K" [33]. CABL  EMBED Equation.3  38?5@3@0D, 345  EMBED Equation.3  - <=>65AB2> 25@H8= (M;5<5=B>2), 0  EMBED Equation.3  - <=>65AB2> @515@ (A>548=5=89). @8G5< ?>4<=>65AB2>  EMBED Equation.3  - <=>65AB2> B5@<8=0;>2, B> 5ABL M;5<5=B>2, ?>;>65=85 :>B>@KE D8:A8@>20=>, 0 :064>5  EMBED Equation.3  - ?>4<=>65AB2> <=>65AB20 V. 0;55, ?CABL  EMBED Equation.3  - :>>@48=0BK 35><5B@8G5A:>3> F5=B@0 M;5<5=B0  EMBED Equation.3 . ">340 4;8=0 @51@0 e, ?@54AB02;O5<0O ?> A;54CNI59 D>@<C;5:  EMBED Equation.3  (1), O2;O5BAO ?>;C?5@8<5B@>< =08<5=LH53> ?@O<>C3>;L=8:0, 2:;NG0NI53> 2A5 25@H8=K 87 e. 1I0O 4;8=0 ?@>2>4>2 1C45B @02=0 AC<<5 2A5E 4;8= 4;O :064>3> @51@0 87 <=>65AB20 E. "0:8< >1@07><, B@51C5BAO <8=8<878@>20BL 25;8G8=C  EMBED Equation.3 . @8<5< B5?5@L, GB> 2KA>B0 2A5E M;5<5=B>2 >48=0:>20 8 G8A;> AB@>: 2 @07<5I5=88 D8:A8@>20=>. @><5 B>3>, >B<5B8<, GB> 4>;6=K >BACBAB2>20BL ?5@5:@KB8O <564C M;5<5=B0<8. (!;54C5B 4>1028BL, GB>, :0: >B<5G05BAO 2 AB0BL5 [33], 0;3>@8B< @01>BK mPL <>65B 1KBL ;53:> <>48D8F8@>20= 4;O @01>BK A M;5<5=B0<8 @07=>9 2KA>BK.) 1KG=> 7040G0 > @07<5I5=88 A2>48BAO : 42C< ?>47040G0<: 3;>10;L=>5 @07<5I5=85 8 45B0;L=>5 @07<5I5=85. @8 3;>10;L=>< @07<5I5=88 M;5<5=BK @0A?@545;ONBAO ?> >1;0AB8 @07<5I5=8O, ?@8G5< =0 MB>< MB0?5 4>?CA:0NBAO ?5@5:@KB8O. >A;5 MB>3> 35=5@8@C5BAO 45B0;L=>5 @07<5I5=85, =5 4>?CA:0NI55 ?5@5:@KB89. @8G5<, >=> B0:65 <>65B C<5=LH8BL 4;8=C ?@>2>4>2. !B0BLO [33] 2 >A=>2=>< >?8AK205B 3;>10;L=>5 @07<5I5=85. $>@<C;8@>2:0 >3@0=8G5=89 <8=8<870F88 (Constraint minimization formulation)  MB>< @0745;5 1C4CB @0AA<>B@5=K 3;04:85 0??@>:A8<0F88 4;8=K ?@>2>4>2 (1) 8 3;04:85 0??@>:A8<0F88 >3@0=8G5=89, A2O70==KE A =54>?CAB8<>ABLN ?5@5:@KB89. ;04:0O 0??@>:A8<0F8O 4;8=K ?@>2>4>2 "0: :0: D>@<C;0 (1) =5 48DD5@5=F8@C5<0, 0 >3@0=8G5=8O =5 O2;ONBAO 2K?C:;K<8, B> ?@>1;5<0 <8=8<870F88 4>AB0B>G=> A;>6=0. "0:8< >1@07><, =5>1E>48<> 8A?>;L7>20BL 3;04:85 0??@>:A8<0F88 4;O (1). '0AB> 8A?>;L7C5BAO :204@0B8G=0O 0??@>:A8<0F8O, 7040205<0O, 2 >1I5< 2845, A;54CNI8< >1@07><:  EMBED Equation.3 , (2) 345 W  <0B@8F0 :>MDD8F85=B>2. @58<CI5AB2> :204@0B8G=>9 0??@>:A8<0F88 A>AB>8B 2 B><, GB> ?@8 55 ?@8<5=5=88 4;O =0E>645=8O <8=8<C<0 2 7040G5 157 >3@0=8G5=89 4>AB0B>G=> @5H8BL ?>;>68B5;L=>->?@545;5==CN A8AB5<C ;8=59=KE C@02=5=89. 4=0:>, 70 MB> ?@8E>48BAO ?;0B8BL 4;8==K<8 A>548=5=8O<8, 8, A>>B25BAB25==>, =5C40G=>9 >F5=:>9 4;8=K ?@>2>4>2. K @0AA<>B@8< ;CGHCN 0??@>:A8<0F8N ?>;C?5@8<5B@0:  EMBED Equation.3  (3), 345  EMBED Equation.3  <0;>. AB5AB25==>, 2 @0<:0E <0H8==>9 0@8D<5B8:8  EMBED Equation.3  =5 <>65B 1KBL 2K1@0=> A:>;L C3>4=> <0;K<, >4=0:>, 8AE>4O 87 [33], <>6=> A:070BL, GB>, 5A;8  EMBED Equation.3 , B> 4>AB0B>G=> 27OBL  EMBED Equation.3 , @02=>5 0,01. @><5 B>3>, <>6=> 8A?>;L7>20BL 4@C3CN 0??@>:A8<0F8N (1), 8A?>;L7CNICN, B.=.  EMBED Equation.3 -=>@<8@>2:C:  EMBED Equation.3 . (4)  M:A?5@8<5=B0E, ?@>2545==KE 02B>@0<8 [33], 1K;> 2K1@0=> p=32. ;04:85 0??@>:A8<0F88 >3@0=8G5=89 B<5B8<, GB> >3@0=8G5=8O, =5 4>?CA:0NI85 ?0@=>3> ?5@5:@KB8O M;5<5=B>2 2 >1;0AB8, 2 >1I5< A;CG05, =5 O2;ONBAO 2K?C:;K<8 8, 8AE>4O 87 :C@A0 <5B>4>2 >?B8<870F88, 8, 2 G0AB=>AB8, B5>@5<K C=0-"0:5@0 4;O 3;04:>9 2K?C:;>9 7040G8, C4>2;5B2>@8BL MB8< >3@0=8G5=8O< 4>AB0B>G=> B@C4=>. 0AA<>B@8<, :0: <>6=> @5H8BL MBC ?@>1;5<C. AE>4O 87 7040G8 @07<5I5=8O =0 >1;0AB8 R, @0745;8< MBC >1;0ABL =0 m*n >48=0:>2KE ?>4>1;0AB59 (bins)  EMBED Equation.3 , ?@8G5<  EMBED Equation.3 . CABL B0:65  EMBED Equation.3  8  EMBED Equation.3  - H8@8=0 8 2KA>B0 ?>4>1;0AB59, A>>B25BAB25==>. ?@545;8< 25;8G8=C  EMBED Equation.3 , :0: A@54=NN ?;>B=>ABL M;5<5=B>2 2 OG59:5  EMBED Equation.3 :  EMBED Equation.3 , (5) 345  EMBED Equation.3  - G0ABL ?;>I048 k->3> M;5<5=B0, ;560I0O 2 ?>4>1;0AB8  EMBED Equation.3 . ".5. A@54=OO ?;>B=>ABL 5ABL >B=>H5=85 AC<<K ?;>I0459 B5E G0AB59 M;5<5=B>2, :>B>@K5 ;560B 2=CB@8 OG59:8 (bin) : ?;>I048 A0<>9 OG59:8. ">340 ?>AB0=>2:0 >?B8<870F8>==>9 7040G8 A >3@0=8G5=8O<8 1C45B 2K3;O45BL B0::  EMBED Equation.3  (6), 345 W(x,y)  4;8=0 ?@>2>4>2,  EMBED Equation.3  70405BAO A ?><>ILN D>@<C;K (5), 0 K  >1I0O ?;>I04L 2A5E M;5<5=B>2, 45;5==0O =0 ?;>I04L >1;0AB8 @07<5I5=8O (02B>@0<8 [33] 1K;> ?@8=OB>, GB>  EMBED Equation.3 ). A=>2=0O ?@>1;5<0  =09B8 @07<5I5=85, <8=8<878@CNI55 4;8=C ?@>2>4>2 W(x,y). 4=0:> ?@8 =53;04:>9 DC=:F88 ?;>B=>AB8 (5) 4>AB0B>G=> B@C4=> @5H8BL MBC ?@>1;5<C, ?>MB><C 2 [33] ?@54;0305BAO 8A?>;L7>20BL >1@0B=>5 ?@5>1@07>20=85 0?;0A0, GB>1K A3;048BL MBC DC=:F8N. !3;06820NI89 >?5@0B>@  EMBED Equation.3  >?@545;O5BAO A ?><>ILN @5H5=8O C@02=5=8O 5;L<3>;LF0:  EMBED Equation.3  (7), 345  EMBED Equation.3 ,  EMBED Equation.3  - 2=5H=OO 548=8G=0O =>@<0;L,  EMBED Equation.3  - 3@0=8F0 >1;0AB8 R, d(x,y)  =5?@5@K2=0O DC=:F8O 3CAB>BK,  EMBED Equation.3  - >?5@0B>@ 0?;0A0. 1@0B=K9 >?5@0B>@ 2?>;=5 >?@545;5=, B.:. C@02=5=85 (7) 8<55B 548=AB25==>5 @5H5=85 ?@8  EMBED Equation.3 . ".:. @5H5=85 C@02=5=8O (7) 8<55B 425 ?@>872>4=K5, B>  EMBED Equation.3  - A3;065==0O DC=:F8O 3CAB>BK.  [33] ?@54;0305BAO 8A?>;L7>20BL, B.=. :>=5G=K9 48DD5@5=F80;L=K9 <5B>4, ?@8<5=ONI89 A5B:C ?>4>1;0AB59 (bin grids), >?8A0==CN 2KH5, 4;O @5H5=8O 7040G8 (7). CABL  EMBED Equation.3  - 7=0G5=85  EMBED Equation.3  2 F5=B@5  EMBED Equation.3 . ">340 <>65< 7040BL AE5<C 0??@>:A8<0F88:  EMBED Equation.3  (9), 345  EMBED Equation.3  (10) 8  EMBED Equation.3  - A@54=OO 3CAB>B0 2  EMBED Equation.3 . CABL  EMBED Equation.3  - <0B@8F0, A>>B25BAB2CNI0O >?8A0==>9 ;8=59=>9 A8AB5<5. ">340 <0B@8F0  EMBED Equation.3  <>65B 1KBL 2KG8A;5=0 A ?><>ILN A;54CNI59 ;8=59=>9 A8AB5<K:  EMBED Equation.3  (11), 345  EMBED Equation.3 . "5?5@L <K <>65< ?5@5D>@<C;8@>20BL 7040GC (6):  EMBED Equation.3  (12), 345  EMBED Equation.3  8  EMBED Equation.3  - ?>AB>O==K9 25:B>@, 0 1=(1& 1). "5?5@L <K <>65< @0AA<>B@5BL @5H5=85 MB>9 7040G8 8 53> ?@8<5=5=85 : @5H5=8N 8AE>4=>9 7040G8. 5H5=85 7040G8 !CI5AB2C5B <=>3> 0;3>@8B<>2 4;O @5H5=8O (12).  [33] ?@54;0305BAO 8A?>;L7>20BL B.=. 0;3>@8B< Uzawa, ?>72>;ONI89 @5H8BL (12) A ?><>ILN <=>68B5;59 03@0=60:  EMBED Equation.3  (13), 345  EMBED Equation.3  - <=>68B5;8 03@0=60 =0 k->9 8B5@0F88,  EMBED Equation.3  - ?0@0<5B@ 4;O :>=B@>;O A:>@>AB8 AE>48<>AB8, 0  EMBED Equation.3  8  EMBED Equation.3  - :>>@48=0BK ?>;>65=8O M;5<5=B0 =0 k->9 8B5@0F88. @0485=B  EMBED Equation.3  >B=>A8B5;L=> M;5<5=B0  EMBED Equation.3  <>65B 1KBL 0??@>:A8<8@>20= ?> A;54CNI59 AE5<5:  EMBED Equation.3  (14), 5A;8 F5=B@  EMBED Equation.3  ;568B 2=CB@8  EMBED Equation.3 , 8 =C;L, 5A;8 A=0@C68. 0 :064>< H035 8B5@0B82=>9 AE5<K (13) =0< B@51C5BAO @5H8BL =5;8=59=>5 C@02=5=85. 5H5=85 =5;8=59=>3> C@02=5=8O  MB> CAB>9G82>5 @5H5=85 A;54CNI53> >1K:=>25==>3> 48DD5@5=F80;L=>3> C@02=5=8O:  EMBED Equation.3  (15), 345  EMBED Equation.3  8  EMBED Equation.3  C:07K20NB =0 @07<5I5=85 2 <><5=B 2@5<5=8 t. ;O MB>3> <>65B 1KBL @0AA<>B@5=0 AE5<0 3@0485=B=>3> A?CA:0 4;O DC=:F88 03@0=60:  EMBED Equation.3 , (16) B0: :0:  EMBED Equation.3 . (17) >6=> A<>B@5BL =0 MB> :0: =0 <8=8<870F8N ?@>2>4>2 8 >4=>2@5<5==> HB@0D ?> >3@0=8G5=8O<. 5H0BL (15) ?@54;0305BAO <5B>4>< -9;5@0. @82545< 0;3>@8B<, @5H0NI89 7040GC (12). -B>B 0;3>@8B< =0720= GFD (Generalized Force Directed). E>4=K<8 40==K<8 4;O =53> O2;ONBAO G8A;> 2=5H=8E 8B5@0F89 (:>;8G5AB2> 2=5H=8E F8:;>2 2 0;3>@8B<5) 8 :@8B5@89 >AB0=>20 4;O 2=CB@5==8E 8B5@0F89.  EMBED Equation.3  - ?0@0<5B@ 4;O CA:>@5=8O AE>48<>AB8,  EMBED Equation.3  - A:>@>ABL @>AB0  EMBED Equation.3 ,  EMBED Equation.3  - ?@>F5=B OG55: (bins) A =5=C;52>9 3CAB>B>9. N  G8A;> ?5@5<5I05<KE M;5<5=B>2, P  G8A;> B5@<8=0;>2. AE>4=>5 @07<5I5=85 206=> 2 A8;C B>3>, GB>, @5H0O (13), <>6=> ?>;CG8BL ;8HL ;>:0;L=K9 <8=8<C<. =5H=85 8B5@0F88 <>6=> @0AA<0B@820BL, :0: ?@>4>;65=85 <5B>40, 345 @5H5=85 =0 :064>9 8B5@0F88 5ABL 8AE>4=>5 @07<5I5=85 =0 A;54CNI59 8B5@0F88. K 8A?>;L7C5< >48=0:>2K5 OG59:8 @07185=8O (bin grids) 8 G8A;> 8E ?@81;878B5;L=> @02=> G8A;C @07<5I05<KE M;5<5=B>2. B<5B8<, GB> 2 =0H5< A;CG05 >1@0B=>5 ?@5>1@07>20=85 0?;0A0 0??@>:A8<8@C5B DC=:F8N 3CAB>BK 3;>10;L=>. "0: :0: 3;>10;L=>5 @07<5I5=85, ?>;CG5==>5 A ?><>ILN 0;3>@8B<0 GFD, <>65B A>45@60BL ?5@5:@KB8O, B> 2 :>=5G=>< 8B>35 B@51C5BAO ?@8<5=OBL B.=. ;530;870F8N @5H5=8O. A?>;L7C5BAO =5A;>6=K9 604=K9 0;3>@8B< 4;O @07<5I5=8O M;5<5=B>2 2 AB@>:0E 157 ?5@5:@KB89. @82545< B5?5@L A0< 0;3>@8B<: GFD(outer_iters, stop_percent) A;8 8AE>4=>5 @07<5I5=85 =5 40=> <A?>;L7>20BL :204@0B8G=CN 0??@>:A8<0F8N 4;8=K ?@>2>4>2  :0G5AB25 8AE>4=>3> @5H5=8O> >=5F 5A;8 KG8A;8BL nnb = <G8A;> OG55: A =5=C;52>9 3CAB>B>9> P = <>1I55 :>;8G5AB2> B5@<8=0;>2> N = <>1I55 :>;8G5AB2> ?5@5<5I05<KE M;5<5=B>2> inner_iters = N // :>;8G5AB2> 2=CB@5==8E 8B5@0F89  EMBED Equation.3  = 1,5 // M:A?5@8<5=BK ?>:07K20NB, GB> MB> E>@>H89 :><?@><8AA // <564C A:>@>ABLN 8 4;8=>9 ?@>2>4>2 ;O i >B 1 4> outer_iters 2K?>;=8BL  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  ;O j >B 1 4> inner_iters 2K?>;=8BL A;8 nnb =5 C25;8G5=>  EMBED Equation.3   EMBED Equation.3  <@5H8BL (15) <5B>4>< -9;5@0> <2KG8A;8BL nnb> A;8 1>;LH5, G5<  EMBED Equation.3  OG55: A =5=C;52>9 3CAB>B>9 KE>4 87 F8:;0 >=5F F8:;0 <2K7>2 0;3>@8B<0 45B0;L=>3> @07<5I5=8O> >=5F F8:;0. =>3>C@>2=520O AB@C:BC@0 =>385 8AA;54>20=8O ?>:07K20NB, GB> <=>3>C@>2=52K9 0;3>@8B< A?>A>15= @07@5H8BL ?@>1;5<C 3@><>74:8E 7040G > @07<5I5=88. @8G5< >= 8A?>;L7C5BAO =5 B>;L:> 4;O CA:>@5=8O @5H5=8O 7040G8, => 8 4;O ;CGH59 >?B8<870F88.  MB>< @0745;5 1C45B @0AA<>B@5=> 2=54@5=85 0;3>@8B<0 GFD 2 <=>3>C@>2=52CN AB@C:BC@C, 8A?>;L7C5<CN 2 mPL. -B0 AB@C:BC@0 2:;NG05B 2 A51O 3;>10;870F8N (coarsening), @5;0:A0F8N (relaxation), 8=B5@?>;OF8N (interpolation) 8 <=>3>C@>2=52K9 ?>B>: (multilevel flow). 0AA<>B@8< :064CN 87 MB8E :><?>=5=B. ;>10;870F8O (coarsening) &5;L 3;>10;870F88  ?>AB@>8BL 85@0@E8N 4;O <=>3>C@>2=52>9 ?0@0483<K. ?@545;8< @>4AB2> <564C 25@H8=0<8 v 8 w, :0:  EMBED Equation.3 , (18) 345  EMBED Equation.3  - 25A, =07=0G5==K9 @51@C e, area(e)  AC<<0 ?;>I0459 M;5<5=B>2, 2E>4OI8E 2 e,  EMBED Equation.3  - :>;8G5AB2> M;5<5=B>2 2 e. !=0G0;0 2A5 25@H8=K C?>@O4>G820NBAO ?> C1K20=8N AB5?5=59. 0B5< 2A5 25@H8=K ?>A;54>20B5;L=> @0AA<0B@820NBAO 8 4;O :064>9 25@H8=K 2KG8A;O5BAO @>4AB2> A> 2A5<8 25@H8=0<8, A :>B>@K<8 >=0 A>548=5=0 @51@0<8. >A;5 MB>3> :>=AB@C8@C5BAO 3@0D @>4AB20: :0640O 25@H8=0 >1J548=O5BAO A >4=>9 87 A>A54=8E 25@H8=, 4;O :>B>@>9 8<55B <5AB> <0:A8<0;L=>5 @>4AB2>. 0640O 3@C??0 A>548=5==KE 25@H8= =07K205BAO 3@>74LN 8 AB0=>28BAO 25@H8=>9 1>;55 3@C1>3> C@>2=O. 8?5@@51@0 2 3@>748 >?@545;ONBAO >G5284=K< >1@07><: :064>5 38?5@@51@> =0 1>;55 2KA>:>< C@>2=5 AB0=>28BAO 38?5@@51@>< =0 3@C1><. ".5. 5A;8 2 42CE 3@>74OE 1K;8 M;5<5=BK, A>548=5==K5 @51@><, B> ACI5AB2C5B @51@>, A>548=ONI55 MB8 425 3@>748.  38?5@@51@0, A>548=ONI85 3@>74L A0<C A A>1>9, ?@>AB> 83=>@8@CNBAO. "0:8< >1@07><, =0 1>;55 3@C1>< C@>2=5 <K ?>;CG05< <5=LH89 38?5@3@0D. 5;0:A0F8O (relaxation) mPL3 @5H05B =5;8=59=K5 7040G8 =0 A0<>< 3@C1>< C@>2=5. 0 ?>A;54CNI8E C@>2=OE 8A?>;L7C5BAO GOTO->1<5= 8 :204@0B8G=0O @5;0:A0F8O =0 ?>4<=>65AB20E A =57=0G8B5;L=K<8 42865=8O<8 M;5<5=B>2, GB>1K C<5=LH8BL 703@C65==>ABL >1;0AB8. K 70<5=O5< MB8 <5B>4K >?B8<870F88 0;3>@8B<>< GFD, >15A?5G820NI8< 1>;LHCN @5;0:A0F8N, B0: :0: >= ?5@5<5I05B 2A5 M;5<5=BK >4=>2@5<5==>, GB>1K C<5=LH8BL 4;8=C ?@>2>4>2. =B5@?>;OF8O (interpolation) =B5@?>;OF8O 8A?>;L7C5BAO 4;O ?5@52>40 @5H5=8O A >4=>3> C@>2=O =0 4@C3>9. "> 5ABL, 5A;8 C =0A 5ABL @07<5I5=85 =0 3@C1>< C@>2=5, B> <K 8A?>;L7C5< 8=B5@?>;OF8N 4;O ?>;CG5=8O @07<5I5=8O =0 1>;55 45B0;L=>< C@>2=5. @0D>20O <>45;L A2O7=>AB8 8A?>;L7C5BAO 4;O >?@545;5=8O 8=B5@?>;OF88: 25A @51@0  EMBED Equation.3   EMBED Equation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quation.3 . =>3>C@>2=52K9 ?>B>: (multilevel flow) >A;5 ?5@2>3> V-F8:;0 4;O C;CGH5=8O @57C;LB0B0 8A?>;L7C5BAO 4>102>G=K9 V-F8:;. 5@5>?@545;8< @>4AB2> 25@H8= v 8 w:  EMBED Equation.3 , (20) 345 dist(v,w)  2:;84>2> @0AAB>O=85 <564C 25@H8=0<8 v 8 w. "0:8< >1@07><, :;0AB5@K @07<5I5=K 2 ?>78F88 A@54=53> 2725H5==>3> 8E :><?>=5=B. 5;0:A0F8O =0 MB>9 87<5=5==>9 85@0@E88 8A?>;L7C5BAO 4;O 40;L=59H53> C<5=LH5=8O 4;8=K ?@>2>4>2. "5?5@L <>65< @0AA<>B@5BL >:>=G0B5;L=K9 0;3>@8B< (>B<5B8<, GB> CA:>@5=85 @01>BK 0;3>@8B<0 <>65B 4>AB830BLAO 70 AG5B (1) 7040=8O stop_percent=95, (2) C25;8G5=8O  EMBED Equation.3  2 0;3>@8B<5, 2=5 7028A8<>AB8 >B B>3>, C25;8G8;>AL ;8 nnb, (3) C<5=LH5=8O G8A;0 OG55: (bins) 4> ?>;>28=K >B =>@<0;L=>3>, (4) C<5=LH5=85 ?5@542865=89 M;5<5=B>2 ?@8 45B0;L=>< @07<5I5=88): <8A?>;L7>20=85 (18) 4;O 3;>10;870F88 3@0D0 4> B5E ?>@, ?>:0 :>;8G5AB2> M;5<5=B>2 =5 AB0=5B <5=LH5 500> nl = <:>;8G5AB2> C@>2=59> stop_percent = 97 // ?@54?>;0305BAO, GB> C@>25=L nl  =081>;55 45B0;L=K9, B> 5ABL A>>B25BAB2CNI89 // 8AE>4=><C 38?5@3@0DC ;O i >B 1 4> nl-1 2K?>;=8BL distri_percent = min(50 + 50*i/nl; 90): =0 C@>2=5 i 2K7>2 GFD(1, distri_percent) <8=B5@?>;8@>20BL @07<5I5=85 A C@>2=O i =0 C@>25=L i+1> >=5F F8:;0 // 0G0;> 2B>@>3> V-F8:;0 <8A?>;L7>20=85 (20) 4;O 3;>10;870F88 3@0D0 4> B5E ?>@, ?>:0 :>;8G5AB2> M;5<5=B>2 =5 AB0=5B <5=LH5 500> <@07<5I5=85 87 ?5@2>3> V-F8:;0 8=B5@?>;8@C5BAO 4> 3@C1>3> C@>2=O A ?><>ILN coarsening> nl = <:>;8G5AB2> C@>2=59> ;O i >B 1 4> nl-1 2K?>;=8BL distri_percent = min(50 + 50*i/nl; 90): =0 C@>2=5 i 2K7>2 GFD(1, distri_percent) <8=B5@?>;8@>20BL @07<5I5=85 A C@>2=O i =0 C@>25=L i+1> >=5F F8:;0 2K7>2 GFD(1, stop_percent) 2K7>2 45B0;L=>3> @07<5I5=8O      Dragon  Dragon  MB> 1KAB@K9, MDD5:B82=K9 standard-cell ?;59A5@ 4;O @5H5=8O variable-die 8 fixed-die 7040G. Dragon ?@>872>48B <8=8<870F8N 4;8=K ?@>2>4>2, :><18=8@CO <>I=K9 8=AB@C<5=B @07185=8O 38?5@3@0D>2 (hMetis) A 8<8B0F859 >B6830. -B>B 8=AB@C<5=B @07@010BK205BAO 2 !525@>-70?04=>< C=825@A8B5B5. 57C;LB0BK Dragon 0 A@02=8<K A @57C;LB0B0<8 :><<5@G5A:8E ?@>4C:B>2, B0:8E :0: ITools (=5:>340 TimberWolf) 8 Cadence QPlace. > A@02=5=8N A iTools1.4.0, Dragon <>65B 4020BL @07<5I5=85 =5<=>38< ;CGH55 (1,4%), 70B@0G820O ?@8 MB>< 7=0G8B5;L=> <5=LH5 2@5<5=8 (CA:>@5=85 2 420 @070). ! >D8F80;L=>3> A09B0 ( HYPERLINK "http://er.cs.ucla.edu/Dragon/" http://er.cs.ucla.edu/Dragon/) 1K;0 A:0G0=0 25@A8O 4.0 40==>3> ?;59A5@0, 2K?CI5==0O 2 2005 3>4C. 1I0O AB@C:BC@0 <5B>40  SHAPE \* MERGEFORMAT  -B0?K >?B8<870F88 ;>10;L=>5 @07<5I5=85 (Global placement) 07=0G5=85 MB0?0: 07185=85 <=>65AB20 M;5<5=B>2 =0 =51>;LH85 3@C??K, 2 :064>9 87 :>B>@KE =5 1>;55 7 M;5<5=B>2, 8 =07=0G5=85 3@C??0< CG0AB:0 (bin) =0 ?;0B5. -B> 45;05BAO 4;O B>3>, GB>1K ?>=878BL @07<5@=>ABL 7040G8 4;O simulated annealing. 07185=85 2A59 ?;0BK =0 <=>65AB2> CG0AB:>2 7=0G8B5;L=> A>:@0I05B ?@>AB@0=AB2> ?>8A:0. E>4: 38?5@3@0D, ?@54AB02;ONI89 @07<5I05<CN AE5<C @07<5@K ?;0BK KE>4: 8=D>@<0F8O > @07185=88 ?;0BK =0 CG0AB:8 8=D>@<0F8O > @07185=88 38?5@3@0D0 =0 ?>43@0DK, 2 :064>< 87 :>B>@KE =5 1>;55 7 M;5<5=B>2 8=D>@<0F8O > @07<5I5=88 ?>43@0D>2 ?> CG0AB:0< ;3>@8B< @07185=8O 38?5@3@0D0: @>872>48BAO 18A5:F8O 38?5@3@0D0 ?@8 ?><>I8 2K7>20 DC=:F88 HMETIS_PartRecursive 181;8>B5:8 hMetis. @8 MB>< A @51@0<8, :>B>@K5 B5@?OB @07@K2, ?>ABC?0NB A>3;0A=> ?>4E>4C B, >?8A0==><C 2 AB0BL5 [34]. 8?5@@51@> =5 C40;O5BAO A>2A5<, 0 70<5=O5BAO A>AB02;ONI8<8 53> ?>4@51@0<8, A>548=ONI8<8 25@H8=K, ?>?02H85 2 >4=C G0ABL @07185=8O.  SHAPE \* MERGEFORMAT  ;3>@8B< @07185=8O ?;0BK =0 CG0AB:8: #G0AB>: 45;8BAO :064K9 @07 ?> 1>;LH5<C 87<5@5=8N. "> 5ABL 2KA>:85 CG0AB:8 45;OBAO ?> 3>@87>=B0;8, 0 4;8==K5 ?> 25@B8:0;8. ;3>@8B< ?@8A2>5=8O ?>43@0DC CG0AB:0: @>872>;L=>. Bin-swapping 0;3>@8B<: @8<5=O5BAO 604=K9 @0=4><878@>20==K9 0;3>@8B<. K18@0NBAO 420 ?@>872>;L=KE CG0AB:0 (2A5-B0:8 4>AB0B>G=> 1;87:8E), ?@>872>48BAO >1<5= ?>43@0D>2 8 2KG8A;O5BAO ?@8@0I5=85 DC=:F88 AB>8<>AB8. A;8 >=> >B@8F0B5;L=>, ?5@5AB0=>2:0 ?@8=8<05BAO, 2 ?@>B82=>< A;CG05  >B25@305BAO. @8B5@89 >AB0=>20: ;81> >3@0=8G5=85 ?> >1I5<C G8A;C ?>?KB>:, ;81> ?> G8A;C =5C40G=KE (?>4@O4?) ?>?KB>:. <8B0F8O >B6830 (Simulated Annealing) 07=0G5=85 MB0?0: #;CGH5=85 ?@5420@8B5;L=>3> @07<5I5=8O, :>B>@>5 1K;> >?@545;5=> =0 MB0?5 Global placement. @>872>48BAO MB> ?CB5< 70?CA:0 <5B>40 simulated annealing, :>B>@K9 >ACI5AB2;O5B ?5@5@0A?@545;5=85 M;5<5=B>2 ?> CG0AB:0< ?;0BK. E>4: A>2?0405B A 2KE>4>< ;>10;L=>3> @07<5I5=8O KE>4: A>2?0405B A 2KE>4>< ;>10;L=>3> @07<5I5=8O, 65;0B5;L=> A <5=LH8< 7=0G5=85< DC=:F88 AB>8<>AB8 ;3>@8B<: 5B>4 35=5@0F88 A;54CNI53> A>A540: A;CG09=K9 2K1>@ M;5<5=B0 8 ?5@5<5I5=85 53> =0 ?@>872>;L=> 2K1@0==K9 CG0AB>: ?;0BK A;CG09=K9 2K1>@ ?0@K M;5<5=B>2 87 @07=KE CG0AB:>2 8 8E >1<5= "5<?5@0BC@=>5 @0A?8A0=85: 35><5B@8G5A:>5 (?>41>@ ?0@0<5B@>2 M:A?5@8<5=B0;L=K< ?CB5<) @8B5@89 >AB0=>20: 4>AB865=85 <8=8<C<0 B5<?5@0BC@K 5B0;L=>5 @07<5I5=85 (Detailed placement) 07=0G5=85 MB0?0: 5@5E>4 >B 3@C1>3> @07<5I5=8O, >?@545;5==>3> =0 MB0?0E ;>10;L=>3> @07<5I5=8O 8 <8B0F88 >B6830 : 45B0;L=><C @07<5I5=8N 2 2845 @O4>2. $0:B8G5A:8 =0 MB>< MB0?5 <K 2?5@2K5 ?>;CG05< @07<5I5=85, :>B>@>5 <>6=> @0AA<0B@820BL :0: :>=5G=>5. E>4: !>2?0405B A 2KE>4>< <8B0F88 >B6830 KE>4: =D>@<0F8O > @07<5I5=88 M;5<5=B>2 ?> @O40< ;3>@8B<: 5@5E>4 >B 18=>2>9 AB@C:BC@K : @O40<: A;8BL 2A5 M;5<5=BK, =0E>4OI85AO 2 >4=>< AB>;1F5 18=>2. 0B5< @02=><5@=> @0A?@545;8BL 8E (M;5<5=BK) ?> @O40< B0:, GB>1K =5 2>7=8:;> ?5@5:@KB89. Cell annealing: ?@8<5=5=85 Simulated Annealing : >B45;L=K< M;5<5=B0< =0 ?;0B5. 04=0O >?B8<870F8O (Greedy optimization) 07=0G5=85 MB0?0: #;CGH8BL :0G5AB2> @07<5I5=8O, 2>7=8:H53> 2 @57C;LB0B5 Detailed placement. #;CGH5=85 ?@>872>48BAO ;>:0;L=K<8 ?@5>1@07>20=8O<8. E>4: KE>4 5B0;L=>3> @07<5I5=8O KE>4: !>2?0405B A 2KE>4>< 5B0;L=>3> @07<5I5=8O, => A 65;0B5;L=> <5=LH8< 7=0G5=85< DC=:F88 AB>8<>AB8 ;3>@8B<: !;CG09=K< >1@07>< 2K18@05BAO 107>2K9 M;5<5=B. K18@05BAO 3>@87>=B0;L=>5 8;8 25@B8:0;L=>5 =0?@02;5=85 >?B8<870F88 2 7028A8<>AB8 >B @53C;8@C5<>3> ?0@0<5B@0  EMBED Equation.3 .  EMBED Equation.3  - MB> 25@>OB=>ABL 25@B8:0;L=>3> ?>8A:0.  AB0BL5 [34] 02B>@K ?@54;030NB 1@0BL  EMBED Equation.3  @02=K< 20%. @8 25@B8:0;L=>< =0?@02;5=88 >?B8<870F88 M;5<5=B, =0E>4OI89AO =5?>A@54AB25==> =04 107>2K< 8;8 ?>4 =8<, <5=O5BAO <5AB0<8 A 107>2K< M;5<5=B><. @8 MB>< A>A54=85 M;5<5=BK <>6=> ?5@542830BL, GB>1K 871028BLAO >B 2>7<>6=KE ?5@5:@KB89 8 ?CAB>B. @8 3>@87>=B0;L=>< =0?@02;5=88 >?B8<870F88 @0AA<0B@820NBAO 4 A>A54=8E M;5<5=B0 (A?@020 8;8 A;520) ?> >B=>H5=8N : 107>2><C, 0 40;LH5 ?@>872>48BAO ?>;=K9 ?5@51>@ 2A52>7<>6=KE 8E ?5@5AB0=>2>: A F5;LN C<5=LH5=8O 4;8=K ?@>2>4>2. 0:;NG5=85 0<8 1K;8 @0AA<>B@5=K ?OBL A>2@5<5==KE 8=AB@C<5=B>2 @07<5I5=8O. A5 >=8 A>740=K 2 0:045<8G5A:8E :@C30E A F5;LN 0?@>10F88 =>259H8E 8459 CG5=KE, @01>B0NI8E 2 >1;0AB8 CAD. >MB><C 2A5 0;3>@8B<K 8=B5@5A=K, ?@5645 2A53>, B5<8 845O<8, :>B>@K5 2 =8E 70;>65=K. '0I5 2A53> ?@8<5=O5BAO 420 ?>4E>40: partition-based 8 simulated annealing. -B8 <5B>4K C65 402=> ?@8<5=ONBAO 2 7040G5 > @07<5I5=88, 8 ?>-?@56=5<C AG8B0NBAO 4>2>;L=> MDD5:B82=K<8. @0B:85 8E E0@0:B5@8AB8:8 B0:>2K: 8=AB@C<5=BK, >A=>2K20NI85AO =0 simulated annealing, 40NB @07<5I5=8O 2KA>:>3> :0G5AB20, => B@51CNB 4;O MB>3> <=>3> 2@5<5=8; partition-based 8=AB@C<5=BK @01>B0NB A@02=8B5;L=> 1KAB@>, => :0G5AB2> @57C;LB0B>2 ?>;CG05BAO =5A:>;L:> =865. -B> >1CA;>28;> B>B D0:B, GB> <=>385 87 ?>A;54=8E 8=AB@C<5=B>2 :><18=8@CNB 2 A515 >10 ?>4E>40, =0?@8<5@ Dragon. >;>68B5;L=K< <><5=B>< O2;O5BAO B>, GB> 2A5 8=AB@C<5=BK @072820NBAO, 8 8E @50;870F88 @53C;O@=> >1=>2;ONBAO. >4>1=0O 0:B82=>ABL 3>2>@8B > B><, GB> 40;5:> =5 2A5 5I5 8725AB=> 2 7040G5 > @07<5I5=88 8 B@51C5BAO 55 BI0B5;L=>5 8AA;54>20=85. A5 MB> 405B >A=>20=8O ?>;030BL, GB> 1C45B 5I5 <=>3> >B:@KB89 8 4>AB865=89 2 MB>9 >1;0AB8. !?8A>: ;8B5@0BC@K VLSI placement [1] S. N.Adya, M. C. Yildiz, I. L. Markov, P. G. Villarrubia, P. N. Parakh, P. H. Madden  Benchmarking for Large Scale placement and beyond , 2003 [2] J. A. Chandy Parallel algorithms for standard cell placement using simulated annealing, 1996 [3] C.-C. Chang, J. Cong, D. Pan, and X. Yuan Multilevel Global Placement with Congestion Control, 2003 [4] C.-C. Chang, J. Cong, M. Romesis, M. Xie Optimality and Scalability Study of Existing Placement Algorithms, 2004 [5] A. E. Dunlop, B. W. Kernighan A Procedure for Placement of Standard Cell VLSI Circuits, IEEE TCAD, Jan 1985 [6] H. Eisenmann, F. Johannes, Generic Global Placement and Floorplanning, DAC-98, p.269 - 274 [7] D. R. Greening Simulated annealing with errors, 1995 [8] Prof. Kurt Keutzer The physical placement problem in integrated circiuts, University of California, Berkeley, CA. [9] J. M. Klienhans, G. Sigl, F. M. Johannes, K. J. Antreich GORDIAN: VLSI placement by quadratic programming and slicing optimization, 1991 [10] G. Lu, S. Areibi An island-based GA implementation for VLSI standard-cell placement, 2001 [11] P. H. Madden Reporting of standard cell placement results, 2002 [12] W.-J. Sun, C. Sechen Efficient and effective placement for very large circuits, 1993 [13] M. Wang, X. Yang, M. Sarrafzadeh Dragon2000: Standard-cell placement tool for large industry circuits, 2000 [14] Z. Yang, S. Areibi Mathematical and heuristic techniques for VLSI global circuit placement, 2003 ><0H=85 AB@0=8FK: Capo  HYPERLINK "http://www.public.iastate.edu/~nataraj/FastPlace.html"  HYPERLINK "http://vlsicad.eecs.umich.edu/BK/PDtools/#PUBS" http://vlsicad.eecs.umich.edu/BK/PDtools/#PUBS FengShui  HYPERLINK "http://vlsicad.cs.binghamton.edu/pubs.html" http://vlsicad.cs.binghamton.edu/pubs.html FastPlace  HYPERLINK "http://www.public.iastate.edu/~nataraj/FastPlace.html" http://www.public.iastate.edu/~nataraj/FastPlace.html MPL  HYPERLINK "http://ballade.cs.ucla.edu/cpmo/" http://ballade.cs.ucla.edu/cpmo/ Dragon  HYPERLINK "http://er.cs.ucla.edu/Dragon/" http://er.cs.ucla.edu/Dragon/ Capo [15] S.N.Adya, S.Chaturverdi, J.A.Roy, D.A.Papa, I.L.Markov, Unification of Partitioning, Placement and Floorplanning, ICCAD, 2004, pp. 550-557  HYPERLINK "http://vlsicad.eecs.umich.edu/BK/PDtools/" \l "PUBS" [16] S. N. Adya, I. L. Markov, Consistent placement of macro-blocks using floorplanning and standard-cell placement (2002)  HYPERLINK "http://vlsicad.eecs.umich.edu/BK/PDtools/" \l "PUBS" [17] S. N. Adya, I. L. Markov and P. G. Villarrubia, "On Whitespace and Stability in Mixed-Size Placement" (2003)  HYPERLINK "http://vlsicad.eecs.umich.edu/BK/PDtools/" \l "PUBS" [18] S. N. Adya, I. L. Markov, Improving Min-cut Placement for VLSI Using Analytical Techniques (2003) [19] A.E. Caldwell, A.B. Kahng, I.L. Markov, Design and Implementation of the Fiduccia-Mattheyses Heuristics for VLSI Netlist Partitioning, Lecture Notes in Computer Science, vol.1619, Springer (1999)  HYPERLINK "http://vlsicad.eecs.umich.edu/BK/PDtools/" \l "PUBS" [20] A.E. Caldwell, A.B. Kahng, I.L. Markov, Can recursive bisection alone produce routable placements? (2000)  HYPERLINK "http://www-users.cs.umn.edu/~karypis/metis/hmetis/" [21] G.Karypis, R.Aggarwal, V.Kumar, and S.Shekar, Multilevel Hypergraph Partitioning: Applications in VLSI design, DAC 97, pp. 526-529. Addictional publications and benchmark results for hMetis-1.5 are available at [22] G.Karypis and V.Kumar, Multilevel k-way Hypergraph Partitioning, DAC 99, pp. 343-348  HYPERLINK "http://vlsicad.eecs.umich.edu/BK/PDtools/" \l "PUBS" [23] A. B. Kahng, I. L. Markov, S. Reda, On Legalization of RowBased Placements (2004) HYPERLINK "http://vlsicad.eecs.umich.edu/BK/PDtools/" \l "PUBS"[24] D. A. Papa, S. N. Adya, I. L. Markov, Constructive Benchmarking for Placement (2004) HYPERLINK "http://vlsicad.eecs.umich.edu/BK/PDtools/" \l "PUBS"[25] J. A. Roy, D. A. Papa, S. N. Adya, H. H. Chan, A. N. Ng, J. F. Lu, I. L. Markov, Capo: Robust and Scalable OpenSource MinCut Floorplacer (2005) FengShui  HYPERLINK "http://vlsicad.cs.binghamton.edu/pubs/Agnihotri05_ispd.pdf" [26] A. R. Agnihotri, S. Ono, P. H. Madden: Recursive bisection placement: Feng Shui 5.0 implementation details, 2005  HYPERLINK "http://vlsicad.cs.binghamton.edu/pubs/Agnihotri03.pdf" [27] A. Agnihotr, M. C. Yildiz, A. Khatkhate, A. M. S. On, P.H. Madden: Fractional cut: improved recursive bisection placement, 2003  HYPERLINK "http://vlsicad.cs.binghamton.edu/pubs/Yildiz010068.pdf" [28] M. C. Yildiz, P. H. Madden: Global objectives for standard cell placement, 2001 FastPlace  HYPERLINK "http://www.public.iastate.edu/~nataraj/FastPlace.html%20-%20Literature_" [29] N. Viswanathan, C. C.-N. Chu FastPlace: Efficient Analytical Placement using Cell Shifting, Iterative Local Refinement and a Hybrid Net Model, 2004 HYPERLINK "http://www.public.iastate.edu/~nataraj/FastPlace.html%20-%20Literature_"[30] N. Viswanathan, M. Pan, C. Chu An efficient and effective detailed placement algorithm, 2005  HYPERLINK "http://www.public.iastate.edu/~nataraj/" [31] Natarajan Viswanathan, Min Pan and Chris Chu FastPlace 2.0: An Efficient Analytical Placer for Mixed-Mode Designs, 2006 mPL  HYPERLINK "http://ballade.cs.ucla.edu/~cong/papers/mpl2.pdf" [32] T. F. Chan, J. Cong, T. Kong, J. R. Shinnerl, K. Sze, An Enhanced Multilevel Algorithm for Circuit Placement, 2003  HYPERLINK "http://www.gigascale.org/pubs/600/46-chan.pdf" [33] T. Chan, J. Cong, K. Szeb Multilevel generalized force-directed method for circuit placement, 2005. Dragon [HYPERLINK "http://er.cs.ucla.edu/Dragon/papers"34] M. Wang, X. Yang, M. Sarrafzadeh "Dragon2000: Standard-cell placement tool for large industry circuits", 2000 [ HYPERLINK "http://er.cs.ucla.edu/Dragon/papers/dragon.pdf" 35] X. Yang, B. Choi, M. 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?#" `2hDtga/{bDu$`!<Dtga/{b @XJ| xcdd``> @c112BYL%bpu u;&g+KRsePdhB 0Cbd`/O$DdTh\  c $A? ?#" `2g_Nkލ̴˸_Cw$`!;_Nkލ̴˸_ @XJ| xcdd``> @c112BYL%bpu 1ܘ!+(|-!T T0[Z 1< 2/$37X/\!(?71y F+A\Ps}6vv0o(L021)W2ePdh,s;.Ff -3NDdTh\  c $A? ?#" `2g_Nkލ̴˸_Cy$`!;_Nkލ̴˸_ @XJ| xcdd``> @c112BYL%bpu 1ܘ!+(|-!T T0[Z 1< 2/$37X/\!(?71y F+A\Ps}6vv0o(L021)W2ePdh,s;.Ff -3NMDyK 6http://www.public.iastate.edu/~nataraj/FastPlace.htmlyK lhttp://www.public.iastate.edu/~nataraj/FastPlace.html5DyK /http://vlsicad.eecs.umich.edu/BK/PDtools/#PUBSyK Thttp://vlsicad.eecs.umich.edu/BK/PDtools/PUBS!DyK +http://vlsicad.cs.binghamton.edu/pubs.htmlyK Vhttp://vlsicad.cs.binghamton.edu/pubs.htmlMDyK 6http://www.public.iastate.edu/~nataraj/FastPlace.htmlyK lhttp://www.public.iastate.edu/~nataraj/FastPlace.htmlDyK !http://ballade.cs.ucla.edu/cpmo/yK Bhttp://ballade.cs.ucla.edu/cpmo/DyK http://er.cs.ucla.edu/Dragon/yK <http://er.cs.ucla.edu/Dragon/DyK  yK Thttp://vlsicad.eecs.umich.edu/BK/PDtools/PUBSDyK  yK Thttp://vlsicad.eecs.umich.edu/BK/PDtools/PUBSDyK  yK Thttp://vlsicad.eecs.umich.edu/BK/PDtools/PUBSDyK  yK Thttp://vlsicad.eecs.umich.edu/BK/PDtools/PUBSDyK yK fhttp://www-users.cs.umn.edu/~karypis/metis/hmetis/DyK  yK Thttp://vlsicad.eecs.umich.edu/BK/PDtools/PUBSDyK  yK Thttp://vlsicad.eecs.umich.edu/BK/PDtools/PUBSDyK  yK Thttp://vlsicad.eecs.umich.edu/BK/PDtools/PUBSDyK yK vhttp://vlsicad.cs.binghamton.edu/pubs/Agnihotri05_ispd.pdfDyK yK lhttp://vlsicad.cs.binghamton.edu/pubs/Agnihotri03.pdfDyK yK nhttp://vlsicad.cs.binghamton.edu/pubs/Yildiz010068.pdfDyK yK http://www.public.iastate.edu/~nataraj/FastPlace.html - Literature_DyK yK http://www.public.iastate.edu/~nataraj/FastPlace.html - Literature_DyK yK Phttp://www.public.iastate.edu/~nataraj/DyK yK bhttp://ballade.cs.ucla.edu/~cong/papers/mpl2.pdfDyK yK \http://www.gigascale.org/pubs/600/46-chan.pdfDyK yK Hhttp://er.cs.ucla.edu/Dragon/papersDyK yK ^http://er.cs.ucla.edu/Dragon/papers/dragon.pdfDdl_11903753816F0֨80֨8Ole CompObj58fObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39qs"Ed 2 .  @& & MathType-OlePres00079Equation Native _11903754084T<F0֨80֨8Ole Times New Romanww w0- 2 `const2 2o2 r1o2 2o2 r1o 2 `cost2 `R o 2 `4TotalSymbolw@ ww w0-2 `+o2 `c+o2 `{+o2 `T +o2 `z=oTimes New Romanww w0-2 `yo2 `Iyo2 `yo2 `mxo2 `F xo2 `xo Times New Romanww w0-2 To2 pyo2 To2 To2 Wxo2  ToTimes New Romanww w0-2 `do2 `5Qo2 `z do2 `2 Qo & "System !tw-f] Total cost=12x T Qx+d x  T x+12y T Qy+d y  T y+const FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj;>fObjInfoOlePres000=?Equation Native    .  &` & MathTypeP-))}Symbolw@ CwCw w0-2 (ySymbolw@ sCwCw w0-2 )y Times New RomanCwCw w0-2 D2y2 l 2yTimes New RomanCwCw w0-2 )y2 (y2  )y2 (y2 2y2 1y2 VI y 2 V:cell2 V e 2 Vandl2 Vp n2 V n 2 Vcell 2 Va between 2 V4net 2 V e 2 V the2 VKof 2 V4Cost e Times New RomanCwCw w0-2 jo2 io2 A jo2 F io2 ijTimes New RomanCwCw w0-2 +yj2 Gyj2 Z xj2 xj2 wj2 Vjj2 V ijSymbolw@ CwCw w0-2 -j2 m+j2  -j2 d=j & "System !)Bw-NANIu  Cost of the net between cell i and cell j=12w ij (x i "x j ) 2 +(y i "y j ) 2 ()_1190375622BF0֨80֨8Ole CompObjADfObjInfo      !#$%&'(* FMicrosoft Equation 3.0 DS Equation Equation.39q' 0  .  @$&$ & MathTypeTimes New RomanCwCw w0- 2 &5ectors 2 & Solution v2 &k,c2 " c OlePres000CErEquation Native 21TableciSummaryInformation(H"2  cell2 u  e 2 Sandl2  n2 K n 2 ;cell 2 between 2 net 2 i e 2 B the2 of 2  Weight 2 ; e 2 +cell2  e2 oof 2 zcenter 2 & e 2  the2 of2 C f2 sf 2  Coordinate 2 :LeteSymbolw@ CwCw w0-2 &=e2 =e2 =eTimes New RomanCwCw w0-2 &ye2 &xeTimes New RomanCwCw w0-2 z#je2 ie2 we2 ie2 )e2 >,y2 (x Times New RomanCwCw w0-2 Fij2 ^ij2 ij & "System !)Bw-NANI Let(x i ,y i )=Coordinates of the center of cell iw ij =Weight of the net between cell i and cell jx,y=Solution vectorsOh+'0t  0 < HT\dl VLSI PlacementOlga_MakoveevalgaNormalk 30Microsoft Word 9.0@:@@e6@̦0  # A2J)(ݵb6d&Ue`!)(ݵb6dd@q0xcdd``6b 2 ĜL0##0KQ* WA^RcgbR x Szjx|K2B* Rj8 :@u!f010m`00~!356A $37X/\!(?71aXr= \@ڈ+4: 3ʇw0SG!P31,.ည1T(l53$$1O>dMa(w tq` ̮%R #ad1T~*?_棺k! q]] s+xO sA 8ߍ‡S?,Ir'z`|(S0FxU}8;0Nȣ3#)2!" ~ "yQ卸 \γM HqĤ\Y\ q1(2t$K1&#E'wV:F"a`WRsbo&h@t\td+-[~J= \-^7l~o?*vmEJt[{.] ՙvJNĊ[NQ\NW4▝ϯ;/zg|Af2Vp7JmX*RDTg3 T_9 g)aĹϔHu>rTBG Rst2hw9Ԟ׷˽pņ|z'|(`uM1CjM3fl%8۽+2e1JZW (Jz Pw*n> m)!lEkUDdxJ  C Aw? 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