ICS 252 Introduction to Computer Design

Transcription

1 ICS 252 Introducton to Computer Desgn Prttonng El Bozorgzdeh Computer Scence Deprtment-UCI

2 Prttonng Decomposton of complex system nto smller susystems Done herrchclly Prttonng done untl ech susystem hs mngele sze Ech susystem cn e desgned ndependently Interconnectons etween prttons mnmzed Less hssle nterfcng the susystems Communcton etween susystems usully costly 2

3 Exmple: Prttonng of Crcut Input sze: 48 Cut 1=4 Sze 1=15 Cut 2=4 Sze 2=16 Sze 3=17 [ Sherwn] 3

4 Herrchcl Prttonng Levels of prttonng: System-level prttonng: Ech su-system cn e desgned s sngle PCB Bord-level prttonng: Crcut ssgned to PCB s prttoned nto sucrcuts ech frcted s VLSI chp Chp-level prttonng: Crcut ssgned to the chp s dvded nto mngele su-crcuts NOTE: physclly not necessry [ Sherwn] 4

5 Dely t Dfferent Levels of Prttons A x B D 10x C PCB1 20x PCB2 5

6 Prttonng: Forml Defnton Input: Grph or hypergrph Usully wth vertex weghts Usully weghted edges Constrnts Numer of prttons (K-wy prttonng) Mxmum cpcty of ech prtton OR mxmum llowle dfference etween prttons Oectve Assgn nodes to prttons suect to constrnts s.t. the cutsze s mnmzed Trctlty Is NP-complete 6

7 Kernghn-Ln (KL) Algorthm On non-weghted grphs An tertve mprovement technque A two-wy (secton) prttonng lgorthm The prttons must e lnced (of equl sze) Iterte s long s the cutsze mproves: Fnd pr of vertces tht result n the lrgest decrese n cutsze f exchnged Exchnge the two vertces (potentl move) Lock the vertces If no mprovement possle, nd stll some vertces unlocked, then exchnge vertces tht result n smllest ncrese n cutsze W. Kernghn nd S. Ln, Bell System Techncl Journl,

8 Kernghn-Ln (KL) Algorthm Intlze Bprtton G nto V 1 nd V 2, s.t., V 1 = V 2 ± 1 n = V Repet for =1 to n/2 Fnd pr of unlocked vertces v V 1 nd v V 2 whose exchnge mkes the lrgest decrese or smllest ncrese n cut-cost Mrk v nd v s locked Store the gn g. Fnd k, s.t. =1..k g =Gn k s mxmzed If Gn k > 0 then move v 1,...,v k from V 1 to V 2 nd v 1,...,v k from V 2 to V 1. Untl Gn k 0 8

9 Kernghn-Ln (KL) Exmple c d e f g h Step No. Vertex Pr Gn Cut-cost { d, g } { c, f } {, h } {, e }

10 Kernghn-Ln (KL) : Anlyss Tme complexty? Inner (for) loop Itertes n/2 tmes Iterton 1: (n/2) x (n/2) Iterton : (n/2 + 1) 2. Psses? Usully ndependent of n O(n 3 ) Drwcks? Locl optmum Add dummy nodes Blnced prttons only No weght for the vertces Replce vertex of weght Hgh tme complexty ω wth ω vertces of sze 1 Only on edges, not hyper-edges 10

11 Fducc-Mttheyses (FM) Algorthm Modfed verson of KL A sngle vertex s moved cross the cut n sngle move Unlnced prttons Vertces re weghted Concept of cutsze extended to hypergrphs Specl dt structure to mprove tme complexty to O(n 2 ) (Mn feture) Cn e extended to mult-wy prttonng C. M. Fducc nd R. M. Mttheyses, 19 th DAC,

12 The FM Algorthm: Dt Structure +pmx Ist Prtton v 1 v 2 -pmx +pmx Vertex v n 2nd Prtton v 2 Lst of free vertces -pmx Vertex n 12

13 The FM Algorthm: Dt Structure Pmx Mxmum gn p mx = d mx. w mx, where d mx = mx degree of vertex (# edges ncdent to t) w mx s the mxmum edge weght Wht does t men ntutvely? -Pmx.. Pmx rry Index s ponter to the lst of unlocked vertces wth gn. Lmt on sze of prtton A mxmum defned for the sum of vertex weghts n prtton (lterntvely, the mxmum rto of prtton szes mght e defned) 13

14 The FM Algorthm Intlze Strt wth lnce prtton A, B of G (cn e done y sortng vertex weghts n decresng order, plcng them n A nd B lterntvely) Itertons Smlr to KL A vertex cnnot move f voltes the lnce condton Choosng the node to move: pck the mx gn n the prttons Moves re tenttve (smlr to KL) When no moves possle or no more unlocked vertces vlle, the pss ends When no move cn e mde n pss, the lgorthm termntes 14

15 Why Hyperedges? For mult termnl nets, K-L my decompose them nto mny 2- termnl nets, ut not effcent! Consder ths exmple: If A = {1, 2, 3} B = {4, 5, 6}, grph model shows the cutsze = 4 ut n the rel crcut, only 3 wres cut Reducng the numer of nets cut s more relstc thn reducng the numer of edges cut 1 m 3 2 q k p m m m 3 q 2 q q k p

16 Hyperedge to Edge Converson A hyperedge cn e converted to clque Rel cut=1 3 w w w 4 net cut=2w 2 2 w=? w=2/(n-1) hs een used, lso w=2/n Best: w=4/(n 2 mod(n,2)) for n=3, w=4/(9-1)=0.5 Alwys necessry to convert hyper-edge to edge? 16

17 Gn Clculton G A n G B Internl cost I = x A C x, E = y B C y Externl cost Lkewse, D D = = E E I I = x A C x y B C y 17

18 Gn Clculton (cont.) Lemm: Consder ny A, B. If, re nterchnged, the gn s Proof: Totl cost efore nterchnge (T) etween A nd B T= Totl cost fter nterchnge (T ) etween A nd B T= Therefore E I + g = D + D 2C + E I C + C D + (cost for ll + (cost for ll D others) others) g = T T = E I + E I 2C 18

19 Gn Clculton (cont.) Lemm: Let D x, D y e the new D vlues for elements of A - { } nd B - { }. Then fter nterchngng &, D D x y = D = D x y + 2C + 2C x y 2C 2C x y,, x A { y B { } } Proof: The edge x- chnged from nternl n D x to externl n D x The edge y- chnged from nternl n D x to externl n D x The x- edge chnged from externl to nternl The y- edge chnged from externl to nternl More clrfcton n the next two sldes 19

20 Clrfcton of the Lemm β α x 20

21 Clrfcton of the Lemm (cont.) Decompose Ix nd Ex to seprte edges from nd : I x = Cx x x + α E = C + β Wrte the equtons efore the move D x = E... And fter the move x I = α + β C x D = ( C x x x D + β ) ( C + α) + C x x = α + β + C = I x = C x x x x + 2C 2C + α E = C + β x C x x x 21

22 FM Gn Clculton: Drect Hyperedge Clc FM s le to clculte gn drectly usng hyperedges ( not necessry to convert hyperedges to edges) Defnton: Gven prtton (A B), we defne the termnl dstruton of n s n ordered pr of ntegers (A(n),B(n)), whch represents the numer of cells net n hs n locks A nd B respectvely (how fst cn e computed?) Net s crtcl f there exsts cell on t such tht f t were moved t would chnge the net s cut stte (whether t s cut or not). Net s crtcl f A(n)=0,1 or B(n)=0,1 22

23 FM Gn Clc: Drect Hyperedge Clc (cont.) Gn of cell depends only on ts crtcl nets: If net s not crtcl, ts cutstte cnnot e ffected y the move A net whch s not crtcl ether efore or fter move cnnot nfluence the gns of ts cells Let F e the from prtton of cell nd T the to : g() = FS() - TE(), where: FS() = # of nets whch hve cell s ther only F cell TE() = # of nets contnng nd hve n empty T sde 23

24 Exmple: KL Step 1 - Intlzton A = {2, 3, 4}, B = {1, 5, 6} A = A = {2, 3, 4}, B = B = {1, 5, 6} Step 2 - Compute D vlues D 1 = E 1 -I 1 = 1-0 = +1 D 2 = E 2 -I 2 = 1-2 = -1 D 3 = E 3 -I 3 = 0-1 = -1 D 4 = E 4 -I 4 = 2-1 = +1 D 5 = E 5 -I 5 = 1-1 = +0 D 6 = E 6 -I 6 = 1-1 = +0 Intl prtton [ Kng] 24

25 Exmple: KL (cont.) Step 3 - compute gns g 21 = D 2 + D 1-2C 21 = (-1) + (+1) - 2(1) = -2 g 25 = D 2 + D 5-2C 25 = (-1) + (+0) - 2(0) = -1 g 26 = D 2 + D 6-2C 26 = (-1) + (+0) - 2(0) = -1 g 31 = D 3 + D 1-2C 31 = (-1) + (+1) - 2(0) = 0 g 35 = D 3 + D 5-2C 35 = (-1) + (0) - 2(0) = -1 g 36 = D 3 + D 6-2C 36 = (-1) + (0) - 2(0) = -1 g 41 = D 4 + D 1-2C 41 = (+1) + (+1) - 2(0) = +2 g 45 = D 4 + D 5-2C 45 = (+1) + (+0) - 2(+1) = -1 g 46 = D 4 + D 6-2C 46 = (+1) + (+0) - 2(+1) = -1 The lrgest g vlue s g 41 = +2 nterchnge 4 nd 1 ( 1, 1 ) = (4, 1) A = A - {4} = {2, 3} B = B - {1} = {5, 6} oth not empty 25

26 Exmple: KL (cont.) Step 4 - updte D vlues of node connected to vertces (4, 1) D 2 = D 2 + 2C 24-2C 21 = (-1) + 2(+1) - 2(+1) = -1 D 5 = D 5 + 2C 51-2C 54 = (0) - 2(+1) = -2 D 6 = D 6 + 2C 61-2C 64 = (0) - 2(+1) = -2 Assgn D = D, repet step 3 : g25 = D 2 + D 5-2C 25 = (0) = -3 g26 = D 2 + D 6-2C 26 = (0) = -3 g35 = D 3 + D 5-2C 35 = (0) = -3 g36 = D 3 + D 6-2C 36 = (0) = -3 All vlues re equl; rtrrly choose g 36 = -3 (2, 2) = (3, 6) A = A - {3} = {2}, B = B - {6} = {5} New D vlues re: D 2 = D 2 + 2C 23-2C 26 = (1) - 2(0) = +1 D 5 = D 5 + 2C 56-2C 53 = (1) - 2(0) = +0 New gn wth D 2 D 2, D 5 D 5 g 25 = D 2 + D 5-2C 52 = (0) = +1 (3, 3) = (2, 5) 26

27 Exmple: KL (cont.) 5 Step 5 - Determne the # of moves to tke g 1 = +2 g 1 + g 2 = +2-3 = -1 g 1 + g 2 + g 3 = = 0 The vlue of k for mx G s 1 X = { 1 } = {4}, Y = { 1 } = {1} Move X to B, Y to A A = {1, 2, 3}, B = {4, 5, 6} Repet the whole process: The fnl soluton s A = {1, 2, 3}, B = {4, 5, 6}

28 Sugrph Replcton to Reduce Cutsze Vertces re replcted to mprove cutsze Good results f lmted numer of components replcted A A B B A A B B C. Krng nd A. R. Newt, ICCAD,

29 Clusterng Clusterng Bottom-up process Merge hevly connected components nto clusters Ech cluster wll e new node Hde nternl connectons (.e., connectng nodes wthn cluster) Merge two edges ncdent to n externl vertex, connectng t to two nodes n cluster Cn e preprocessng step efore prttonng Ech cluster treted s sngle node , , ,4 29

30 Other Prttonng Methods KL nd FM hve ech held up very well Mn-cut / mx-flow lgorthms Ford-Fulkerson for unconstrned prttons Rto cut Genetc lgorthm Smulted nnelng 30