Probabilistic Analysis of Rectilinear Steiner Trees

Transcription

1 Probabilistic Aalysis of Rectiliear Steier Trees Chuhog Che Departet of Electrical ad Coputer Egieerig Uiversity of Widsor, Otario, Caada, N9B 3P4 E-ail: Abstract Steier tree is a fudaetal proble i the autoatic itercoect optiizatio for VLSI desig. We preset a probabilistic aalysis ethod for costructig rectiliear Steier trees. The best solutio uder statistical sese is obtaied for ay give set of N poits. Experiets show that our results are better tha those by the previous techique or very close to the optia. 1. Itroductio Oe of the key probles i VLSI itercoect desig is the topology costructio of sigal ets with the iiu cost. The Steier tree proble is to fid the tree structure which coects all pis of the sigal et such that the wire legth (i.e., cost) ca be iiized. If all edges of the tree are restricted to the horizotal ad vertical directios as are the case i VLSI desig, the proble is called rectiliear Steier tree (RST). I geeral, the RST ca cotai, i additio to the pis of the et, soe ore poits which are called s. I particular, the RST without Steier poits is called rectiliear iiu spaig tree (RMST) which has bee well studied [1]. While the RST ca lead to better results tha the RMST i ters of wire legth, it has bee show that the RST proble is NP-coplete [2]. Several effective heuristic approaches have bee proposed towards the optial or sub-optial solutios. For exaple, Haa [3] showed a optial algorith whe the et cotais o ore tha four pis. Cohoo [4] proposed a optial algorith whe the pis of a et lie o the perieter of a rectagle. Hwag [6] proved that the ratio of tree legths betwee a RMST ad a RST is o worse tha 3/2. A o(n log N) algorith for the RST was also proposed i [5], while the results were far fro the optial solutio. A good survey o Steier tree probles ca be foud i [7]. For a coprehesive survey of the itercoect desig, the readers are referred to [10]. I this paper we provide probabilistic aalysis for the rectiliear Steier tree proble. By cosiderig all possible topologic structures coectig every pair of pis, we ca calculate the probability of the structures passig over idividual edges. The optial Steier tree uder statistical sese is the tree with axiu su of the probabilities for all edges of which the tree is coprised. Experiet shows that the obtaied tree topology is very close to the optial RST. I the ext sectio, soe backgroud together with a probabilistic odel is described. The, we show the Steier tree costructio algorith i Sectio 3, followed by the experiets give i Sectio 4. Fially, Sectio 5 cocludes the paper. 2. Probabilistic odel for rectiliear steier trees I this sectio, we itroduce soe preliiaries ad propose a probabilistic odel for rectiliear Steier trees Grid graph Cosider a set of N poits, P {,,, p N } i a plae, where the locatio of p i is deoted by (x i, y i ). Assuig x i x j ad y i y j for i j (discussios will be give later o if this is ot the case), we ca costruct a grid graph which cosists of the itersectios (or, segets) of horizotal ad vertical lies through all poits. It was show [3] that oly those segets withi the sallest rectagle eclosig all poits eed to be cosidered i obtaiig the RST. A optial RST is a subset of segets, T, such that T is a tree for give poits ad the total wire legth over all segets i T is iiu. Figure 1 illustrates the grid graph for a set of three poits, P {,, }. A optial Steier tree of Figure 1 is show i Figure 2, where S 1 is a. colu # l 1 l row # Figure 1. The grid graph for a set of three poits If we uber the colus ad rows of the grid graph, the sybol R(i, j) ca be used to represet the horizotal seget which lies o row i ad betwee colus j ad j+1. S 1 Figure 2. A optial Steier tree of Figure 1

2 Siilarly, we use C(i, j) to represet the vertical seget which lies o colu j ad betwee rows i ad i + 1. For istace, the segets l 1 ad l 2 i Figure 1 ca be deoted by R(2, 1) ad C(2, 2), respectively. Note that the rows are ubered fro botto to top i the graph, ad the colus are ubered fro left to right. Before goig further, we have the followig defiitios. Defiitio 1: Give two poits p i, p j P, the distace c(i) c(j) is called the horizotal grid-distace betwee the, where c(i) ad c(j) are the colu ubers of p i ad p j, respectively. Siilarly, the vertical grid-distace betwee the is defied to be r(i) r(j), where r(i) ad r(j) are the row ubers of p i ad p j, respectively. Defiitio 2: If 0 c(j) c(i) 1 for poits p i, p j P, the H( x j x i is called the k-th horizotal physicallegth of the grid graph, where k c(i). If 0 r(j) r(i) 1 for poits p i, p j P, the V(l) y j y i is defied to be the l- th vertical physical-legth of the graph, where l r(i) Probabilistic odel Cosider two poits p i, p j P i the grid graph as show i Figure 3. Without loss of geerality, we assue c(i) < c(j) ad r(i) > r(j). Let I r(j), ad J c(j). The horizotal ad vertical grid distaces betwee p i ad p j are c(j) c(i), ad r(i) r(j), respectively. Let M be the uber of all possible shortest paths fro p i to p j. The uber of those paths which pass through the seget R(I, J 1) (i.e., R 2 i Figure 3) oly depeds o ad, ad is deoted by. The uber of those paths which pass through the seget C(I, J) (i.e., C 2 i Figure 3) is also a fuctio of ad, deoted by. Obviously, we have M. I particular, for ay positive iteger q, we have 1, q) q, 1) 1, ad q, 1) 1, q) q. Fro a statistical poit of view, the probability of a shortest path betwee the two poits passig through R(I, J 1) (or, C(I, J)) is give by /M (or, /M). Furtherore, fro Figure 3, ad ca be writte as: (1) ad G ( (2) or, F ( (3) ad F ( (4) Theore 1: If we defie q, 0) 0, q) 1 for ay iteger q > 0, the ad ca be coputed recursively as follows: K colu c(i) Figure 3. Probabilistic aalysis for the segets through which the shortest paths betwee poits (p i ad p j ) pass k 0 k 1 ad M F ( + 1, + 1) where > 1, ad,, ad M are defied as earlier. Proof : The secod part of the theore is straightforward, ad we prove the first part oly. Addig equatios (3) ad (4) gives 1) (5) or F ( + 1, 1) (6) Fro (1) ad (5), we have + 1) + 1) + 2) L 1) + k 2 Sice 1) ad q, 0) 1, equatio (7) ca be rewritte as Fro (6) ad (8), we have p i R 1 C 1 k 0 colu J c(j) R 2 C 2 p j row r(i) row I r(j), K vertical grid-distaces, L horizotal grid-distaces C 1 vertical seget C(I + K, J L), C 2 vertical seget C(I, J) R 1 horizotal seget R(I + K, J L 1), R 2 horizotal seget R(I, J 1) L (7) (8)

3 + 1, 1) 1 k 0 1 k 0 k 1 k + 1) The values of ad for 6 are show i Table 1 ad Table 2, respectively. Table Table More geerally, let us cosider the specific horizotal seget R (I + k, J l 1) i Figure 3, where 0 k, 0 l 1. Aog all M shortest paths fro p i to p j, the uber of paths passig through this seget is give by l, [l, l, ] l, l + 1, Thus, the probability of a shortest path passig through this seget is expressed as F ( l, k ) F ( l + 1, k ) PR ( I + k, J l 1) (10 ) F ( + 1, Siilarly, the probability of a shortest path betwee p i ad p j passig through the specific vertical seget C (I + k, J l) (refer to Figure 3) is expressed as G ( l, k ) G ( l, k + 1) PC ( I + k, J l) G ( + 1) where 0 k 1, ad 0 l. (9) (11) Whe we accout for the shortest paths for all N (N 1)/2 pairs of poits i the grid graph, two probability atrices, (deoted by PR ad PC), ca be defied to represet the probabilities of all horizotal ad vertical segets, respectively, through which the shortest paths would pass. The eleet PR(i, j) i PR (or PC(i, j) i PC) correspods to a horizotal (or vertical) seget R(i, j) (or C(i, j)) i the graph. PR is a N (N 1) atrix, ad PC is a (N 1) N atrix. The cotributios of each pair of poits to the atrices are deteried by the equatios (10) ad (11). Ituitively, the greater value of a eleet iplies higher probability that the correspodig seget is to be used i obtaiig a shortest path/tree. Therefore, oe ca costruct a optial tree uder statistical sese by fidig a tree such that the su of probabilities of all segets i the tree is axiized. 3. Algorith Based o the probabilistic odel of RST, we describe a algorith for Steier tree costructio as follows. Probabilistic Aalysis Algorith: Ste: Give P {,,, p N }, costruct its grid graph, ad copute the row uber r(i) ad colu uber c(i) for p i, i 1, 2,, N; Ste: Copute the horizotal ad vertical physicallegths, i.e., H( ad V( for k 1, 2,, N 1; Ste: Copute ad, ) for 1, 2,, N, ad 0, 1,, N 1; Ste: Obtai the probability atrices, PR ad PC, usig equatios (10) ad (11) for all N (N 1)/2 pairs of poits; Ste: Noralize all eleets of PR ad PC by settig PR(i, j) PR(i, j) / H(j), 1 i N, 1 j N 1, ad PC(i, j) PC(i, j) / V(i), 1 i N 1, 1 j N ; Ste: Costruct a tree T by selectig the segets oe by oe i the decreasig order of their correspodig probabilities i PR ad PC, ad perforig the followig three operatios durig the selectio: (i) Igore the curret seget if selectig it would leads to a loop which cotradicts the tree defiitio; (ii) Delete all redudat segets oce all poits i P have bee coected by the tree; (iii) Calculate the total wire legth of the obtaied Steier tree after the selectio is copleted. Most of the above algorith is self-explaatory. I particular, the atrix oralizatio i Ste is ecessary sice the segets with sae probability eed to be treated differetly, depedig o their physical legths. The shorter

4 seget is selected first i the tree costructio so that the total wire legth ca be iiized. If there are poits with sae X-coordiate, i.e., H(j) 0, which iplies that o horizotal segets i the j-th colu are required, the we delete the j-th colu of PR (istead of dividig it by H(j) as show i Ste). Siilarly, if soe poits have sae Y- coordiate, i.e., V(i) 0, eaig that o vertical segets i the i-th row are required, the i-th row of PC ca be deleted. I this algorith the ost expesive coputatio occurs i Ste which takes o(n 4 ) tie. We have the followig theore without proof: Theore 2: The tie coplexity of the probabilistic aalysis algorith for Steier trees is o(n 4 ), where N is the uber of give poits. 4. Experiets ad discussios We ipleeted the proposed algorith ad carried out experiets with differet uber of poits for Steier tree costructio. The results idicate that the obtaied Steier trees are better tha those by the previous ethod, or very close to the optial solutios. Several of our test exaples are show i Figures 4 through 6. While the optial RST is ukow i geeral, especially, for the big probles with large value of N, the effectiveess of our approach ca still be evideced by ispectio of these sall-size probles. Particularly, for the case of Figure 6 which was take fro [5], the result due to the algorith fro [5] is show i Figure 6(a) with the total wire legth of 32, copared to the legth of oly 30 by our algorith as show i Figure 6(b). I the followig, we use Figure 6 to deostrate the procedure of our algorith preseted i Sectio 3. Fro Steps 1 ad 2, we ca obtai four vectors which deote the row uber, colu uber, horizotal physical-legth ad vertical physical-legth. They are respectively: r [ ], c [ ], H [ ], ad V [ ]. The values of ad, ) have bee show i Tables 1 ad 2. After perforig Steps 4 ad 5, we have the probability atrices: Figure 5. A exaple with N 11 (ote that replacig R 1 ad C 1 with R 2 ca lead to a better result) p 7 0 R 2 p PR p C 1 R 1 1 (a) total wire legth: 32 (b) total wire legth: 30 Figure 6. Copariso of the results by (a) algorith fro [5] ad (b) our algorith based o probabilistic aalysis Figure 4. A exaple with N PC

5 Fially, Ste gives the Steier tree T {R(2, 1), R(2,2), R(2, 3), R(3, 3), R(3, 4), R(4, 5), R(5, 2), C(1, 4), C(2, 4), C(3, 3), C(3, 5), C(4, 3), C(5, 2)}, as show i Figure 6(b) which turs out to be a optial RST. We clai that while the proposed algorith produces proisig results, it is geerally ot optial. I Figure 5, for istace, a better solutio could be foud by replacig the segets R 1 ad C 1 with the seget R 2 (show as dotted lie). The ai characteristics of our approach are two-fold. First, it is siple to ipleet ad applicable to Steier trees with ay give set of poits. Secod, it ca be exteded easily to the ore geeral Steier probles with the obstacles or blockages (withi their grid graphs) where ay seget is prohibited [8], or with the routig cogested regios where the segets are discouraged [9]. This ca be doe by properly assigig additioal weights to the related eleets of the probability atrices, so that the segets fallig ito these restricted areas would be less likely to be selected i the tree costructio. 5. Coclusios We have described a approach to Steier tree probles based o probabilistic aalysis. The best solutio uder statistical sese has bee obtaied for ay give set of poits. The results are better tha those by the previous techique or very close to the optia. Further work icludes coprehesive study ad test o our odel, so as to obtai the average perforace of the algorith o a large uber of becharks. We also would like to exted the probabilistic odel to applicatios for tiig-drive itercoect optiizatio. 6. Refereces [1] F. K. Hwag, A o(n log N) Algorith for Rectiliear Miial Spaig Trees, Joural of the ACM, vol. 26, pp , [2] M. R. Gray ad D. S. Johso, The Rectiliear Steier Tree Proble is NP-Coplete, SIAM J. Appl. Math, vol. 32, pp , [3] M. Haa, O Steier s Proble with Rectiliear Distace, SIAM J. Appl. Math, pp , [4] J. P. Cohoo, D. S. Richards, ad J. S. Salowe, A Optial Steier Tree Algoriths for a Net whose Terials Lie o the Perieter of a Rectagle, IEEE Tras. Coputer-Aided Desig, vol. 9, o. 4, pp , April [5] R. Codaoor ad I. G. Tollis, A New Heuristic for Rectiliear Steier Trees, i Proceedigs of Iteratioal Syposiu o Circuits ad Systes, 1990, pp [6] F. K. Hwag, O Steier Miial Trees with Rectiliear Distace, SIAM J. Appl. Math, pp , [7] F. K Hwag ad D. S. Richards, Steier Tree Probles, Networks 22, pp , [8] C. J. Alpert, et al, Steier Tree Optiizatio for Buffers, Blockages, ad Bays, IEEE Tras. Coputer-Aided Desig of Itegrated Circuits ad Systes, vol. 20, o. 4, pp , April [9] E. Bozorgzadeh, R. Kaster, ad M. Sarrafzadeh, Creatig ad Exploitig Flexibility i Steier Trees, i Proceedigs of 38 th IEEE/ACM Desig Autoatio Coferece, Jue 2001, pp. -. [10] J. Cog, L. He, C. K. Koh, ad P. H. Madde, Perforace Optiizatio of VLSI Itercoect Layout, INTEGRATION, the VLSI Joural, 21, pp. 1-94, 1996.