Efficient Obstacle-Avoiding Rectilinear Steiner Tree Construction

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1 Efficint Obstacl-Avoiding Rctilinar Stinr Tr Construction Chung-Wi Lin, Szu-Yu Chn, Chi-Fng Li, Yao-Wn Chang, and Chia-Lin Yang Graduat Institut of Elctronics Enginring Dpartmnt of Elctrical Enginring Dpartmnt of Computr Scinc and Information Enginring National Taiwan Univrsity, Taipi 106, Taiwan ABSTRACT Givn a st of pins and a st of obstacls on a plan, an obstacl-avoiding rctilinar Stinr minimal tr (OARSMT) conncts ths pins, possibly through som additional points (calld Stinr points), and avoids running through any obstacl to construct a tr with a minimal total wirlngth. Th OARSMT problm bcoms mor important than vr for modrn nanomtr IC dsigns which nd to considr numrous routing obstacls incurrd from powr ntworks, prroutd nts, IP blocks, fatur pattrns for manufacturability improvmnt, antnna jumprs for rliability nhancmnt, tc. Consquntly, th OARSMT problm has rcivd dramatically incrasing attntion rcntly. Nvrthlss, considring obstacls significantly incrass th problm complxity, and thus most prvious works suffr from ithr poor quality or xpnsiv running tim. Basd on th obstacl-avoiding spanning graph (OASG), this papr prsnts an fficint algorithm with som thortical optimality guarants for th OARSMT construction. Unlik prvious huristics, our algorithm guarants to find an optimal OARSMT for any -pin nt and many highrpin nts. Extnsiv xprimnts show that our algorithm rsults in significantly shortr wirlngths than all stat-ofth-art works. Catgoris and Subjct Dscriptors B.7. [Intgratd Circuits]: Dsign Aids [Placmnt and Routing] Gnral Trms Algorithms, Prformanc, Dsign Kywords Physical dsign, routing, Stinr tr, spanning tr This work was partially supportd by National Scinc Council of Taiwan undr Grant No s NSC 95-1-E-00-37, NSC 95-1-E , and NSC E PAE. Prmission to mak digital or hard copis of all or part of this work for prsonal or classroom us is grantd without f providd that copis ar not mad or distributd for profit or commrcial advantag and that copis bar this notic and th full citation on th first pag. To copy othrwis, to rpublish, to post on srvrs or to rdistribut to lists, rquirs prior spcific prmission and/or a f. ISPD 07, March 18 1, 007, Austin, Txas, USA. Copyright 007 ACM /07/ $ INTRODUCTION Givn a st of n pins and a st of obstacls on a plan, an obstacl-avoiding rctilinar Stinr minimal tr (OARSMT) conncts ths pins, possibly through som additional points (calld Stinr points), and avoids running through any obstacl to construct a tr with a minimal total wirlngth. Th OARSMT problm bcoms mor important than vr for modrn nanomtr IC dsigns which nd to considr numrous routing obstacls incurrd from larg-scal powr ntworks, prroutd nts, IP blocks, fatur pattrns for manufacturability improvmnt, antnna jumprs for rliability nhancmnt, tc. Consquntly, th OARSMT problm has rcivd dramatically incrasing attntion rcntly [3, 7, 8, 11, 1]. Th rctilinar Stinr minimal tr problm, vn without obstacl considration, is a wll-known NP-complt problm [5]. Th prsnc of obstacls furthr incrass th complxity, and thus most prvious works on th OARSMT problm suffr from ithr poor quality or xpnsiv running tim. As a fundamntal problm with xtnsiv practical applications to routing and wirlngth/congstion/timing stimations in arly IC dsign stags, such as floorplanning and th placmnt, it is dsird to dvlop an ffctiv and fficint algorithm for th OARSMT problm to facilitat th IC dsign flow. Prvious mthods for th OARSMT problm can b classifid into four major catgoris: (1) th maz-routing basd approach, () th nondtrministic approach, (3) th constructionby-corrction approach (calld th squntial approach in [11]), and (4) th connction graph basd approach. Maz routing, first proposd in [9], can optimally rout -pin nts. Howvr, its tim complxity and mmory usag grow prohibitivly hug as th routing ara bcoms largr. Furthr, its multi-pin variants [6, 10] incur unsatisfiabl solution quality sinc thy ar initially dsignd for -pin nts. As a rsult, th abov drawbacks mak th maz-routing basd approach lss popular for modrn applications. Basd on ant colony optimization, Hu t al. [8] prsntd a nondtrministic local sarch huristic to handl small-scal OARSMT problms with complx obstacls of both concav and convx polygons. Although this nondtrministic approach is flxibl in handling complx obstacls, it incurs prohibitivly xpnsiv running tim for larg-scal dsigns. Th construction-by-corrction approach constructs a Stinr or a spanning tr for a multi-pin nt first and thn rplacs th dgs ovrlapping obstacls with dgs around th obstacls. This approach is popular in industry du to its simplicity and fficincy. Howvr, th first stp for th tr construction may not hav th global viw of th obstacls, and thus th scond stp might only rmov th ovrlaps locally around th obstacls. As a rsult, th solution quality may b limitd, as pointd out in [11]. Exampl works in th catgory includ [13] and [3]. Yang t al. [13] prsntd a huristic to rmov th ovrlaps. Vry

2 rcntly, Fng t al. [3] constructd an obstacl-avoiding Stinr tr for an arbitrary λ-gomtry by Dlaunay triangulation. Th last catgory is basd on th connction graph. This approach is to first construct a connction graph by pins and obstacl boundaris, which guarants at last a dsird OARSMT is mbddd in th graph. Thn, som sarch tchniqus ar applid to find th dsird OARSMT from th connction graph. Unlik th construction-by-corrction approach, this approach has a mor global viw of both pins and obstacls. Consquntly, this approach can oftn obtain much bttr solution quality. Nvrthlss, thr xists a trad-off btwn ffctivnss and fficincy in this approach; th largr siz of th connction graph, th highr probability that a bttr OARSMT is mbddd in th connction graph, but th mor xpnsiv running tim. Clarkson t al. [1] considrd only -pin nts and prsntd an O(n(lg n) )-tim algorithm to comput a rctilinar shortst path btwn two pins through polygonal obstacls, whr n is th numbr of pins and obstacl boundaris. Latr, Ganly and Cohoon [4] prsntd an algorithm to find an optimal OARSMT with thr or four pins, but its tim complxity is O(n 4 ). Hu t al. [7] dvlopd an fficint hirarchical huristic to partition all pins into substs, thn connct pins in ach subst, and finally construct an OARSMT using a connction graph-lik approach. Basd on th spanning graph [14] that dos not considr obstacls, Shn t al. [11] rcntly proposd a clvr huristic to construct an OARSMT. In this huristic, an obstacl-avoiding spanning graph (OASG) was first constructd and thn transformd into an OARSMT. Th tim complxity of th OASG construction is O(n lg n), and that of th OARSMT transformation is Ω(n lg n) though not analyzd or xplicitly statd in [11]. This work [11] is ffctiv in gnral, but w obsrv that it misss many ssntial dgs which can lad to mor dsird solutions in th construction of th OASG, rsulting in significant dgradation in th solution quality for many practical cass. Furthr, its OARSMT transformation procdur could also b significantly improvd. In this papr, w construct an OASG with ssntial dgs and prov th xistnc of a rctilinar shortst path btwn any two pins, which is not guarantd in th OASG constructd by [11]. With this proprty, our algorithm guarants to find an optimal OARSMT for any -pin nt and many highr-pin nts. Aftr constructing an initial OARSMT, w dvlop an ffctiv rfinmnt schm for th U-shapd connction in th OARSMT to furthr rduc th total wirlngth. Empirical rsults basd on th last-squar analysis show that our algorithm run in about O(n 1.46 ) tim whil th thortical tim complxity is O(n 3 ). Extnsiv xprimnts basd on tst cass (5 industrial dsigns, 1 tst cass from [3], and 5 largr random dsigns) show that our algorithm significantly outprforms all stat-of-th-art works in th total wirlngth and rquirs comparabl running tim to [11] for practical-sizd problms. Considring th diffrncs from th half-primtr of th bounding box of all pins (which is a lowr bound of th optimal OARSMT solution), th rspctiv avrag improvmnts ar 7.79%, 6.66%, and 5.79%, compard with th rcnt works [3], [1], and [11]. With th compltnss of th OASG construction, in particular, our algorithm also provids ky insights into th sarch for mor dsirabl OARSMT solutions. Th rst of this papr is organizd as follows. Sction formulats th OARSMT problm. Sction 3 prsnts our OARSMT algorithm and its tim complxity. Sction 4 rports th xprimntal rsults. Finally, w conclud our work in Sction 5.. PROBLEM FORMULATION W dfin an obstacl and a as follows: Dfinition 1. An obstacl is a rctangl on th xy-plan. No two obstacls ovrlap with ach othr, but two obstacls could b point-touchd at th cornr or lin-touchd at th boundary. (c) (d) () (f) Figur 1: Any two obstacls cannot ovrlap ach othr, but two obstacls could b point-touchd at th cornr or lin-touchd at th boundary. (c) A pinvrtx may not locat insid any obstacl, but (d) it could b at th cornr or on th boundary of an obstacl. () Any dg of th OARSMT cannot intrsct any obstacl, but (f) it could b point-touchd at th cornr or lin-touchd on th boundary of an obstacl. S Figur 1 for two ovrlappd obstacls, and Figur 1 for point-touchd and lin-touchd obstacls. Dfinition. A is a vrtx on th xy-plan. A must not locat insid any obstacl, but it could b at th cornr or on th boundary of an obstacl. S Figur 1 (c) for an illgal instanc with two pin-vrtics insid an obstacl, and Figur 1 (d) for a lgal instanc with a at th cornr and anothr on th boundary of an obstacl. Lt P = {,,..., p m } b a st of pin-vrtics for an m- pin nt, O = {o 1, o,..., o k } b a st of k obstacls, and V = {v 1, v,..., v n } = P {cornrs in O} b th st of n vrtics for th problm, whr v i has th coordinat (x i, y i ). W hav n m + 4k sinc ach obstacl has four cornrs. Th rctilinar (Manhattan) distanc btwn v i and v j can b computd by x i x j + y i y j. W considr rctilinar (vrtical and horizontal) routs and dfin th obstacl-avoiding rctilinar Stinr minimal tr (OARSMT) problm as follows: Problm: Obstacl-Avoiding Rctilinar Stinr Minimal Tr: Givn a st P of pins and a st O of obstacls on a plan, construct a rctilinar Stinr tr to connct th pins in P, possibly through som additional points (calld Stinr points), such that no tr dg intrscts an obstacl in O and th total wirlngth of th tr is minimizd. Not that no dg of th OARSMT can intrsct with any obstacl, but an dg could b point-touchd at th cornr or lin-touchd on th boundary of an obstacl. S Figur 1 () for a rctilinar Stinr tr intrscting an obstacl, and Figur 1 (f) for tr dgs bing lin-touchd on th boundary of an obstacl. Throughout this papr, w rprsnt th bottom-lft, top-lft, top-right, and bottom-right cornr-vrtics of an obstacl o i by c i,1, c i,, c i,3, and c i,4 with thir coordinats bing (x i,min, y i,min ), (x i,min, y i,max ), (x i,max, y i,max ), and (x i,max, y i,min ), rspctivly. Bsids, C = k i=1 {c i,j}, j = 1,, 3, and ALGORITHM W now prsnt our algorithm. Our algorithm consists of th following four stps: 1. Obstacl-Avoiding Spanning Graph (OASG) Construction: In this stp, an OASG conncting all vrtics in P C is constructd. This stp nsurs that th following stps, xcpt th oprations in Sction 3.4.3, can ignor th obstacls without violating th obstacl-avoiding proprty. S Figur for an xampl OASG construction.. Obstacl-Avoiding Spanning Tr (OAST) Construction: An OAST conncting all pin-vrtics is constructd by slcting dgs from th OASG constructd in St. S Figur (c) for an xampl OAST construction.

3 obstacl turning-vrtx Stinr-vrtx obstacl wast wast (c) (d) (c) (d) () Figur 4: A comparison btwn our OASG and that of Shn t al. Our OASG has th dg (, ) and rsults in an optimal rctilinar connction. (c) Th OASG of Shn t al. dos not contain th dg and (d) rsults in two wastd sgmnts. Figur : Th four stps ( ()) for OARSMT construction. obstacl R of c i, R 3 of c i, R of c i,3 R 3 of c i,3 R of c i,1 c i, c i,3 R 4 of c i,3 R 1 of c i, c i,1 c i,4 R 3 of c i,4 R 1 of c i,1 R 4 of c i,1 R 1 of c i,4 R 4 of c i,4 R R 3 R 1 R 4 obstacl Figur 3: Th dividd rgions for ach of an obstacl and a. 3. Obstacl-Avoiding Rctilinar Spanning Tr (OARST) Construction: An OARST is constructd by transforming ach slant dg of th OAST in St to rctilinar (vrtical and horizontal) dgs. S Figur (d) for an xampl OARST construction. 4. Obstacl-Avoiding Rctilinar Stinr Tr (OARSMT) Construction: Finally, an initial OARSMT is constructd by introducing Stinr points and rmoving ovrlapping dgs of th OARST in St. Thn, a rfinmnt schm for som particular routing shaps is applid to find an OARSMT with a smallr total wirlngth. S Figur () for an xampl OARSMT construction. Th following subsctions dtail th four stps. 3.1 OASG Construction In this stp, w construct an obstacl-avoiding spanning graph (OASG) which is dfind as follows: Dfinition 3. An obstacl-avoiding spanning graph (OASG) is an undirctd connctd graph on th vrtx st P C, whr no dg intrscts with an obstacl in O. W xtnd th spanning graph proposd by Zhou [14] to considr obstacls for th OASG construction. For ach vrtx in P C, w divid th plan into four rgions, R 1, R, R 3, and R 4, as shown in Figurs 3 and. Th division is similar to that in [11], but w construct an OASG with mor ssntial dgs to improv th solution quality. As an xampl shown in Figur 4, our OASG contains th dg (, ) (s Figur 4 ) whil that in [11] dos not (s Figur 4 (c)). Aftr transforming thm to rctilinar connctions, w can obtain an optimal connction as shown in Figur 4, whil th work [11] rsults in a suboptimal solution as shown in Figur 4 (d). As th xampl shown in Figur 5 with r + 1 pin-vrtics, ach obstacl is of -unit high, and th dg (p i, p i+1 ), 1 i r, is of 4-unit long. For this cas, w can rduc th total wirlngth by about 33% ovr th algorithm in [11] and obtain an optimal p r p r+1 4r p r p r+1 (c) (d) Figur 5: Anothr comparison btwn our OASG and that of Shn t al. Our OASG has th dgs (p i, p i+1 ), 1 i r, and rsults in an optimal rctilinar connction with th total wirlngth of 4r. (c) Th OASG of Shn t al. dos not contain ths dgs and (d) rsults in th connction with th total wirlngth of 6r +. solution. In Figur 5, our OASG contains th dgs (p i, p i+1 ), 1 i r, rsulting in an optimal rctilinar connction with th total wirlngth of 4r, as shown in Figur 5. Howvr, th OASG constructd by [11] is illustratd in Figur 5 (c), which dos not contain th dgs (p i, p i+1 ), 1 i r, rsulting in th connction with th total wirlngth of 6r +, as shown in Figur 5 (d) OASG Construction within a Rgion For th OASG construction within a rgion, th nighbors of a vrtx ar dfind as follows: Dfinition 4. A vrtx f P C is a nighbor of a vrtx v P C if no othr vrtx in P C or obstacl is insid or on th boundary of th bounding box of v and f. As shown in Figur 6, c 4,4, c,4, and c 5,4 ar th nighbors of c 6,, but is not bcaus c 5,4 is on th boundary of th bounding box of c 6, and. Our OASG construction is to construct dgs btwn a vrtx v P C and ach of its nighbors. W will focus on R of a vrtx in P C for th discussion, whil th othr rgions ar similarly handld. Not that if th vrtx is at th cornr or on th boundary of an obstacl, it is clar that no dg will b constructd within th rgions blockd by th obstacl. Th algorithm of th OASG construction for R of a vrtx is summarizd in Figur 7. Figur 6 shows an xampl to construct th OASG within th R of c 6,. Aftr th initialization stps (lins 1 3), lin swping is prformd from lft to p r p r+1 4r p r p r+1

4 o c p 3 3,4 o 5 o 1 o 4 o c,4 c 1,4 c 4,4 c 5,4 c 6, o 6 obstacl o c p 3 3,4 o 5 o 1 o 4 c1,4 c 4,4 o c,4 c 5,4 c 6, o (c) obstacl Figur 6: An xampl instanc and th OASG construction for th vrtx c 6, in R of a vrtx. Algorithm: OASG-R (O, P, v, E) Input: O /* th st of obstacls */ P /* th st of pin-vrtics */ v = (x, y) /* OASG is for th R of v */ Output: E /* dgs addd to OASG */ 1 E = A = /* candidat st */ 3 I = /* intrval st as th blocking information */ 4 Prform lin swping from lft to right 5 if it mts l lft boundaris of obstacls, o α1, o α,..., o αl 6 I = I {[y α1,min, y α1,max],..., [y αl,min, y αl,max]} 7 if it mts r right boundaris of obstacls, o β1, o β,..., o β r 8 I = I \ {[y β1,min, y β1,max],..., [y β r,min, y β r,max]} 9 for j = 1 to l 10 if c αj,1 R of v and [y, y αj,min] is not blockd by I 11 A = A {c αj,1} 1 for j = 1 to r 13 if c βj,4 R of v and [y, y βj,min] is not blockd by I 14 A = A {c βj,4} 15 ls if c βj,3 R of v and [y, y βj,max] is not blockd by I 16 A = A {c βj,3} 17 if it mts i pin-vrtics, p γ1, p γ,..., p γi 18 for j = 1 to i 19 if p γj R of v and [y, y γj ] is not blockd by I 0 A = A {p γj } 1 if th swping lin mts v Go to lin 3 3 Sort vrtics in A in th non-dcrasing y-coordinat ordr (For vrtics with th sam y-coordinat, sort thm with th non-dcrasing x-coordinat ordr.) 4 for ach vrtx v A 5 if th vrtx v is a nighbor of v 6 E = E {(v, v )} 7 Rturn E Figur 7: Th algorithm of th OASG construction for th R of a vrtx. right. Whn th lin mts th lft boundary of o 1, th intrval [y 1,min, y 1,max ] is insrtd into th intrval st I as th blocking information (lins 5 6). Whn th lin mts th lft boundary of o, th intrval [y,min, y,max ] is also insrtd into th intrval st I as th blocking information (lins 5 6). At th sam tim, th swping lin mts th, but is not insrtd into th candidat st A du to th intrsction of th blocking information (lins 17 19). Whn th swping lin mts th right boundary of o 1, [y 1,min, y 1,max ] is dltd from th intrval st I (lins 7 8), and c 1,4 is insrtd into th candidat st A (lins 1 14). Similarly, c 4,4, c,4 ar insrtd into th candidat st A (lins 1 14), whil c 3,4 is not du to th intrsction of th blocking information (lins 1 13). Thn, whn th swping lin mts th and th right boundary of o 5, and c 5,4 ar insrtd into th candidat st A (lins 17 0 and lins 1 14). Thrfor, whn th swping lin mts th lft boundary of o 6, th swping lin halts (lins 1 ), and th candidat st A is {c 1,4, c 4,4, c,4,, c 5,4 }. Aftr th sorting (lin 3), th candidat st A bcoms {c 1,4, c 4,4, c,4,, c 5,4 }. Thrfor, c 4,4, c,4, and c 5,4 can asily b dtctd as th nighbors of c 6, (lins 4 5). Finally, (c 4,4, c 6, ), (c,4, c 6, ), and (d) () (f) Figur 8: An xampl OAST construction. (c 5,4, c 6, ) ar insrtd into th st E (lin 6), and th OASG within th R of c 6, is constructd as shown in Figur Proprtis of Pin-Vrtx Shortst Paths W claim that th OASG implis a rctilinar shortst path of any two vrtics in P C, i.., a rctilinar shortst path of any two vrtics can b obtaind by transforming som dgs in th OASG to rctilinar (vrtical and horizontal) dgs. Bsids, ach slant dg is transformd into only on vrtical dg and on horizontal dg. W first dfin th trritory of a vrtx in P C as follows: Dfinition 5. A vrtx g on th xy-plan is in th trritory of a vrtx v P C if no othr vrtx in P C or obstacl is insid th bounding box of v and g. Not that th trritory of a vrtx is not ncssarily a clos rgion. Lmma 1. Givn a sourc s P C, a targt t P C (s t), and any of thir rctilinar shortst paths, RSP (s, t), thr must xist a nighbor f of s such that th rctilinar shortst lngth δ r (s, t) = δ r (s, f) + δ r (f, t). Lmma. Givn a vrtx v P C, for any nighbor f of v, thr must xist an dg btwn v and f in th OASG, i.., a rctilinar shortst path of v and f is implid by th OASG. Du to th limitation of spac, w omit th proofs of Lmma 1, Lmma, and othr thorms throughout this papr. Thorm 1. Th OASG implis a rctilinar shortst path of any two vrtics in P C. 3. OAST Construction W first dfin an obstacl-avoiding spanning tr (OAST) as follows: Dfinition 6. An obstacl-avoiding spanning tr (OAST) is an undirctd tr conncting all pin-vrtics without intrscting with any obstacl. W construct an OAST by slcting som dgs from th givn OASG. As illustratd in Figur 8, th OAST construction consists of thr stps: (1) pin-vrtics shortst path computation, () initial OAST construction, and (3) local rfinmnt Pin-Vrtics Shortst Path Computation For ach dg in th givn OASG, its lngth is dfind as th Manhattan distanc of its two nd vrtics. W apply Dijkstra s shortst-path algorithm [] for ach pair to comput thir distanc, as illustratd in Figur 8.

5 v c v b v a v c v v d v a v b (c) (d) () (f) v v d v c v b v a Figur 9: Thr cass in th OARST construction for a slant dg and its nighboring dg. Th graphs in, (c), and () ar transformd into thos in, (d), and (f), rspctivly. Algorithm: OARST (E i, V o, E o) Input: E i /* th dg st of OAST */ Output: V o /* th vrtx st of OARST */ E o /* th dg st of OARST */ 1 V o = E o = 3 A = E i /* unprocssd dg st */ 4 whil A 5 Slct th longst dg in A /* = (v, v ) */ 6 if is a vrtical dg or a horizontal dg 7 V o = V o {v, v } 8 E o = E o {} 9 ls 10 Slct a nighboring dg of with longst sharing lngth 11 if = NULL or th rlation of and is Cas 1 /* = NULL if has no nighboring dg */ /* Cas 1, Figurs 9 and */ 1 A = A \ {} 13 Dcid v a, v b, and v c 14 V o = V o {v a, v b, v c } 15 E o = E o {(v a, v b ), (v b, v c )} 16 ls if th rlation of and is Cas /* Cas, Figur 9 (c) and (d) */ 17 A = A \ {, } 18 Dcid v a, v b, v c, v d, and v 19 V o = V o {v a, v b, v c, v d, v } 0 E o = E o {(v a, v b ), (v b, v c), (v c, v d ), (v d, v )} 1 ls /* Cas 3, Figurs 9 () and (f) */ A = A \ {, } 3 Dcid v a, v b, v c, v d, and v 4 V o = V o {v a, v b, v c, v d, v } 5 E o = E o {(v a, v b ), (v b, v c ), (v c, v d ), (v d, v )} 6 Rturn (V o, E o ) Figur 10: Th algorithm for th OARST construction. 3.. Initial OAST Construction W thn construct a complt graph for th P pin-vrtics. Th dg wight is dfind as th distanc of its two nd vrtics computd in Sction W thn apply Prim s algorithm [] on th complt graph to obtain a minimum spanning tr (s Figur 8 (c)). By th shortst paths computd in Sction 3..1, w can map ach dg in th minimum spanning tr to a shortst path in th spanning graph, so th initial spanning tr on th spanning graph is constructd (s Figur 8 (d)). It should b notd that shortst paths may shar a common dg. In such a cas, th initial spanning tr on th spanning graph will count it only onc Local Rfinmnt In th initial OAST, thr could b som pairs of vrtics whos corrsponding dgs ar in th OASG, but not in th initial OAST. W add such dgs into th OAST (s Figur 8 ()) and comput th minimum spanning tr on it to rmov unwantd cycls (s Figur 8 (f)). This local rfinmnt may lad to a nw OAST with a smallr total wir lngth. 3.3 OARST Construction In this stp, w transform ach slant dg of th givn OAST into vrtical and horizontal dgs to obtain an obstacl-avoiding rctilinar spanning tr (OARST). c 1 c t p3 p 5 t 1 p 5 p4 p 4 c 1 turning-vrtx c t 5 c 1 c t 5 t 3 t 4 t t p3 3 t 4 t p3 t 1 t 1 p 5 p 4 (c) (d) () Figur 11: An xampl OARST construction. Dfinition 7. An obstacl-avoiding rctilinar spanning tr (OARST) is an undirctd graph conncting all pin-vrtics with vrtical and horizontal dgs. W thn dfin a nighboring dg and its sharing lngth in an OAST as follows: Dfinition 8. A nighboring dg of an dg is an dg which has a common nd vrtx with. Dfinition 9. Th sharing lngth of two dgs 1 and is th summation of th ovrlapping lngths whn 1 and ar projctd to th x- and th y-axs. Thr cass in th OARST construction for a slant dg and its nighboring dg nd to b considrd, in which w tak th common vrtx as th origin on th xy-plan: Cas 1. Th two dgs ar in opposit rgions (s Figur 9 ). In this cas, is transformd into a vrtical dg and a horizontal dg (s Figur 9 ). Thr ar two possibl transformations, so w randomly choos on. Cas. Th two dgs ar in nighboring rgions (s Figur 9 (c)). In this cas, both and ar transformd into a vrtical dg and a horizontal dg. Thr ar svral possibl transformations, so w choos th on with dg ovrlap (s Figur 9 (d)). Cas 3. Th two dgs ar in th sam rgion (s Figur 9 ()). In this cas, using Figur 9 (f) as an xampl, and ar transformd into (v a, v b ) and (v b, v c ), rspctivly. Thr ar two possibl transformations for (v c, v ), and w randomly choos on. Figur 10 summarizs th algorithm for th OARST construction. W us th xampl shown in Figur 11 to xplain th procss. Aftr th initialization stps (lins 1 3), th unprocssd dg st A is {(, c 1 ), (, c 1 ), (c 1, c ), (c, ), (, p 4 ), (, p 5 )} as shown in Figur 11, and th st E o is. In th first itration, (, p 5 ) is slctd as (lin 5), and (, p 4 ) is slctd as (lin 10). Thn, Cas 3 (s Figur 9 ()) is applid, and thy ar transformd into (t 1, p 4 ), (t 1, p 5 ), (t 1, t ), and (t, ) as shown in Figur 11 (lins 1 5). Aftr th first itration, th unprocssd dg st A is {(, c 1 ), (, c 1 ), (c 1, c ), (c, )}, and th st E o is {(t 1, p 4 ), (t 1, p 5 ), (t 1, t ), (t, )}. In th scond itration, (, c 1 ) is slctd as (lin 5), and (, c 1 ) is slctd as (lin 10). Thn, Cas is applid (s Figur 9 (c)); (, c 1 ) is transformd into (, t 3 ) and (t 3, c 1 ), and (, c 1 ) is transformd into (, t 4 ) and (t 4, c 1 ) (s Figur 11 (c)) (lins 16 0). Aftr th scond itration, th unprocssd dg st A is {(c 1, c ), (c, )}, and th st E o is {(t 1, p 4 ), (t 1, p 5 ), (t 1, t ), (t, ), (, t 3 ), (t 3, c 1 ), (, t 4 ), (t 4, c 1 )}. In th third itration, (c, ) is slctd as (lin 5), and (c 1, c ) is slctd as (lin 10). Thn, Cas 1 is applid (s Figur 9 ), and (c, ) is transformd into (c, t 5 ) and (t 5, ) (s Figur 11 (d)) (lins 11 15). Aftr th third itration, th unprocssd dg st A is {(c 1, c )}, and th st E o is {(t 1, p 4 ), (t 1, p 5 ), (t 1, t ), (t, ), (, t 3 ),

6 v v (c) (d) () (f) v 3 v 3 v 4 4 (g) (h) (i) (j) (c) (d) Figur 1: Fiv cass of th ovrlapping dg rmoval. Th graphs in, (c), (), (g), and (i) ar transformd into thos in, (d), (f), (h), and (j), rspctivly. Figur 14: Two cass of th U-shapd pattrn rfinmnt. Th graphs in and (c) ar transformd into thos in and (d) rspctivly. turning-vrtx Stinr-vrtx Stinr-vrtx (c) (d) Figur 15: Whn m = 3, a rctilinar Stinr tr is on of th two topologis: two simpl paths btwn pinvrtics, or thr pin-vrtics connctd to a singl Stinr-vrtx. Figur 13: Th OARSMT construction of Figur 11 (). (t 3, c 1 ), (, t 4 ), (t 4, c 1 ), (c, t 5 ), (t 5, )}. In th fourth itration, (c 1, c ) is slctd as (lin 5). Sinc (c 1, c ) is a horizontal dg, it is transformd into (c 1, c ) dirctly (lins 6 8). Aftr th fourth itration, th unprocssd dg st A is, and th st E o is {(t 1, p 4 ), (t 1, p 5 ), (t 1, t ), (t, ), (, t 3 ), (t 3, c 1 ), (, t 4 ), (t 4, c 1 ), (c, t 5 ), (t 5, ), (c 1, c )}. Finally, th OARST is constructd as shown in Figur 11 (). 3.4 OARSMT Construction In this stp, w construct an obstacl-avoiding rctilinar Stinr tr (OARSMT). Th construction consists of thr stps: (1) ovrlapping dg rmoval, () rdundant vrtx rmoval, and (3) U-shapd pattrn rfinmnt Ovrlapping Edg Rmoval For ach pair of dgs in th OARST, w classify thir rlation into fiv cass as shown in Figur 1, (c), (), (g), and (i), and thn transform thm into thos in Figur 1, (d), (f), (h), and (j), rspctivly. Using Figur 11 () as an xampl, th rsult aftr ovrlapping dg rmoval is shown in Figur Rdundant Vrtx Rmoval A rdundant-vrtx is dfind as follows: Dfinition 10. A rdundant-vrtx is a non- with th dgr of, and th two dgs conncting to it ar paralll. For a rdundant-vrtx, w mrg th two dgs conncting to it. Using Figur 13 as an xampl, two vrtics ar rmovd as shown in th Figur U-Shapd Pattrn Rfinmnt Th total wirlngth can b furthr improvd by som local rfinmnts. Considring th trad-off btwn solution quality and fficincy, w spcially rfin U-shapd pattrns. Th U- shapd pattrn rfinmnt ruls ar dfind as follows: Dfinition 11. A vrtx satisfis th U-shapd pattrn rfinmnt ruls if it is not a, and its dgr is. W nd to considr two cass for th U-shapd pattrn rfinmnt: Cas 1. Svral dgs form th shap as shown in Figur 14. On of th vrtics v 1 and v must satisfy th rfinmnt rul. In this cas, without intrscting any obstacl, th dg is movd as right as possibl, whil dgs 1 and 3 ar still connctd by it. Edgs connctd to a vrtx satisfying th rfinmnt rul ( 1 in Figur 14 ) ar shortnd. Th rsulting rfinmnt is shown in Figur 14. Cas. Svral dgs form th shap as shown in Figur 14 (c). Both vrtics v 1 and v must satisfy th rfinmnt ruls. In this cas, without intrscting any obstacl, th dgs and 3 ar movd as right as possibl, whil dgs 1 and 4 ar still connctd by thm. Th dg 5 is strtchd, but th two dgs connctd to a vrtx satisfying th rfinmnt rul ( 1 and 4 in Figur 14 (c)) ar shortnd. Th rsulting rfinmnt is shown in Figur 14 (d). Aftr th U-shapd pattrn rfinmnt, th rdundant vrtx rmoval is applid to nsur that thr is no rdundant-vrtx in th OARSMT. Using Figur 13 as an xampl, th rsulting rmoval is shown in Figur 13 (c). A Stinr-vrtx is a vrtx which is not a, and its dgr is mor than. W also mark Stinr-vrtics. As an xampl shown in Figur 13 (c), two Stinr-vrtics ar markd (s Figur 13 (d)). Thorm. Th ovrall tim complxity of our algorithm is O(n 3 ) in th worst cas and O(n lg n) for practical applications. Not that n is th total numbr of pin-vrtics and cornrvrtics Optimality W can construct an optimal OARSMT whn th pin numbr m =. Evn for nts with m 3, our algorithm can still achiv optimal solutions in many cass. In th following, w giv thorms for th optimality of our algorithm. Not that ths thorms giv th sufficint but not ncssary conditions for an optimal solution, i.., mor optimal solutions may still b gnratd in othr cass. Bsids, th U-shapd pattrn rfinmnt is not ncssary for ths thorms, implying that our OASG is indd complt to gnrat ths optimal solutions. Thorm 3. If m =, our constructd OARSMT is an optimal solution. Whn m = 3, a rctilinar Stinr tr is on of th two topologis: two simpl paths btwn pin-vrtics as shown in Figur 15

7 Tst HPBB Total Edg-Lngth Improvmnt (%) ( X E X / X E X A ) Cass m k (A) [1] (B) [3] (C) [11] (D) Ours (E) [1] (X = B) [3] (X = C) [11] (X = D) ind / 9.66 ind ,00 10,100 9, / 6.3 ind / 8.00 ind ,11 1, / 0.00 ind ,39 1, / 4.4 rc ,890 6,970 30,410 7,730 6, / / / 8.43 rc ,470 41,700 45,640 4,840 4,10-1. / / /.70 rc ,380 6,380 58,570 56,440 55, / / / 1.86 rc ,850 66,560 63,340 60,840 60, / / / 1.0 rc ,600 80,010 83,150 76,970 76, / / / 1.1 rc , ,750 86,403 83, / / 4.55 rc ,88 181, ,47 113, / / 4.7 rc ,803 0,741 13, , / / 4.46 rc9 00 1,000 19,964 14, , , / / 3.58 rc , , , , / / 0.50 rc11 1, ,984 50,570 38,111 36, / / 0.69 rc1 1,000 10,000 65,4 1,73, ,59 789, / / 7.00 rt ,363,438, / rt ,80 51,981 48, / 9.9 rt ,996 8,783 8, / 6.11 rt ,000 1,985 10,619 10, / 3.63 rt5 00,000 8,097 55,557 53,993.8 / 3.30 Avrag 4.7 / / / 5.79 Tabl 1: Th comparison on th total dg-lngth, whr HPBB is th half-primtr of th bounding box of all pin-vrtics, and mans that th rsult is not availabl. Th rsults bfor / ar th improvmnts on th total dg-lngth, whil thos aftr / ar th improvmnts on th diffrnc from th half-primtr of th bounding box of all pin-vrtics., or thr pin-vrtics connctd to a singl Stinr-vrtx as shown in Figur 15. W can construct an optimal OARSMT for th first topology. Thorm 4. If m = 3 and th topology of an optimal solution contains two simpl paths btwn pin-vrtics, our constructd OARSMT is an optimal solution. Not that non of th aformntiond proprtis is guarantd by th algorithm in [11] du to th missing ssntial dgs, so [11] cannot guarant optimal solutions vn for m =, as illustratd in Figur 4. Also, most nts in a ral cas ar - pin nts or 3-pin nts, which maks th abov proprtis mor important for practical applications. Furthrmor, rgardlss of th topology, w can construct an optimal OARSMT for a 3-pin nt if thr is no obstacl. Thorm 5. If m = 3 and thr is no obstacl, our constructd OARSMT is an optimal solution. Whn m 4, w can also construct an optimal OARSMT which contains only simpl paths btwn pin-vrtics. Thorm 6. If m 4 and th topology of an optimal solution contains only simpl paths btwn pin-vrtics, our constructd OARSMT is an optimal solution. Similarly, this proprty is not guarantd by th algorithm in [11]. 4. EXPERIMENTAL RESULTS W implmntd our algorithm in th C/C++ languag on a Ghz AMD-64 machin with 8 GB mmory undr Ubuntu 6.06 oprating systm. Thr ar totally bnchmark circuits, fiv industrial tst cass (ind1 ind5) from Synopsys, twlv tst cass usd in [3] (rc1 rc1), and fiv random tst cass (rt1 rt5) gnratd by us. W rmovd an ovrlap of two obstacls in rc1 bcaus it is invalid. On th othr hand, th numbr of obstacls is usually much largr than that of pin-vrtics in a ral dsign, so w st th ratios of k and m to 5, 10, and 50 to gnrat th fiv larg random cass. Givn th constraints on th aras and th aspct ratios of obstacls, thir positions, lngths, and widths wr randomly gnratd without ovrlapping ach othr. Bsids, th positions of pin-vrtics wr also randomly gnratd without locating insid any obstacl. W compard our algorithm with thos prsntd in [1], [3], and [11]. Th rsults of [1] ar providd by th authors, and wr gnratd from a Unix workstation with.66 GHz CPU and 1 GB mmory. Th rsults of [3] ar dirctly quotd from th papr, whr th algorithm was prformd on a Sun V880 fir workstation with 755 MHz CPU and 4 GB mmory. W also implmntd th algorithm in [11]. Diffrnt from our OASG graph construction, it only constructs an dg within ach rgion. In addition, it oprats without th U-shapd pattrn rfinmnt as dscribd in Sction W also vrifid th gnratd OARSMTs by anothr program to nsur that all pin-vrtics wr connctd without intrscting any obstacl. Tabl 1 lists th total wirlngths of ths algorithms without any scaling. Considring th diffrncs from th half-primtr of th bounding box of all pin-vrtics, th rspctiv avrag improvmnts on th total wirlngth ar 6.66%, 7.70%, and 5.79%, compard with th algorithms in [1], [3], and [11]. Furthrmor, th improvmnt ovr [11] can b up to 6.3% (for ind). Sinc th half-primtr of th bounding box of all pin-vrtics is a lowr bound for an optimal solution for this OARSMT problm, ths improvmnts ar vry significant. (If w considr th diffrncs from an optimal solution, th improvmnt is vn largr.) In largr tst cass, sinc th half-primtrs of ths cass ar far from thir optimal solutions, th improvmnts sm to b lss than thos of small cass. In fact, considring th prcntags of th rducd lngth, th algorithm is still vry ffctiv, indpndnt of th sizs of tst cass. Figur 16 shows th rsulting layout for th tst cas rt3. Tabl compars th CPU tims of ths algorithms. Our algorithm is sufficintly fficint. For xampl, whn th numbrs of pin-vrtics and obstacls rach 00 and 1,000 rspctivly (rc9), our algorithm taks only 0.91 sconds and achivs 3.58% improvmnt ovr th algorithm in [11]. As shown in Figur 17,

8 Figur 16: Th final routing rsult of rt3, whr a pinvrtx is rprsntd by a solid circl. Tst CPU Tim (scond) # Edgs Cass [1] [3] [11] Ours in our OASG ind1 < 0.01 < ind < 0.01 < ind3 < 0.01 < ind4 < 0.01 < ind5 < rc < 0.01 < 0.01 < rc 1.03 < 0.01 < 0.01 < rc < 0.01 < 0.01 < rc < 0.01 < 0.01 < rc < rc ,58 rc ,99 rc ,33 rc ,505 rc ,445 rc ,546 rc ,091 rt ,358 rt ,44 rt ,447 rt ,83 rt ,938 Tabl : Th comparison on th CPU tim, whr mans that th rsult is not availabl. CPU Tim (scond) CPU Tim of Our Algorithm CPU Tim of [11] Th Last Squars Lin of Ours Th Last Squars Lin of [11] Numbr of Pin-Vrtics and Cornr-Vrtics (thousand) Figur 17: Th CPU tim is plottd as a function of n. th CPU tims of [11] and ours ar plottd as functions of th input siz n. By th last squars fitting on th log-log-axs, th rspctiv slops of th fitting lins ar 1.40 and 1.46, implying that th mpirical tim complxity of our algorithm is clos to O(n 1.46 ) whil that of [11] is about O(n 1.40 ). Not that this is rasonabl sinc w add mor dgs into our OASGs to guarant th optimality dscribd in Sction 3.4.4, whil th work [11] dos not. Furthr, th mpirical tim complxity is far undr th thortical worst-cas complxity of O(n 3 ) in Thorm. Th much lowr mpirical tim complxity can b xplaind by th sizs of our OASGs. Th numbrs of dgs in our OASGs ar listd in th last column of Tabl. By th last squars fitting on th log-log function of th numbr of dgs to th circuit siz, th numbr of dgs in our OASG grows only about O(n 1.03 ) mpirically in th input siz n, which is far undr th thortical worst-cas complxity of O(n ). Th xprimntal rsults show that our algorithm is vry ffctiv and fficint. 5. CONCLUSIONS W hav proposd an algorithm to construct an obstacl-avoiding rctilinar Stinr tr (OARSMT). W can achiv an optimal solution for any -pin nt and nts with mor pins in many cass. Exprimntal rsults hav shown that our algorithm is vry ffctiv and fficint. With th compltnss of th OASG construc- tion, in particular, our algorithm also provids ky insights into th sarch for mor dsirabl OARSMT solutions. 6. REFERENCES [1] K. L. Clarkson, S. Kapoor and P. M. Vaidya, Rctilinar shortst paths through polygonal obstacls in O(nlog 3/ n) tim, Proc. SCG, pp , [] T. Cormn, C. Lisrson, R. Rivst, and C. Stin, Introduction to Algorithms, nd dition, Th MIT Prss, 001. [3] Z. Fng, Y. Hu, T. Jing, X. Hong, X. Hu, and G. Yan, An O(nlogn) algorithm for obstacl-avoiding routing tr construction in th lambda-gomtry plan, Proc. ISPD, pp , 006. [4] J. Ganly and J. P. Cohoon, Routing a multi-trminal critical nt: Stinr tr construction in th prsnc of obstacls, Proc. ISCAS, vol. 1, pp , [5] M. Gary and D. Johnson, Th rctilinar Stinr tr problm in NP-Complt, SIAM Journal of Applid Mathmatics, pp , [6] D. W. Hightowr, A solution to th lin routing problm on th continous plan, Proc. of th 6th Dsign Automation Workshop, pp. 1 4, [7] Y. Hu, Z. Fng, T. Jing, X. Hong, Y. Yang, G. Yu, X. Hu, and G. Yan, FORst: a 3-stp huristic for obstacl-avoiding rctilinar Stinr minimal tr construction, Journal of Information and Computational Scinc, pp , 004. [8] Y. Hu, T. Jing, X. Hong, Z. Fng, X. Hu, and G. Yan, An-OARSMan: obstacl-avoiding routing tr Construction with good lngth prformanc, Proc. ASP-DAC, pp. 7 1, 005. [9] C. Y. L, An algorithm for connctions and its application, IRE Trans. on Elctronic Computr, pp , [10] K. Mikami and K. Tabuchi, A computr program for optimal routing of printd circuit connctors, Proc. IFIPS, pp , 1968, H47. [11] Z. Shn, C. Chu, and Y. Li, Efficint rctilinar Stinr tr construction with rctilinar blockags, Proc. ICCD, pp , 005. [1] Y. Shi, T. Jing, L. H, and Z. Fng, CDCTr: novl obstacl-avoiding routing tr construction basd on currnt drivn circuit modl, Proc. ASP-DAC, pp , 006. [13] Y. Yang, Q. Zhu, T. Jing, X. Hong, and Y. Wang, Rctilinar Stinr minimal tr among obstacls, Proc. ASIC, pp , 003. [14] H. Zhou, Efficint Stinr tr construction basd on spanning graphs, IEEE Trans. Computr-Aidd Dsign, Vol. 3, No. 5, pp , May 004.