Single-Layer Trunk Routing Using 45-Degree Lines within Critical Areas for PCB Routing

Transcription

1 SASIMI 2010 Proeedings (R3-8) Single-Lyer Trunk Routing Using 45-Degree Lines within Critil Ares for PCB Routing Kyosuke SHINODA Yukihide KOHIRA Atsushi TAKAHASHI Tokyo Institute of Tehnology Dept. of Communitions nd Integrted Systems S3 58 Ookym, Meguro ku, Tokyo , Jpn shinod@l.ss.titeh..jp The University of Aizu Shool of Computer Siene nd Engineering, Tsurug, Ikki-mhi, Aizu-Wkmtsu, Fukushim , Jpn kohir@u-izu..jp Osk University Division of Eletril, Eletroni nd Informtion Engineering 2 1 Ymdok, Suit, Osk , Jpn tsushi@si.eei.eng.osk-u..jp Astrt In reent Printed Ciruit Bords (PCB), the design size nd density hve inresed, nd the improvement of routing tools for PCB is required. Although there re severl routing tools tht generte high-qulity glol routings when only horizontl nd vertil segments re used, PCB designers re not stisfied with these tools euse high-density PCBs require segments tht hve ritrry diretions. In this pper, we propose routing method tht mintins the dvntges of tools tht use horizontl nd vertil segments only, while hndling higher density designs y using 45-degree segments to lolly relx the routing density. I. Introdution Printed iruit ord routing is neessry in order to relize onnetions etween elements, nd to hieve the speifitions relted to dely nd noise, for exmple. Sine the qulity of the routing otined y utomti routing tools for PCBs is inferior to the routing hieved y designers, high-density PCBs re still designed y hnd. However, the numer of nets on PCB hs inresed due to the emergene of BGA pkges, whih hve numerous I/O pins, nd the design sle pprohes the limit of mnul design. As the design sle inreses, the time required to omplete PCB design y hnd inreses. Therefore, the qulity enhnement of utomti PCB routing tools is needed in order to shorten the design period. In typil PCB routing design flow, first, in the glol routing phse, the glol struture of the route is determined for eh net group, whih onsists of relted nets s the result of lyer ssignment, re ssignment, et. Then, in the detiled routing phse, the detiled struture of route of eh net group is determined within the ssigned routing re. In the detiled routing phse, the detiled routing prolem of net group is usully divided into two types of suprolems, nmely, espe routing nd river routing. Espe routing is primrily used to find n espe route for eh net from highly ongested res where I/O pins re losely rrnged [1 5]. On the other hnd, river routing is used to onnet the terminls of eh net, whih re rrnged on the oundry of less ongested routing re of single lyer, s well s to hieve vrious speifitions. A typil river routing prolem is modeled s trunk routing prolem in whih the routing re n e ifurted suh tht eh re ontins one terminl of net [6]. The trunk routing prolem is key prolem euse this prolem orresponds to suprolem for typil net group tht onsists of nets onneting two modules. The formultion of the trunk routing prolem is simple, ut n inferior routing pttern, s ompred to mnul design, is often otined y urrent utomti PCB tools. The qulity improvement in the trunk routing prolem is needed in order to improve the qulity of utomti PCB routing tools. In the trunk routing prolem, routing pttern should hieve vrious speifitions nd relize onnetions [6 12]. There re severl routing tools tht generte highqulity glol routings when only horizontl nd vertil segments re used. However, non-orthogonl segments re often essentil in order to relize the onnetion requirements in trunk routing prolem. Tools tht hndle only horizontl nd vertil segments nnot e used for tul PCB routing design. There exist severl routing methods tht n hndle non-orthogonl segments in multi-lyer VLSI routing [13 15]. However, in the routing methods proposed in [13 15], non-orthogonl segments re primrily used to shorten the wire length. These methods do not effetively use nonorthogonl segments to relize onnetion requirements

2 K. Shinod, Y. Kohir, A. Tkhshi when the routing resoures re limited. In the trunk routing prolem, fesile routing pttern tht relizes the onnetion requirement will e otined if the route of net is determined long the oundry of the routing re, inluding non-orthogonl segments, so tht the design rule is stisfied. However, the otined routing pttern will ontin numer of unneessry ends, nd the totl length will e lrge. The qulity might e improved y post-proessing, ut this would require too muh time to otin stisftory routing pttern if the initil routing pttern is too poor. In this work, we propose routing method for trunk routing prolem in whih the onnetion requirements re relized in one routing lyer using 45-degree segments s well s horizontl nd vertil segments. In the proposed routing method, 45-degree segments re used only when they re essentil to relize the onnetion requirements. In the trunk routing prolem, the routing re is divided into ritil zones nd non-ritil zones, where ritil zones require 45-degree segments in order to relize the onnetion requirements, ut non-ritil zones do not. The proposed method identifies the ritil zones y flow nd determines the routing pttern in these zones using 45-degree segments s well s horizontl nd vertil segments without loss of onnetivity. The onnetion requirements re then relized y generting routing pttern of the remining zones, i.e., non-ritil zones, using only horizontl nd vertil segments. The proposed method effiiently genertes fesile routing pttern. Although the hievement of vrious speifitions is eyond the sope of this pper, the proposed method will e helpful for improving vrious indies. Although the 45-degree segments in the ritil zone re invrint in most ses, the segments in nonritil zone re not invrint. The routing pttern in non-ritil zone hs lrge flexiility. The improvement of vrious indies will e hieved y pplying routing tools tht effetively into ount tke vrious indies in non-ritil zones. The routing tools tht hndle horizontl nd vertil segments only re pplile euse 45- degree segments re not required. Of ourse, the routing tools tht hndle 45-degree segments n e used. These routing tools n e used to relize vrious speifitions without worrying out onnetivity issues. II. Preliminries In this pper, single-lyer routing re tht ontins severl ostles is onsidered. In the prolem formultion, the width of eh wire is onsidered to e 0, nd the minimum distne etween djent wires is set to one unit length. In ddition, the minimum distne etween wire nd n ostle is set to one unit length. In the routing re, the trks on whih wires run re defined. All wires run only on these trks. No wire runs on prt of trk on whih n ostle exists. Three types of trks, nmely, horizontl trks (H trks), vertil trks (V trks), nd ±45-degree trks (X trks) re defined. Both H trks nd V trks re defined s hving seprtion of one unit length nd re defined over the entire routing re. The intersetion of n H trk nd V trk is referred to s grid-point. Both H trks nd V trks re referred to s HV trks, nd X Fig. 1. Single-lyer multiple-nets routing prolem trks re defined lolly with the sme seprtion, whih is t lest one unit length. In other words, n HV trk lwys ends t the oundry of the routing re ut n X trk usully ends inside the routing re. The X trks re defined in detil in Setion III. An ostle is ssigned set of grid-points suh tht no wire n seprte the grid-points without violting the design rule. A grid-point is lled off-point if it is ontined in n ostle nd is otherwise lled n on-point. A net onsists of two terminls referred to s soure terminl nd sink terminl. An exmple of routing re is shown in Fig. 1. In this exmple, H trks, V trks, nd X trks re represented y the horizontl, vertil, nd 45-degree lines, respetively. An off-point tht orresponds to n ostle is represented y lk dot. A wire nnot use trk if the trk is in gry region tht surrounds lk dot. A terminl of net is represented y white irle in whih the nme of the net is written. Flow grphs re defined to help nlyze given routing prolem. The flow grph G with edge pity is defined s follows. The vertex set of G onsists of the primry soure, the primry sink, nd verties tht orrespond to on-points. For eh on-point in the routing re, two verties lled the in-vertex nd the out-vertex re inluded in the vertex set of G. The edge set of G onsists of the following four types of direted edges: type-1) direted edge from the in-vertex of n on-point to the out-vertex of the on-point; type-2) direted edge from the out-vertex of n onpoint to the in-vertex of n djent on-point; type-3) direted edge from the primry soure to the in-vertex of the on-point to whih soure terminl is ssigned; type-4) direted edge from the out-vertex of the onpoint to whih the sink terminl is ssigned to the primry sink. The pity of n edge is one if the edge is type-1 edge, nd is infinite otherwise. Unit flows in G do not interset euse the pity of eh type-1 edge is set to one. For exmple, the flow grph with edge pity orresponding to the routing re shown in Fig. 1 is shown in Fig. 2. A prtition (V s,v t )oftheedgesetofv is lled ut if V s nd V t ontin the primry soure nd the primry sink, respetively. The pity of the ut (V s,v t )isthe

3 SASIMI 2010 Proeedings (R3-8) primry soure primry sink Fig. 2. Flow grph with edge pities Fig. 4. An exmple of HVX routing for the prolem of Fig. 1 primry soure Fig. 3. Flow grph with vertex pities primry sink pss long the H, V, nd X trks only nd do not interset. This routing pttern is fesile. In the trunk routing prolem, eh unit flow on the flow grph G orresponds to the route from soure terminl to sink terminl using only H nd V trks. The mximum numer of routes using only H nd V trks n e otined using set of unit flows with the mximum numer. sum of the pities of the edges tht onnet from vertex in V s to vertex in V t. The flow grph G with vertex pity tht orresponds to G is defined in order to simplify the grph drwing. The flow grph G is otined from G y ontrting type-1 edges. The pity of vertex is one if the vertex orresponds to type-1 edge, nd is infinite otherwise. A ut (V s,v t )ofg, hving finite pity is represented y the set of verties of G tht orrespond to the edges of G tht onnet from vertex in V s tovertexinv t. The set of suh verties of G is lled ut of G. For exmple, the flow grph with vertex pity orresponding to the flow grph with edge pity shown in Fig. 2 is shown in Fig. 3. The trunk routing topology ondition is defined in [6] s follows: Trunk routing topology ondition 1. Eh net onsists of two terminls, nmely, soure terminl nd sink terminl. 2. All terminls of nets re set on the oundry of the routing re. 3. The sequene of terminls long the oundry onsists of the sequene S of soure terminls of ll nets nd the sequenet of sink terminls of ll nets, where T is the reverse of S, ndvievers. A routing prolem tht stisfies this ondition is lled trunk routing prolem. The trunk routing prolem is defined s follows: Trunk routing prolem Input: Routing re with ostles. A set of nets tht stisfies the trunk routing topology ondition. Output: Routing pttern tht onnets ll nets without violting the design rule (minimum distne etween wires nd ostles). For exmple, routing pttern tht stisfies the design rule is shown in Fig. 4. In this routing pttern, ll nets III. X trk The X trks defined in prt of the routing re re speified y the pith nd the offset. The routing re is modeled on the XY oordinte plne. The pith of the X trks is the horizontl distne etween X trks, nd the offset of the X trks is the reminder of the y-interepts of the X trks divided y the pith of the X trks, s if they re extended to interset to y-xis. The pith nd offset of the X trks should e defined refully euse they ffet the onnetivity. In multiple lyer routing, n X trk is often defined suh tht it intersets n HV trk t grid-point. It is esy to ple vis tht onnet wires on different lyers if trks interset t grid-points. However, in single lyer routing, n X trk does not neessrily interset n HV trk t grid-point. The pith must e t lest , so tht the distne etween X trks is t lest one. Although the onnetivity eomes mximum when the pith is set to 2, the routing tool my use mlfuntion due to rounding error, for exmple. In the proposed method, the pith is set to 1.5 so tht n X trk intersets n HV trk either t grid-point or t the middle point of djent grid-points. Even though the distne etween X trks is greter thn one, in most ses, mximum onnetivity is hieved. The offset of the X trks lso ffets the onnetivity. Let us onsider the numer of wires tht runs etween the two ostles shown in Fig. 5 for the se in whih the pith of the X trks is 1.5. The routing results for the ses in whih the offset of the X trks is set to 0.0, 0.5, nd 1.0 re shown in Fig. 6, 7, nd 8, respetively. The onnetion requirements re relized only for the se in whih the offset is set to 0.0, in whih se, three wires run etween ostles. In generl, the numer of wires etween ostles t (x, y)nd(x+w+1,y+h+1)(w, h 1) is mximized if the offset is set to (3 (x+y))mod3 2. The numer of wires tht n run etween the ostles using the X trks when the offset is set ppropritely is

4 K. Shinod, Y. Kohir, A. Tkhshi Fig. 5. HV routing. #net=2 pith 1.5 offset 0.5 Fig. 7. HVX routing. offset=0.5, #net=2 pith 1.5 offset 0 Fig. 6. HVX routing. offset=0.0, #net=3 pith 1.5 offset 1.0 Fig. 8. HVX routing. offset=1.0, #net=2 2(w+h) 1 3, wheres the numer of wires tht n run etween the ostles using HV trks is mx(w, h). The onnetivity etween these ostles is improved when 2(w+h) 1 3 > mx(w, h) if the offset of the X trk is set ppropritely. An pproprite offset of X trks depends on the pir of ostles. An offset of X trks tht mximizes the numer of wires etween pir of ostles does not neessrily mximize the numer of wires etween nother pir of ostles. The offset of X trk should e defined lolly in order to hieve the mximum onnetivity. In the proposed method, su-re, referred to s the ritil zone, in whih X trks re essentil in order to relize the onnetion requirements, is identified, nd the offset of the X trk is defined ppropritely for eh su-re. The identifition of the ritil zone is disussed in Setion IV. IV. Critil zone In the following, we onsider routing prolem tht stisfies the trunk routing topology ondition. A line tht divides the routing re into two su-res suh tht one re ontins ll soure terminls nd the other re ontins ll sink terminls is referred to s dividing line. A dividing line orresponds to ut of flow grph G. In fesile routing pttern, the wire of ny net psses ny dividing line. The numer of wires tht n pss dividing line depends on the design style. The mximum numers of wires tht n pss dividing line L using HV trks nd HVX trks re referred to s the pities of L in the HV nd HVX design styles, respetively. Let C HV (L) ndc HV X (L) e the pities of dividing line L in the HV nd HVX design style, respetively. Let n e the numer of nets. If C HV (L) n for ll dividing lines L, then ll nets n e onneted using only HV trks. Wheres, if there exists dividing line L, the pity of whih in the HV design style is less thn n (C HV (L) <n), then not ll nets n e onneted using only HV trks. Even though not ll nets n e onneted using only HV trks, ll nets n e onneted using HVX trks if C HV X (L) n for ll dividing lines L. A dividing line L is sid to e ritil if C HV (L) <n. Not ll nets n pss dividing line L using only HV trks if L is ritil. In order to otin fesile routing pttern, the pities of ll ritil dividing lines should e inresed. However, the numer of ritil dividing lines would eome exponentil. Thus, it is imprtil to inrese the pities of ll ritil dividing lines individully. A dividing line my pss over ostles. A dividing intervl is the intervl of dividing line suh tht oth ends re loted in n ostle or on the oundry of the routing re, whih does not pss over ny ostle. A dividing line onsists of severl dividing intervls. The pity of dividing intervl is defined nlogously. Let C HV (I) ndc HV X (I) e the pity of dividing intervl I in the HV nd HVX design styles, respetively. Let L e ritil dividing line tht onsists of dividing intervls I 0,I 1,...,I m,wherei i is n intervl etween ostles B i nd B i+1 (0 i m). The sequene of ostles B 0,B 1,...,B m+1 is derived from Lf. Theremy exist severl ritil dividing lines, where the sequene of ostles derived is the sme s L. The pities of ritil dividing lines from whih the sequene of ostles B 0,B 1,...,B m+1 is derived re inresed simultneously. Let L e ritil dividing line in whih the order tht psses ostles is the sme s L nd whose pity in the HV design style is minimum mong them. By dopting the HVX design style, if ll nets n pss L, then ll nets n pss dividing lines from whih the sequene of ostles B 0,B 1,...,B m+1 is derived. Let I 0,I 1,...,I m e the dividing intervls of L. Assume tht n>c HV (L) ndn C HV X (L), nd ssume tht the numer of X trks used t L is minimized. The numer of nets tht n pss I i in the HVX design style is t most C HV X (I i ). If the mximum numer of nets pss dividing intervls of L, exept for I i in the HV design style, then the numer of nets tht must pss I i is n C HV (L)+C HV (I i ). Let K(I i )=mx(c HV X (I i ),n C HV (L)+C HV (I i )). The numer of nets tht pss I i is t most K(I i ). Let d HV (g, B i )ethemximumofx-distne nd y- distne etween grid-point g nd ostle B i. Note tht d HV (g, B i ) orresponds to the numer of wires tht n pss etween g nd B i in the HV design style. Let the pity C HV (B i,g,b i+1 ) of grid-point g in terms of ostles B i nd B i+1 inthehvdesignstyleethe mximum numer of wires tht n pss ny dividing intervl etween B i nd B i+1 tht psses g in the HV design style. Note tht C HV (B i,g,b i+1 )=d HV (g, B i )+ d HV (g, B i+1 ) 1. If C HV (B i,g,b i+1 ) K(I i ), then ll nets n pss ny dividing intervl etween B i nd B i+1 tht psses g without n X trk. Otherwise, there exists fesile routing pttern tht uses n X trk in order to pss the dividing intervl etween B i nd B i+1 tht psses g. The set of grid-points g hving pity of C HV (B i,g,b i+1 ) is less thn or equl to K(I i ) is defined s the ritil zone etween B i nd B i+1. If X trks re ppropritely defined in the ritil zone etween B i nd B i+1 (0 i m), then ll of the nets n pss the ritil dividing lines tht pss over

5 SASIMI 2010 Proeedings (R3-8) primry soure primry sink Fig. 9. Flow grph modified from the flow grph shown in Fig. 3 B 0,B i,...,b m+1. V. Proposed HVX routing The proposed routing method onsists of three phses. In the first phse, the proposed method nlyzes the given trunk routing prolem nd extrts ritil zones from the routing re. In the seond phse, routing pttern in eh extrted ritil zone is determined using H, V, nd X trks. In the third phse, routing pttern in the remining routing re is determined using only H nd V trks. A. Extrting ritil zones In the first phse, the proposed method nlyzes the given trunk routing prolem to determine whether fesile routing pttern n e otined. If fesile routing pttern n e otined, the proposed method extrts ritil zones from the routing re. First, flow grph with vertex pity is onstruted from given routing re. Then, minimum ut is otined using flow method[16]. If the pity of the otined minimum ut is lrger thn or equl to the numer of nets, then the proposed method proeeds to the seond phse. Otherwise, the pity of the orresponding dividing line in the HVX design style is lulted. If the lulted pity is less thn the numer of nets, then the proposed method stops nd fils to otin fesile routing pttern. Otherwise, ritil zones in terms of the orresponding dividing line re extrted. For exmple, the minimum ut otined y the flow method tht onsists of verties indited y striped lines is shown in Fig. 3. In this exmple, the two verties re extrted s ritil zone. The pity of the otined minimum ut is then inresed y modifying the flow grph to find other ritil dividing lines. In order to inrese the pity of the otined minimum ut, the flow grph is modified s follows. 1. The verties orresponding to ritil zones nd the edges inident to the verties re deleted. 2. Edges with infinite pity re inserted etween verties tht re djent to the deleted verties. As n exmple, the modified flow grph otined from theflowgrphinfig.3isshowninfig.9. Then, minimum ut in the modified flow grph is otined. This proedure is repeted until the pity of minimum ut of the flow grph eomes t lest equl to the numer of nets or dividing line tht nnot e pssed is found. Fig. 10. An exmple of ritil zone routing B. Critil zone routing In the seond phse, routing pttern is generted for eh ritil zone. In eh ritil zone, X trks re first generted so tht the numer of wires tht n pss the ritil zone is mximized. Let us onsider ritil zone in whih k nets pss (1 k n). The oundry of the ritil zone is divided into the soure terminl side oundry, the sink terminl side oundry, nd the remining oundry, whih orresponds to the oundry of the routing re or ostles. The soure terminl side oundry hs k on-points. The numer of H trks, V trks, nd X trks tht pss the soure (sink) terminl side oundry is k. On-points in the soure terminl side oundry re leled s 1,s 2,...,s k long the oundry. Similrly, the i-th H trk, i-th V trk, nd i-th X trk tht pss the soure terminl side oundry re leled h s i, vs i,ndxs i, respetively. The route of net tht psses the s i is then determined so tht the route onnets s i nd t i using h s i, vs i, x i, h t i, nd vi t. The wire of eh net uses prt of n X trk in the zone nd uses HV trks if neessry. An exmple of routing pttern in ritil zone is shown in Fig. 10. In the first phse, eh ritil zone orresponding to dividing intervl I i is defined ssuming tht K(I i )nets pss I i. However, the numer of wires tht pss I i,whih orresponds to the mount of flow tht psses I i in the finl flow grph, might e less thn K(I i ). If the numer of wires tht pss I i is less thn K(I i ), then eh ritil zone is redefined using the numer of wires tht pss I i insted of K(I i ). C. Non-ritil zone routing In the third phse, the routing pttern outside the ritil zones is determined. In this phse, the routing prolem is divided into severl suprolems y regrding ritil zones s ostles nd y rrnging terminls on the oundries of ritil zones. Eh suprolem stisfies the trunk routing topology ondition nd the onnetion requirement n e relized using only HV trks. Although the routing method used in this phse is not speified, ny routing method n e pplied, even methods tht nnot hndle X trks ppropritely. Finlly, the results for the routing ptterns re merged. The otined routing pttern does not tke vrious indies into ount. Therefore, it might e neessry to modify the otined routing pttern in post proessing, ut this is eyond the sope of this pper

6 K. Shinod, Y. Kohir, A. Tkhshi TABLE I Experimentl results Rtio Design style d1 d2 d3 d4 d HV HVX HV HVX HV HVX HV HVX VII. Conlusion nd future reserh In this pper, we proposes routing method for trunk routing prolem in whih the onnetion requirements re relized in one routing lyer using 45-degree segments s well s horizontl nd vertil segments. In the proposed routing method, 45-degree segments re used in ritil zones only. The proposed method extrts every ritil zone effiiently nd then otins route in the ritil zone using horizontl segments, vertil segments, nd 45-degree segments. In the non-ritil zone, ny routing tool tht n hndle horizontl nd vertil segments only n e used to improve vrious speifitions euse the proposed method gurntees the onnetivity. In the future, we intend to enhne the proposed method to enle vrious evlutions, suh s length, ends, nd dely, to e tken into ount. In ddition, we will develop PCB routing system tht uses the proposed method s suroutine. Fig. 11. An exmple of the routing pttern VI. Experimentl results The proposed routing method is implemented in C++, whih is ompiled with g4.2.4 nd exeuted on PC hving 2.66-GHz Intel Core2 CPU nd 4 GB of RAM. Theproposedmethodisppliedto20smpledtsets in whih ostles re rndomly generted in the middle of the routing re. The ostle rtio in the middle of the routing re is hnged from 0.05 to 0.20 in 0.05 inrements. The numer of nets in eh smple dt set is 20. In Tle I, for eh smple dt set, the numers of nets tht re ompleted in HV design style nd HVX design style y the proposed method re shown. Bold font is used if the numer of nets tht re ompleted in the HVX design style is lrger thn tht in the HV design style. Figure 11 shows the routing pttern of d2, the ostle rtio of whih is 0.05, otined y the proposed method. In the figure, grid-points tht orrespond to the minimum ut nd grid-points tht orrespond to the oundry of the orresponding ritil zone re represented y smll irle nd lrge irle, respetively. If the ostle rtio is etween 0.05 nd 0.15, then the numer of nets tht n onnet often inrese with the use of X trks. The effet of the X trk inreses when ostles re pled digonlly nd when the distne etween the ostles is reltively lrge. In this experiment, smll ostles re rndomly rrnged in smll re. The ondition of this experiment ppers to e reltively severe ompred to the requirements in tul PCB design. These results indite tht the proposed method is promising for pplition in tul PCB design. Referenes [1] Y. Kjitni, The Potentil Router, Tehnil Report of IEICE, VLD , vol.106, no.548, pp.61 66, (In Jpnese). [2] M. Ingi, Y. Tkshim, nd Y. Kjitni, Espe Fitting etween Pir of Pin-Sets, Tehnil Report of IEICE, VLD , vol.106, no.548, pp.67 72, (In Jpnese). [3] T. Yn nd M. Wong, Untngling Twisted Nets for Bus Routing, Proeedings of Interntionl Conferene Computer-Aided Design, pp , [4] L. Luo nd M. Wong, Ordered Espe Routing Bsed on Boolen Stisfiility, Proeedings of Asi nd South Pifi Design Automtion Conferene (ASP-DAC), pp , [5] L. Luo nd M. Wong, On using st to ordered espe prolems, ASP-DAC 09: Proeedings of the 2009 Asi nd South Pifi Design Automtion Conferene, Pistwy, NJ, USA, pp , IEEE Press, [6] Y. Kohir nd A. Tkhshi, CAFE router: A Fst Connetivity Awre Mutiple Nets Routing Algorithm for Routing Grid with Ostles, Proeedings of Asi nd South Pifi Design Automtion Conferene(ASP-DAC), pp , [7] R. Pinter, On Routing Two-Point Nets Aross Chnnel, Proeedings of Design Automtion Conferene, pp , [8] C. Hsu, Chnnel River Routing Algorithm, Proeedings of Design Automtion Conferene, pp , [9] T. Lenguer, Comintoril lgorithms for integrted iruit lyout, John Wiley & Sons, In., New York, NY, USA, [10] M. Ozdl nd M. Wong, A Length-Mthing Routing Algorithm for High Performne Printed Ciruit Bords, IEEE Trnstions on Computer-Aided Design of Integrted Ciruits nd Systems, vol.25, no.12, pp , [11] M. Ozdl nd M. Wong, Algorithmi Study of Single-Lyer Bus Routing for High-Speed Bords, IEEE Trnstions on Computer-Aided Design of Integrted Ciruits nd Systems, vol.25, no.3, pp , [12] T. Yn nd M. Wong, BSG-route: length-onstrined routing sheme for generl plnr topology, IEEE Trnstions on Computer-Aided Design of Integrted Ciruits nd Systems, vol.28, no.11, pp , [13] N. Ito nd H. Ktgiri, Digonl Routing in High Performne Miroproessor Design, Proeedings of Asi nd South Pifi Design Automtion Conferene (ASP-DAC), pp , [14] C. Ching, Q. Su, nd C. Ching, Wirelength Redution y Using Digonl Wire, Proeedings of Gret Lkes Symposium on VLSI, pp , [15] S.L. Teig, The X rhiteture: not your fther s digonl wiring, SLIP 02: Proeedings of the 2002 interntionl workshop on System-level interonnet predition, New York, NY, USA, pp.33 37, ACM, [16] J.A. Bondy nd U.S.R. Murty, Grph Theory with Applitions, Elsevier Siene Ltd,