Routing on Switch Matrix Multi-FPGA Systems

Transcription

1 Routng on Swtch Matrx Mult-FPGA Systems Abdel Enou and N. Ranganathan Center for Mcroelectroncs Research Department of Computer Scence and Engneerng Unversty of South Florda Tampa, FL Abstract In ths paper, we address the problem of routng nets on mult-fpga systems nterconnected by a swtch matrx. Swtch matrces were ntroduced to route sgnals gong from one channel to another nsde the FPGA chps. We extend the swtch matrx archtecture proposed by Zhu et al. [] to route nets between FPGA chps n a mult-fpga system. Gven a lmted number of routng resources n the form of programmable connecton ponts wthn a twodmensonal swtch matrx, ths problem examnes the ssue of how to route a gven net traffc through the swtch matrx structure. Frst we formulate the problem as a general undrected graph n whch each vertex has one sngle color. Snce there can be at most sx colors n the entre graph, the problem s defned as a search for at most sx ndependent vertex sets of each color n the graph. We propose an exact soluton for ths problem that s sutable only for small sze swtch matrces. For large sze swtch matrces used n mult-fpga systems, we convert the graph-theoretc formulaton to a constrant satsfacton problem. Due to ts large sze, we then model the constrant satsfacton problem as a 0- mult-dmensonal knapsack problem for whch a fast approxmate soluton s appled. Experments were conducted on swtch matrces of varous szes to measure the performance of the proposed approxmate soluton. The results show that the performance of our proposed heurstc mproves wth the ncreasng sze of the swtch matrces. Index Terms -Swtch routng, FPGA archtecture, Interconnecton structures, Global routng, Feld programmable gate arrays, Mult-FPGA systems. Address of Correspondence N. Ranganathan, Ph.D. Dept. of Computer Scence & Engneerng Unversty of South Florda 4202 E. Fowler Avenue, ENB 258 Tampa, FL Offce : (83) E-mal : ranganat@csee.usf.edu

2 . Introducton Wth the ncrease n complexty of modern electronc crcuts, logc verfcaton has become the worst bottleneck n desgn automaton. In spte of the constant mprovements n ther performance, software smulators are stll unable to offer acceptable smulaton tmes for these new large crcuts [2]. To overcome ths dffculty, several researchers proposed hardware acceleraton of software smulaton n order to acheve an ncrease n speed and capacty of smulaton up to 00 tmes [2]. Lately, hardware emulaton was proposed to overcome the verfcaton bottleneck [3]. The ncrease n speed and capacty of smulaton provded by these emulators can reach,000 to 0,000 tmes [4]. Most emulaton engnes consst of an array of FPGA chps nterconnected by a routng structure [5, 6]. Early mult-fpga systems reled on drect routng archtectures where several fxed sets of wres were used to nterconnect the FPGA chps [7]. Other early systems used actual FPGA chps solely for routng nstead of packng logc crcutry, whch dmnshed overall logc utlzaton n these systems. Delay and routng completon s dffcult to predct on these early mult-fpga systems. Lately, several ndrect archtectures were proposed n order to overcome routng unpredctablty n mult-fpga systems. Among these archtectures are the partal crossbar [3] and the swtch matrx [8, 9]. These archtectures are characterzed by a unform and predctable delay. As of routng predcton on these archtectures, ts outcome depends heavly on the routng algorthms used to solve the routng problems assocated wth these archtectures. In ths paper, we present the archtecture of a swtch matrx and examne ts related routng problem. Frst, we formulate the problem as a search for at most sx ndependent vertex sets n an undrected graph. We propose an exact soluton for ths graph-theoretc problem that s sutable only for small matrces. Based on ths graph formulaton, we model the routng problem as a constrant satsfacton problem where a feasble soluton s sought. Snce ts dmensons are too large, we convert the constrant satsfacton problem to a 0- multdmensonal knapsack problem for whch a fast approxmate soluton s proposed. The organzaton of ths paper s as follows: Secton 2 descrbes prevously proposed swtch matrx archtectures and the approaches used to solve the routng problem on these archtectures. Secton 3 ntroduces varous concepts used to defne the graph-theoretcal formulaton of the problem. The 0- mult-dmensonal knapsack formulaton of ths problem s proposed n Secton 4. Secton 5 descrbes the concepts behnd the proposed routng soluton whle Secton 6 explans the conducted experments and the obtaned results. Conclusons are drawn n Secton Related Work Routng structures can have dfferent archtectures. The most popular ones are the partal crossbar [3] and the swtch matrx [9] whose delays are constant. The frst swtch matrx archtecture, known as FPIC, was proposed n [8]. The matrx conssts of an array of horzontal and vertcal channels populated by segmented tracks that span the full wdth of the array. A swtch cell s located at the ntersecton of a vertcal and horzontal segmented tracks. In order 2

3 to establsh a connecton across the array, a seres of up to three swtch cells may be necessary. The swtches have dfferent sze dependng on ther locaton. Ther densty has been optmzed for performance and routablty. The delay n ths archtecture s not necessarly constant but t seems to be upper bounded. Later, ths archtecture was mproved by reducng the sze of the reprogrammable ponts [0]. Ths reducton was acheved by replacng the nne-transstor reprogrammable SRAM cell by a smaller fve-transstor reprogrammable SRAM cell. Ths archtecture s known as FPIC2 where the de area s 40% smaller compared to FPIC s area. In addton, RAM cells used to program the swtch cells were reduced n number by sharng them among sets of swtch cells. It was dscovered that for N swtches n the array, only a small set among the 2 N possble ways of confgurng the array swtches was useful for routng n practce. Ths archtecture uses orthogonal decodng whereby a swtch cell s on only f ts two decoder lnes are hgh. Ths archtecture was later mproved further n [9] by usng non-orthogonal decodng whch results n an ncrease n the number of nets that can be routed through a set of swtch cells. Ths scheme has the advantage of ncreasng routablty throughout the array and decreasng further the SRAM cells to swtch count rato. In [], the authors proposed a swtch matrx archtecture to support routng n segmented channels wthn an FPGA chp. Ths archtecture has the advantage of allowng a connecton from one sde of the matrx to another through only one programmable pont to mnmze delay. The authors studed the desgn of such matrces wthn the context of segmentaton desgn n symmetrcal FPGAs. In [], the authors showed that the dstrbuton of the connecton ponts wthn a swtch matrx has an effect on routng, whch led to address the queston of how to desgn effcent swtch matrces for maxmum routablty and mnmum delay. In order to evaluate the effcency of such archtectures, the authors had to compare routng outcomes for varous net traffcs on these archtectures. Ths requred the desgn of good routng algorthms. To solve routng on these swtch matrces, an approxmate soluton based on a network flow formulaton of the routng problem was proposed [,, 2]. Ths soluton tends to overestmate the routablty of the matrx. Later, the authors n [2] proposed an exact soluton based on an nteger lnear programmng formulaton that s sutable only for small sze matrces. Several classes of swtch matrces were studed for whch effcent approxmate solutons exst [3]. In ths paper, we extend ths archtecture to large matrces to accommodate routng between several FPGA chps n order to emulate or prototype large desgns on mult-fpga systems [4]. 3. Problem Defnton In ths secton, we descrbe the terms and concepts assocated wth the swtch matrx routng problem. We wll adhere to the same termnology used n [] as much as possble. A swtch matrx s a rectangular or square grd of W horzontal and W vertcal tracks. A 5 x 5 swtch matrx s shown n Fgure. Each track conssts of a sngle wre wth termnals at both ends. There are 4W termnals around the grd, whch could be vewed as a box wth W termnals on each sde of the box. Wthn ths grd, there are two types of programmable connecton ponts: the Crossng and the Separatng ponts. 3

4 Separatng pont Crossng pont Fgure. A 5 x 5 swtch matrx. Defnton : A crossng pont s a connecton pont located at the ntersecton of a horzontal and a vertcal track wthn the swtch matrx. The purpose of a crossng pont s to connect a horzontal track wth a vertcal track. Ths can be acheved by turnng on the crossng pont. As long as the crossng pont s programmed n the off state, the two ntersectng tracks are electrcally non-nteractng. Defnton 2: A separatng pont s a connecton pont located anywhere on a sngle horzontal or vertcal track. The purpose of a separatng pont s to connect two segments of the same track. As long as the separatng pont s n the off state, the two segments of the track are electrcally non-nteractng. In order to establsh a connecton between two termnals on dfferent sdes of the matrx, a connecton pont must be turned on. Defnton 3: A straght connecton s a connecton between two termnals on the opposte sdes of the swtch matrx, whch can be establshed by turnng on a separatng pont. There are two types of straght connectons: and 2 types that can be establshed by turnng on the separatng ponts located on the horzontal and the vertcal tracks respectvely. The straght connectons are shown n Fgure 2. 4

5 Fgure 2. Connecton types. Defnton 4: A bent connecton s a connecton between two termnals on the adacent sdes of the swtch matrx, whch can be establshed by turnng on a crossng pont. There are four types of bent connectons: types 3, 4, 5 and 6, whch are shown n Fgure 2. The placement of both types of connecton ponts wthn a gven W x W grd represents the specfcaton of a swtch matrx M. It s requred that at most one separatng pont can be located on any gven track and that an establshed connecton can only use one connecton pont, be t a crossng or separatng pont. In ths context, a connecton forms a path between two termnals. If a set of connectons needs to be establshed across a gven swtch matrx, the set wll be represented by a routng requrement vector (rrv). Defnton 5: A routng requrement vector (rrv) s a sx-tuple n = (n, n 2, n 3, n 4, n 5, n 6 ) where 0 n W and each n represents the number of connectons needed of type for 6. The rrv represents actually the traffc needed to go through the swtch matrx. It s addtonally requred that any two connectons n the rrv across the swtch matrx must not be electrcally nteractng. Ths means that no two connectons should share a termnal. The paths establshed by such connectons must be dsont. Defnton 6: A routng s the establshment of a set of connectons represented by a routng requrement vector n across a gven swtch matrx M wthout volatng the stated requrements. In Fgure 3, an example of a routng for the rrv n = (,,, 2,, ) on the swtch matrx of Fgure 2 s shown. Gven a specfed swtch matrx M and a rrv n, how can one determne whch connecton ponts should be turned on n order to obtan a routng? Ths was defned as the Swtch Matrx Routng Problem(SMRP) n []. 5

6 Separatng pont Crossng pont Fgure 3. Routng of n = (,,, 2,, ). 3. Neghborhood Graph Formulaton In ths secton, we propose a formulaton of the problem based on the ndependence property of vertex sets n undrected graphs. Gven a swtch matrx M and a rrv n = (n, n 2, n 3, n 4, n 5, n 6 ), we buld frst ts neghborhood graph, G = (V, E). A connecton pont can yeld one or more connectons of dfferent types. Defnton 7: The flexblty of a connecton pont s the number of connectons t can yeld. The flexblty of a separatng pont can only be whle that of a crossng pont can be, 2, or 4. In Fgure, the separatng pont on track 0 yelds only one connecton of type and subsequently has a flexblty of. On the other hand, the pont (0, 0) n Fgure yelds two connectons of types 3 and 6 and has a flexblty of 2. Defnton 8: The flexblty set of a connecton pont s the set of connecton types t yelds. For example, the pont (, ) shown n Fgure has a flexblty of 4 and ts flexblty set s {3, 4, 5, 6}. On the other hand, the pont (0, 4) has a flexblty of and ts flexblty set s {4}. We construct the neghborhood graph of a gven swtch matrx M as follows: for each connecton pont, we frst determne ts flexblty set and for each element n ths set, we add a vertex. If there s more than one vertex, we lnk them by edges parwse. The result s a complete graph for each connecton pont. Defnton 9: The graph of a connecton pont s the complete graph resultng from ts flexblty set. 6

7 We wll assocate wth each vertex n the graph of a connecton pont a color defned as follows: Defnton 0: The color of a vertex n a connecton pont s graph represents an element n the connecton pont s flexblty set. Ths means that a vertex can have only one color. Snce there are at most sx possble types of connectons and consequently sx possble elements that can belong to a flexblty set, then there can be at most sx possble colors n the entre graph. Defnton : The complete graph consstng of p vertces, denoted by K p, contans an edge between every par of vertces. For example, the pont (0, 0) n Fgure 2 can be represented by a graph K 2 wth two colors snce ts flexblty set s {3, 6}. It s obvous that the sze of the flexblty set of a gven connecton pont s equal to the number of vertces n ts graph, whch s also equal to the number of colors n the same graph. Defnton 2: Let G =(V, E ) and G 2 =(V 2, E 2 ) be two undrected graphs. The on operaton of G and G 2, denoted by G + G 2, s the graph consstng of the vertex set (V V 2 ) and the edge set (E E 2 (V x V 2 )) where (V x V 2 ) represents the Cartesan product of V and V 2. Ths means that the on operaton of the two graphs forms a new graph whose vertex set has a sze ( V + V 2 ) and edge set has a sze ( E + E 2 + V V 2 ). For clarty, we call a track segmented by a separatng pont as a twosegment track. Otherwse, t s a one-segment track. When two connecton ponts share the same segment of a track, we draw an edge from each vertex n the complete graph representng the frst pont to every vertex n the complete graph representng the second pont. The result s a complete graph (K p + K q ) representng both ponts where () K p and K q represent the complete graphs of the frst and second pont respectvely, and () p and q represent the szes of the flexblty sets of the frst and the second pont respectvely. Ths process s repeated for every par of connecton ponts n M that share a track segment. The resultng graph s the Neghborhood Graph of M denoted by G = (V, E). For llustraton purposes, we descrbe only the neghborhood graph of the connecton ponts (0,0) and (,0). The neghborhood graph of the swtch matrx shown n Fgure conssts of 30 vertces n total and s cumbersome to draw. As was mentoned above, the graph of the pont (0, 0) s K 2. Snce the flexblty set of the pont (, 0) s {3, 4, 5, 6}, then ts graph s K 4. Because ponts (0, 0) and (, 0) are located on the same vertcal track 0, then ther complete graph s (K 2 + K 4 ). Ths graph has (4 + 2) = 6 vertces and ( (4 x 2)) = 5 edges. Table shows the complete graphs of each connecton pont n the swtch matrx shown n Fgure. 7

8 Connecton pont Flexblty set Complete graph Horzontal separatng pont 0 {} K Horzontal separatng pont 2 {} K Vertcal separatng pont {2} K Vertcal separatng pont 3 {2} K Vertcal separatng pont 4 {2} K Crossng pont (0, 0) {3, 6} K 2 Crossng pont (0, 2) {4, 5} K 2 Crossng pont (0, 3) {4} K Crossng pont (0, 4) {4} K Crossng pont (, 0) {3, 4, 5, 6} K 4 Crossng pont (, ) {3, 4} K 2 Crossng pont (, 2) {3, 4, 5, 6} K 4 Crossng pont (2, 0) {3, 6} K 2 Crossng pont (2, 3) {4} K Crossng pont (3, 4) {5, 6} K 2 Crossng pont (4, ) {5, 6} K 2 Crossng pont (4, 4) {5, 6} K 2 Table. Connecton ponts of the swtch matrx shown n Fgure and ther flexblty sets and complete graphs. When a bent connecton s establshed by turnng on a crossng pont C, the two horzontal and vertcal ntersectng segments of C are not avalable anymore for future connectons that may use those two segments. Ths elmnates every connecton pont on those two segments for future connectons. Turnng C on for a connecton of type s equvalent to selectng a vertex v of color n the complete graph of C. By elmnatng every connecton pont on the ntersectng segments of C, we elmnate every adacent vertex to v n the entre neghborhood graph of the swtch matrx M. On the other hand, when a straght connecton s establshed by turnng on a separatng pont S on a horzontal or vertcal track, ths elmnates the two segments on each sde of S from beng used n future connectons. By turnng S on, we select the vertex u n the complete graph of S of color or 2, dependng on whether S s located on a horzontal or vertcal track. Note that the complete graph of a separatng pont s always K, a trval graph consstng of one sngle vertex. When every connecton pont on the track where S s located s no more avalable for future connectons, then every neghbor of u s dsregarded from consderaton for future use. Establshng two connectons of types and n M s equvalent to selectng two non adacent vertces of colors and n the neghborhood graph G = (V, E) of M. Gven a swtch matrx M and a rrv n = (n, n 2, n 3, n 4, n 5, n 6 ), the swtch matrx routng problem s equvalent to fndng sx ndependent vertex sets of color and sze equal to or greater than n, 6, n G = (V, E) of M. 8

9 4. Proposed Soluton of the Neghborhood Graph Formulaton In ths secton, we explan the algorthm used to solve the neghborhood graph formulaton of the routng problem. We are nterested n fndng sx ndependent vertex sets of color and sze n, 6, n the neghborhood graph of M. The pseudocode for the algorthm s lsted below: Input : G = (V, E) the neghborhood graph of M and rrv n = (n, n 2, n 3, n 4, n 5, n 6 ) Output : routng of n on M. Let m = n + n 2 + n 3 + n 4 + n 5 + n 6 Repeat Generate the next vertex set S of sze m from V If S s ndependent then If S contans a subset U where U = n for each 6, then Stop: S s a routng of n on M EndIf EndIf EndRepeat In each teraton, we generate a vertex set of sze m. Then we check f ths set s ndependent. If t s, we check f t has sx subsets of at least sze n for each color. If t does, we have a routng of the rrv n on M. Any algorthm that generates a subset of sze m from a set of sze n can be used to generate the vertex set at each teraton. We used one of the fastest loop-based and easest to mplement algorthms to generate such subsets [5]. If an ndependent vertex set exsts that satsfes a routng of n on M, ths algorthm wll fnd t. However, ths algorthm does not have any way to reach ths set very quckly. If the sets are generated n lexcal order for example, t may have to check a large number of sets before t can fnd a routng. It s also possble that a routng soluton may not exst at all for a gven n and M. In that case, the algorthm wll have to generate each vertex set of sze m from the neghborhood graph. The number of such sets n relatvely small graphs can be qute large. Let the swtch densty be the number of connecton ponts wthn a swtch matrx. After generatng a swtch matrx of sze 5 x 5 wth a swtch densty of 2 x 5 = 30, we obtaned a neghborhood graph whose vertex set V conssts of 76 vertces. If we try to route the rrv n = (2, 5, 6, 4,, 7) on ths matrx, the possble number of vertex sets of sze ( ) = 25 can easly reach = 783,672 x 0 4. It s obvous that ths algorthm s not sutable for large swtch matrces. However, the neghborhood graph formulaton can be easly mapped to a constrant satsfacton problem. 5. Constrant Satsfacton Formulaton In ths secton, we descrbe the constrant satsfacton formulaton of our problem. Let G = (V, E) be the neghborhood graph of the swtch matrx M. Defnton 3: The set V s the set of vertces of color n G where V = U V. 6 = 9

10 When a connecton of type s needed, a vertex from V can be selected. Obvously, the number of connecton of type that can be establshed s lmted. Defnton 4: A bnary varable x s assocated wth each v V n G where: () x = ff v s selected for a connecton () x = 0 otherwse. If n s the number of connectons needed for connecton type, then a set of vertces of sze n from the vertex set V s needed. Ths requrement s the foundaton for a set of constrants called the routng demand constrants: x n, x ³ V, ˆ ˆ 6 ( ) = Snce there are sx possble types of connectons, the constrant satsfacton formulaton requres at most sx routng demand constrants. Defnton 5: The neghborhood of a vertex v V n a graph G = (V, E), denoted by N(v), s the set of vertces adacent to v n G. When a vertex s selected to establsh a connecton, all ts neghbors n the neghborhood graph must be elmnated from consderaton for future connectons as was stated n Secton 3. Ths requrement s the foundaton for a set of constrants called the neghborhood constrants: N ( v ) x + x ˆ N( v ), v ³ V ( 2) v ³ N ( v ) N(v ) represents the cardnalty of the neghborhood of v n G. When a vertex v s selected for a connecton, ts correspondng varable x s set to whle the varables of ts neghbors are set to 0 n order to satsfy the neghborhood constrant. Ths formulaton requres one constrant for every vertex n V that has neghbors. At most V constrants are needed n ths formulaton. Snce there s a varable for each vertex n V, the number of columns s at most V n the constrant satsfacton formulaton. Ths type of constrant generates a square matrx n the problem formulaton, whch s very sutable for the approxmate soluton we use to solve ths routng problem [6]. One can easly generate both set of constrants gven a neghborhood graph G = (V, E) of a swtch matrx M and a rrv n = (n, n 2, n 3, n 4, n 5, n 6 ),. Fndng a routng of n on M s equvalent to fndng a feasble soluton for the constrant satsfacton formulaton of the problem. A varety of approaches from computatonal numercal analyss can be used to solve ths problem formulaton [7, 8]. However, these approaches are useful only for formulatons wth a small number of columns and they may not be able to fnd strctly nteger solutons. If we extend the swtch 0

11 matrx archtecture to mult-fpga systems, a sutable sze can easly start from 00 x 00. For a 200 x 200 swtch matrx wth a 50% swtch densty, the number of columns can easly reach 46,000. At ths range, there are very few commercal lnear solvers that can solve such large problems wthn reasonable tme. Instead we choose to modfy the constrant satsfacton formulaton on whch we apply a fast heurstc to solve the routng problem Multdmensonal Knapsack Formulaton In ths secton, we explan how we modfy the constrant satsfacton formulaton of the routng problem to change t nto a 0- multdmensonal knapsack formulaton. Although the number of neghborhood constrants s much hgher than the number of routng demand constrants, these constrants can be easly satsfed by usng a varety of heurstc approaches for solvng multdmensonal knapsack problems. In addton, the standard multdmensonal knapsack problem does not have constrants smlar to the routng demand constrants [9, 20]. So, we decde to take them off the constrant satsfacton formulaton and express them as coeffcents n an obectve functon. In ths fashon, we transform a feasblty problem nto an optmzaton problem. It s obvous that f n < V, the constrant satsfacton problem s nfeasble. A problem may be feasble f each n V for 6. Defnton 6: The excess of a routng demand constrant s e = (n - V ), for 6. Clearly as e gets smaller, the th routng demand constrant gets tghter and becomes dffcult to satsfy. Ths means that the varables representng the vertces of V n the th routng demand constrant ought to have each a hgher coeffcent n the obectve functon of the mult-dmensonal knapsack formulaton. An approprate yet smple way to represent ths coeffcent s the rato ( / e ). Defnton 7: The obectve coeffcent of the varable x s c = ( / e ) where e s the excess of routng demand constrant and x s the varable representng v V. The constrant satsfacton formulaton can be converted to the followng multdmensonal knapsack formulaton: max = c x N ( v ) x + x ˆ N( v ), v ³ V ( 2) v ³ N ( v )

12 x ³{ 0, } 7. Proposed Soluton of the 0- Multdmensonal Knapsack Formulaton In ths secton, we descrbe the detals of the approach used to solve the multdmensonal knapsack formulaton of the problem. For smplcty, we wll represent the formulaton of the problem n the prevous secton by the followng standard formulaton: max The approach used to solve ths formulaton s based on an early approach proposed to solve the famlar oneconstrant knapsack problem [2]. In ths approach, c s vewed as the proft ganed by choosng the th column, whch s equvalent to settng x =. If we consder a as the weght of column n the capacty constrant, the rato (c / a ) represents ts proft per unt of weght. In that case, choosng the columns n decreasng order of ther (c / a ) ratos wthout volatng the capacty constrant leads to an optmal soluton. A note of cauton s n order at ths pont: ths approach works well for the relaxed verson of the knapsack problem. Stll, t was consstently exploted n later decades as an upper bound for the nteger versons of the problem [20]. Usng ths approach n our formulaton s complcated because the standard formulaton has more than one constrant. = c x a x ˆ b =, 2,..., V = x ³ { 0, } =, 2,..., V In order to solve effcently the standard formulaton, we rewrte t nto the followng canoncal formulaton: where F = a / b. max = c x F x ˆ =, 2,..., V = x ³ { 0, } =, 2,..., V In order to explan the approach used to solve the canoncal formulaton, we can magne each column or x as proect, a as the quantty of resource consumed by proect, and b as the avalable quantty of resource. We are 2

13 nterested n maxmzng the total proft by selectng a set of proects among all proects wthout volatng the resource constrants. At each teraton, we have to select a column to ncrease the total proft untl there are no more columns to select. Ths means that the procedure stops when there are no more resources left to consume [22, 23]. Defnton 8: The necessary resource-quantty vector of column s the vector P = (F, F 2,, F V ) T where F = (a / b ). Intally, no column has been selected yet. If T and T A are the set of all columns and accepted columns respectvely, then T = {, 2,, V } and T A =. In each teraton, we have to select a column that maxmzes the total proft from the set (T T A ). However, we have to keep track of the amount consumed for each resource by the accepted columns n order to mantan each resource constrant n a bndng state. Defnton 9: For a gven resource, the total consumed quantty of that resource by all the accepted proects s C = F. ³ T A For all resources, we need to represent the quantty C n vector format as follows: Defnton 20: The cumulatve total quantty vector of all accepted columns s the vector P A = (C, C 2,, C V ) T = ( F, F 2,, F )T. ³ T A ³ T A ³ T A We can consder the vector P A as a penalty vector to help us select a column. For example f we have two columns, column wth P = (0., 0.3) T and column 2 wth P 2 = (0.3, 0.) T, t s clear that column s preferable to column 2 wth respect to resource snce column consumes less of resource, that s only 0., compared to column 2 whch consumes 0.3. In addton, f the penalty vector n the current teraton P A = (0.5, 0.2) T, then we wll ncur a penalty of 0.5 for every consumed unt of resource. The total penaltes for columns and 2 can be calculated as follows: Penalty for column = P P A = (0., 0.3) T (0.5, 0.2) T = (0.)(0.5) + (0.3)(0.2) = = 0. Penalty for column 2 = P 2 P A = (0.3, 0.) T (0.5, 0.2) T = (0.3)(0.5) + (0.)(0.2) = = 0.7 Thus, the penalty of a column can be determned by smply calculatng the scalar product of (P P A ). Obvously, a column wth a lower ncurred penalty s preferable to another wth a hgher penalty. The prevous example shows that column s preferable to column 2 snce ts total penalty s lower than the total penalty of column 2. But, calculatng the penalty of a column n ths manner depends on the drecton and the magntude of the penalty vector 3

14 P A. Actually, the magntude of P A s not necessary. All we need s ts drecton, whch can be obtaned by calculatng ts unt vector E = (P A / P A ) where P A s ts magntude. Defnton 2: The aggregate necessary resource quantty of a column s U = P E. For column, U represents geometrcally the proected length of the column penalty P on the penalty vector P A. Fgure 3 shows the two aggregate necessary resource quanttes, U and U 2, for the two columns and the penalty vector used for llustraton n the prevous example. Resource U P P A P Resource U 2 Fgure 3. Aggregate necessary resource quanttes for columns and 2. For a column, U can be calculated as follows: U = P E = P (P A / P A ) = (P P A ) / P A V = = [(F, F 2,, F V ) T (C, C 2,, C V ) T ] / C 2 = = F C = C 2 Gven the proft c and the aggregate necessary resource quantty U of column, we can easly calculate ts proftablty. 4

15 Defnton 22: The proftablty or the effectve gradent of column s G = c / U, whch s the proft per unt of aggregate necessary resource. In order to solve the canoncal formulaton of the multdmensonal knapsack problem, one smply has to evaluate the proftablty or the effectve gradent of each column n (T-T A ) and accept the most proftable n every teraton untl there are no more columns to be accepted. Ths algorthm s called the effectve gradent method [23]. The algorthm lstng s as follows: Input : Canoncal form of the multdmensonal knapsack formulaton of the routng problem Output: Set of selected columns ) Let T = {, 2,. V } and T A = 2) Let P A = (0, 0,, 0) T 3) Let α = (,,, ) T 4) Repeat 5) For each column n T whose P α - P A c V 6) If P A = (0, 0,, 0) T then set G = F 7) Else set G = c U = c = = C F C 2 = 8) EndIf 9) EndFor 0) Let column k be the column who has the largest G k. ) Set x k = 2) Remove k from T and nsert t nto T A 3) Set P A = P A + P k 5) EndRepeat 6) If the soluton s not routable, mprove the soluton In the begnnng, the penalty vector P A = (0, 0,, 0) T snce no resource has been consumed yet. In ths case, ts magntude s 0 and subsequently E cannot be computed. So, we set P A = (,,, ) T only for the frst teraton to c gve equal penalty to each resource, whch makes the gradent of each column G = It s possble that the soluton obtaned by the gradent method may not be routable, n whch case the algorthm enters an mprovement phase. The approach behnd ths phase s to check whether t s possble to exchange a column from the set of columns of over-routed connectons for a column from the set of columns of under-routed connectons. Let S be the set of columns that yeld connectons of type. = F. 5

16 Defnton 23: The connecton type s over-routed ff x > n and under-routed ff x < n. ³ S Let O and U be the set of columns for whch x = over all over-routed connecton types and the set of columns for whch x = 0 over all under-routed connecton types respectvely. The mprovement phase proceeds as follows: ³ S Repeat Pck a column from O and set x = 0 Pck a column from U and set x = If at least one constrant s volated then Reset x = and x = 0 EndIf Untl no mprovement s possble If O s an empty set, then there are no columns to exchange and subsequently no mprovement can be realzed. On the other hand, f U s empty, then the obtaned soluton s routable. 8. Experments and Results In ths secton, we descrbe the experments we conducted to test the routng algorthm on large swtch matrces. A set of swtch matrces of varous szes were generated wth a densty of 0.5 to 0.6W 2 each. We selected ths hgh densty only to constran the routng algorthm. In practce, t s possble to obtan good routablty wth lower denstes from 33 to 40% [0]. For each matrx of a gven sze, a set of routng requrement vectors (rrvs) were generated. Then we ran the algorthm on each swtch of a gven sze for each rrv. If the routng algorthm fals to complete the routng, we solved the multdmensonal knapsack formulaton of the routng problem usng the lnear solve XPRESS-MP [] to get an exact soluton. We kept only the routable rrvs. Table shows the swtch matrces and the number of rrvs. Sze Programmable Ponts Columns n canoncal formulaton Rows n canoncal formulaton Number of rrvs 0 x x x x Table. Swtch matrces used n the experments. Column shows the sze of each swtch matrx whle column 2 shows the number of programmable ponts placed wthn the matrx. Columns 3 and 4 show the number of columns and rows n the canoncal multdmensonal knapsack formulaton of the routng problem respectvely. Column 5 shows the number of routng requrement vectors generated and tested on each swtch matrx. 6

17 As the value of each n n each rrv ncreases, routng completon becomes dffcult. For example f (n + n 3 + n 6 ) > W, then routng s obvously nfeasble snce there are only W termnals on the left sde of the swtch matrx []. Smlar reasonng can be appled to () the upper sde of the matrx f (n 2 + n 3 + n 4 ) > W, () the rght sde f (n + n 4 + n 5 ) > W, and () the bottom sde f (n 2 + n 5 + n 6 ) > W. Based on these observatons, we set the values of each n /3W n each rrv. Let m = (m, m 2, m 3, m 4, m 5, m 6 ) be the routng result obtaned by the routng heurstc and d = (n m ) be the routng defct of a connecton type f n > m for a gven rrv n = (n, n 2, n 3, n 4, n 5, n 6 ). Snce our routng problem s ntally a feasblty problem, we gnore the case where the routng heurstc over-routes a connecton type whereby n m and focus only on the under-routed connectons. The relatve routng defct of a connecton n m = - = 6 n type s defned as the rato δ d = = 6. We can measure how close the routng algorthm comes to n completng the routng of the rrv n = (n, n 2, n 3, n 4, n 5, n 6 ) on a gven matrx by computng the total relatve routng defct δ = δ for every n > m. It s obvous that as δ gets close to 0, m = (m, m 2, m 3, m 4, m 5, m 6 ) gets closer to routng feasblty. After gatherng the total relatve routng defcts for each set of rrvs routed on each swtch matrx of a gven sze, the obtaned results are used n a lnear regresson model to derve an estmate of the total relatve routng defcts for swtch matrces of larger szes [24]. One can choose to compare the routng results of our heurstc wth those obtaned by a lnear solver. However, addressng the routng problem on a large matrx by usng a lnear solver to solve ts correspondng multdmensonal knapsack formulaton s qute mpractcal. Below are lsted three hstograms for each swtch sze showng the total relatve routng defct δ for each generated rrv. Hstogram of 0 x 0 swtch Hstogram of 20 x 20 swtch Total Relatve Routng Defct Routng Requrement Vectors Total Relatve Routng Defct Routng Requrement Vectors 7

18 Hstogramof 30 x 30 swtch Total Relatve Routng Defct Routng Requrement Vectors Fgure 4. Hstograms of the total routng defct δ for each rrv and for each swtch matrx sze. The hstogram of the 40 x 40 swtch matrx s not shown because the total routng defct for each generated rrv s 0. Two observatons can be made based on these hstograms. Frst, the rato of nonzero δs to the number of rrvs tends to decrease as the swtch matrx ncreases. For the 0 x 0 swtch matrx, ths rato s (0 / 40) = ( / 4) whle t s (4 / 20) = ( / 5) for the 20 x 20 matrx. Fnally, t drops to (2 / 20) = ( / 0) for the 30 x 30 matrx. Snce δ s 0 n every generated rrv for the 40 x 40 matrx, ths rato s (0 / 20) = 0. Second, the maxmum δ n each hstogram tends to decrease overall as the matrx sze grows. It goes from 0.4 for the 0 x 0 matrx to 0 for the 40 x 40 matrx. Ths suggests that as the problem sze grows, the obtaned routng soluton by the routng algorthms gets closer to the optmal soluton. However, we need to quantfy ths decrease. A v e r a g e T o t a l R e l a t v e R o u t n g D e f c t s Total Relatve Routng Defct y = x R 2 = S w t c h S z e Fgure 5. Plot of the average total relatve routng defct wth regard to the swtch sze. 8

19 Fgure 5 shows a lnear regresson model of how the average δ relates to the problem sze and ts trend equaton. The szes shown n Fgure 5 are multples of 0. For each swtch sze, δ s averaged over all the generated rrvs and plotted for that sze. As s obvous from ths fgure, there s a clear trend whereby δ decreases when the sze of the swtch matrx grows. Ths trend s hghly correlated as R 2 clearly llustrates n the fgure, whch n turn confrms the observaton already made from the hstograms shown above. Table 2 shows the routng results obtaned by routng a set of 20 rrvs on a 50 x 50 swtch matrx whose canoncal formulaton conssts of 25,505 columns and 25,5 rows. RRV δ Total Iteratons Improvement Iteratons CPU Tme (seconds) Average Table 2. Routng results of the 50 x 50 swtch matrx. Column shows the routng requrement vectors generated to test routng on ths swtch matrx whle column 2 shows the relatve routng defct for each rrv. Column 3 shows the total number of teratons t took to route the rrv 9

20 where a column whose gradent s maxmum s selected n each teraton. In column 4, only the teratons nvolved n the mprovement phase of the algorthm are shown. Column 5 shows the CPU tme n seconds to fnd a routng soluton. All these columns are averaged n the bottom row of Table 2. Out of 20 rrvs, routng faled only n the case of rrv 0. It s mpossble to determne feasblty n ths case snce an exact soluton to the routng problem of rrv 0 s dffcult to obtan wth an exact solver. On the overall, the average δ s so small that t llustrates how hghly accurate ths algorthm s. Note that ths average s skewed compared to the medan of δ over 20 rrvs. If the medan s consdered nstead of the average, t wll be 0 n ths case. The number of teratons s very small compared to the number of columns n the problem. It takes around as many teratons as.05% of the number of columns n the canoncal formulaton of the problem to fnd a routng soluton. In addton, only 8.43% of the total teratons are spent n the mprovement phase of the algorthm. Ths shows that ths routng algorthm s very mnmal n terms of the amount of work t expands to reach the soluton, whch explans ts relatvely small CPU tme consderng the sze of the problem formulaton. Intally the tme t takes to execute one teraton s relatvely longer snce the gradent of each column n the problem formulaton has to be computed. As more columns are elmnated, the computaton tme of a sngle teraton becomes ncreasngly shorter and the algorthm becomes faster. Ths routng algorthm can be mproved further f t s used to route nets through extremely large swtch matrces. Instead of selectng only one column from the canoncal formulaton n each teraton, one can choose to select k columns nstead whose gradents are the k largest amongst the computed gradents of all columns. However, the accuracy of the procedure may decrease somewhat f more than one column s selected n each teraton. Ths wll be translated by a notceable ncrease n δ. We conecture that δ wll be very senstve for the frst few values of k >. For larger k, accuracy wll be so degraded that δ wll be constantly hgh. How fast δ reacts to k remans to be verfed through expermentaton. 9. Conclusons In ths paper, we addressed the problem of routng nets through a swtch matrx n mult-fpga systems. Frst, we formulated the routng problem as a search for up to sx ndependent vertex sets n an undrected graph. The soluton of ths frst formulaton s useful only for small matrces. Based on the graph-theoretcal formulaton, we derved a second formulaton dentcal to the mult-dmensonal knapsack formulaton, for whch a fast heurstc based on the gradent method s appled. Ths heurstc s very sutable for larger matrces. The obtaned results from the conducted experments show that ths heurstc procedure s hghly accurate and predctable consderng the sze of the formulaton. It can be easly ntegrated n a physcal layout envronment for mult-chp systems usng the swtch matrx as an nterconnect structure. References [] K. Zhu, D. F. Wong, and Y. W. Chang, Swtch module desgn wth applcaton to two-dmensonal segmentaton desgn, Proc. Internatonal Conference on Computer-Aded Desgn, 993, pp

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