Average Distance and Routing Algorithms in the Star-Connected Cycles Interconnection Network

Transcription

1 Average Dstance and Routng Algorthms n the Star-Connected Cycles Interconnecton Network Marcelo M. de Azevedo, Nader Bagherzadeh, and Martn Dowd Department of Electrcal and Computer Engneerng Unversty of Calforna, Irvne Irvne, CA Shahram Latf Department of Electrcal and Computer Engneerng Unversty of Nevada, Las Vegas Las Vegas, NV Aprl Techncal Report ECE

2 Average Dstance and Routng Algorthms n the Star-Connected Cycles Interconnecton Network Marcelo Moraes de Azevedo, Nader Bagherzadeh, and Martn Dowd Dept. of Electrcal and Computer Engneerng Unv. of Calforna Irvne, CA mazevedo, nader, Phone: (714) FAX: (714) Shahram Latf Dept. of Electrcal and Computer Engneerng Unv. of Nevada Las Vegas, NV latf@jb.ee.unlv.edu Phone: (70) FAX: (70) Abstract The star-connected cycles (SCC) graph was recently proposed as an attractve nterconnecton network for parallel processng, usng a star graph to connect cycles of nodes. Ths paper presents an analytcal soluton for the problem of the average dstance of the SCC graph. We dvde the cost of a route n the SCC graph nto three components, and show that one of such components s affected by the routng algorthm beng used. Three routng algorthms for the SCC graph are presented, whch respectvely employ random, greedy and optmal routng rules. The computatonal complextes of the algorthms, and the average costs of the paths they produce, are compared. Fnally, we dscuss how source-based and dstrbuted versons of the algorthms presented n ths paper can be used n assocaton wth wormhole routng. Key words Star-connected cycles graph, average dstance, routng, nterconnecton networks, parallel processng. 1 Introducton An nterconnecton network s characterzed by four dstnct aspects: topology, routng, flow control, and swtchng [13]. The topology of a network defnes how the nodes are nterconnected by lnks, and s usually modelled by a graph. Routng determnes the path selected by a packet to reach ts destnaton, and s usually specfed by means of a routng algorthm. Flow control deals wth the allocaton of lnks and buffers to a packet as t s routed through the network. Swtchng determnes the mechansm by whch data s moved from an ncomng lnk to an outgong lnk of a node (e.g., store-and-forward, crcut swtchng, vrtual cut-through, and wormhole routng are examples of swtchng technques found n parallel archtectures). In ths paper, we contnue the study of topologcal and routng aspects of the star-connected cycles (SCC) nterconnecton network [1], whch was recently proposed as an attractve extenson of the star graph [, 3]. An SCC graph s related to a star graph n the same way a cube-connected Ths research was supported n part by Conselho Naconal de Desenvolvmento Centífco e Tecnológco (CNPq - Brazl), under the grant No. 0039/9-1. 1

3 cycles graph [14] s related to a hypercube [15]. Namely, an SCC graph s formed from a star graph by replacng the nodes of the latter wth cycles or rngs of nodes. The SCC graph consttutes an effcent archtecture for executon of parallel algorthms, whch nclude broadcastng [4] and FFT [16]. Mesh algorthms are also supported n SCC graphs va embeddngs [5]. The SCC graph nherts many of the nterestng propertes of the star graph [3], whle employng at most three I/O ports per node. Ths last aspect categorzes the SCC graph as a bounded-degree network (other examples are n [14, 17]). Networks wth bounded degree favor area-effcent VLSI layouts, and scale more easly than varable-degree networks. Prevously known topologcal aspects of SCC graphs nclude degree, symmetry, dameter, and fault-dameter, and were derved n [6, 1]. Here, we contnue the study of these by nvestgatng the average dstance (or average dameter) of SCC graphs. Our nterest n ths property s twofold: 1) to obtan a metrc for comparng the performance of routng algorthms, and ) to provde contnued characterzaton of the graph theoretcal aspects of SCC networks. In the absence of other network traffc, modern swtchng technques (e.g., wormhole routng [8]) acheve a communcaton latency whch s vrtually ndependent of the selected path length [13]. In ths deal envronment, the two factors whch contrbute to the communcaton latency experenced by a packet are the start-up latency and the network latency [13]. In a realstc envronment n whch congeston occurs, however, a thrd factor known as blockng tme also contrbutes to the communcaton latency. Regardless of the flow control and swtchng mechansms beng used n the network, congeston can usually be mnmzed f fewer lnks are used when routng a packet [7]. For communcatonntensve parallel applcatons, the blockng tme (and, consequently, the communcaton latency) s expected to grow wth path length [7]. In such cases, a routng algorthm should deally compute paths whose average cost matches the average dstance of the network. In ths paper, we show that routes n an SCC graph may contan up to three classes of lnks, whch we refer to as lateral lnks, local lnks, and local lnks (see Secton 3 for defntons). Exact expressons for the average number of lateral lnks and local lnks between two nodes n an SCC graph, and an upper bound on the average number of local lnks, are derved. When combned, these expressons produce a tght upper bound on the average dstance of the SCC graph. We show that the number of local lnks s affected by the routng algorthm beng used, and propose three dfferent algorthms for the SCC graph: random, greedy, and optmal routng. The proposed routng algorthms are compared accordng to crtera such as computatonal complexty (whch affects ther mplementaton n hardware) and average routng cost, for whch fgures were obtaned by means of smulaton programs. The results obtaned wth the optmal routng algorthm provde exact numerc solutons for the average dstance of SCC graphs. Our smulatons ndcate that the greedy routng algorthm performs close to the optmal routng algorthm, whle requrng a smaller complexty. We show that the random routng algorthm presents the smallest complexty among the three algorthms descrbed n ths paper, and provde average and worst-case routng cost metrcs for t. Fnally, we dscuss how the three algorthms can be mplemented n combnaton wth wormhole routng [8].

4 Background.1 The star graph An n-dmensonal star graph, denoted by, contans nodes whch are labeled wth the possble permutatons of dstnct symbols. In ths paper, we use the ntegers 1,,, n to label the nodes of. A node s connected to! dstnct nodes, respectvely labeled wth permutatons "#$% '&(*), +-,/.0,/ (.e., 1 s the permutaton resultng from exchangng the symbols occupyng the frst and the.('3 poston n ) [3, ]. Each of these 45! possble exchange operatons s referred to as a generator of. Two nodes and 1 of " are connected by a lnk ff there s a generator 67 such that 98:6;<. The lnk connectng and s referred to as an. '3 -dmenson lnk and s labeled. =. has 0-! "8 *>!+7 lnks. Fgure 1 shows "?. " c d b a c b d Fgure 1: A 4-star graph = 4? 3 3 s a regular graph wth 0ABC and dameter D 0FEHGBI!J>!+LK. s vertex- and edge-symmetrc, and has herarchcal structure. The degree and dameter of are sublogarthmc on the sze of the graph [3], whch makes the star graph compare favorably wth the hypercube.. The star-connected cycles (SCC) graph An n-dmensonal SCC graph, denoted by NM0MO, s a bounded-degree varant of = NM0MN [1]. s obtaned by replacng each node of wth a rng of C! nodes, whch we refer to as a supernode. The connectons between nodes nsde the same supernode are referred to as local lnks. Each supernode s connected to <! adjacent supernodes, usng lateral lnks accordng to the topology of. Fgure shows NM0MP?. 3 4 a "

5 The nodes n each rng are dentfed by a label., where. s an nteger such that +,.N, and s a permutaton of symbols. Then two nodes. and. are connected by a lnk (.. ) n NM0MN ff ether 1. (.. ) s a local lnk,.e. < and mn.=5. 9#.=5. C, or. (.. ) s a lateral lnk,.e..#. and dffers from only n the frst and the.'3 symbols, such that J! < '. and. 'J!. For smlarty wth, the label of the supernode contanng nodes + s. Also, the lateral lnk connected to node. s labeled.. For smplcty, supernode and lateral lnk labels are not shown n Fg.. NM0MN contans NM0MP B!8 nodes, B!8 = local lnks, and B!8 *>!+7 lateral lnks. Thus, the sze of s comparable to that of ) NM0MO. Local lnks account for /3 of the lnks of, and can be lad out very effcently due to the rng topology of the supernodes. Moreover, NM0M has about NM0MN = tmes fewer lateral lnks than ), whch further reduces the complexty of a VLSI layout for when compared to ) NM0MN NM0M. s vertex-symmetrc, and has <+ (for NM0MN <G ), N<G (for ). In addton, the dameter of NM0M s gven by [1]: NM0MN D N for <G 9! for even G;9#"7 for odd %$ (1) c (4,314) (4,134) (3,134) (,134) (,134) (4,314) (3,314) (4,134) (4,431) (4,134) b (,314) (,314) (3,134) (,341) (3,314) (4,134) (,431) (3,431) (3,134) (,314) (4,341) (,134) (4,314) (3,134) (3,314) (4,314) (,314) (,134) (3,341) (3,143) (,143) (,413) d (4,143) a (4,413) (3,413) (3,314) (3,431) (3,341) (4,341) b (4,341) (,341) (,341) (3,341) (3,431) (,413) (,413) (4,413) c (3,413) (3,143) (4,143) (,143) (,143) (4,431) (,431) (,431) (3,431) (3,341) (,341) (4,431) (4,413) (3,143) (,431) (3,413) (,143) (,413) (3,413) (4,143) (3,143) a (4,431) (4,341) (4,143) (4,413) d Fgure : A 4-SCC graph ( NM0MP? ) 4

6 % 3 Average dstance of the SCC graph 3.1 Prelmnares Let the cost of a route between node. and the dentty node. N + + n NM0MN be denoted by 4, where and respectvely denote the number of lateral lnks and the number of local lnks n. Because NM0MO s vertex-symmetrc, ts average dstance can be computed by fndng mnmal cost routes to the dentty from every node n the graph, and averagng those over 9! 8!. Before we can derve the average dstance of NM0MO, some defntons related to lateral lnks are needed. We may organze the symbols of permutaton as a set of r-cycles.e. cyclcally ordered sets of symbols wth the property that each symbol s desred poston s that occuped by the next symbol n the set. We assume n ths paper that all r-cycles are wrtten n canoncal form [10],.e. the smallest symbol appears frst n each r-cycle. A permutaton C+ $ 7G labelng a supernode of NM0M, for example, can be wrtten n cyclc format as (1 6)(3 5)(4). Note that any symbol M already n ts correct poston appears as a 1-cycle. Let #.#. #. be an r-cycle ncluded n permutaton (+B,B, ). Let B8 be the permutaton produced from M by movng the symbols n (.e.,.. N. ) to ther correct postons. The executon of an r-cycle M s, by defnton, a mnmal sequence of lateral lnks, leadng from supernode to supernode 98. can be expressed by [9, 11]: In the case.... f. C ()...J&.. f. C C M, can actually be executed wth dfferent sequences of lateral lnks [9, 11]: C...J&..F.....F8 8 8F...J&.J&. (3) As shown n [3], the mnmum number of lateral lnks n a route from supernode to s: f the frst symbol n N s 1 (4) -+ f the frst symbol n s not 1, where s the number of r-cycles of length at least n and s the total number of symbols n these r-cycles. It s shown n [3, 9] that does not depend on the order chosen to execute the r-cycles n. Routes n NM0MP often consst of sequences of lateral lnks nterleaved wth local lnks. In what follows, we gve some defntons that relate to local lnks. Recall that denotes the contrbuton of the local lnks to the total cost of a route from. to.. can be further dvded nto two components, whch we denote by and, and defne as follows: r-cycles provde a convenent means to represent permutatons [10] and should not be confused wth physcal cycles or rngs, whch consttute the supernodes of!"!$#. Note that local lnks are not an ssue here. 5

7 the number of move-n (MI) local lnks exstng n the route from. to.. By defnton, these are local lnks that must be traversed between two lateral lnks belongng to the executon sequence of an r-cycle n. the number of move-between (MB) local lnks exstng n the route from. to.. By defnton, local lnks are: 1) local lnks that must be traversed between the executons of two consecutve r-cycles n, ) local lnks that must be traversed n supernode, and are requred to move from. to the lateral lnk that ntates the executon of the frst r-cycle of, and 3) local lnks that must be traversed n supernode, and are requred to move from the lateral lnk that fnshes the executon of the last r-cycle of to.. Thus, B. As an example, consder routng from G G + $ to + +;G $ n NM0M. The cyclc representaton of permutaton 3415 s (1 3)( 4)(5). One possble route uses the sequences of lateral lnks + +7 and G7. Fgure 3 shows the lnks and the local lnks n such a route Supernode labels Legend: Source node Destnaton node Lateral lnk MI local lnk MB local lnk Fgure 3: and local lnks n a route n NM0M 5 local Note that from the topologcal vewpont there s no dstncton between NM0M and local lnks. A partcular local lnk exstng n the route between two nodes of s consdered to be ether an or an local lnk, dependng on the condtons stated above. Therefore, the same local lnk can be classfed as an local lnk for some routes, and as an local lnk for others. The cost components,, and NM0M exst n the route between any node n and the dentty node (although n some short routes one NM0M# or more of these components may be null). Therefore, one can derve the average dstance of by computng the average numbers of lateral lnks, local lnks, and local lnks n a route from. to.. We denote such average numbers by,, and, respectvely. The average dstance of NM0M, NM0MP denoted by D, can then be expressed by: NM0MN D (5) Fnally, the average number of local lnks exstng n a route from. to. n NM0MN s, by defnton,. 6

8 M M M M M 3. Average number of lateral lnks The number of lateral lnks n the route between any node of NM0M# and the dentty node s exactly equal to the cost of the correspondng route n the underlyng n-star graph [1]. Therefore, s exactly equal to the average dstance of, whch s gven by [, 3]: < +! where 3.3 Average number of local lnks The number of s the nth Harmonc number [10]. (6) local lnks n the route between two nodes n NM0M# can be calculated as follows. Consder routng from. to the dentty node., and let the number of r-cycles of length at least n be. Let M A.#.B #. be one of these r-cycles, and let be an executon sequence for M (Eq. ). Movng between two consecutve lateral lnks.,. n requres.. local lnks, where [1]: The total number of by... N5. 9. N5. (7) local lnks that must be traversed durng the executon of M, denoted H, s therefore the sum of the dstances.. between all pars of consecutve lateral lnks.. n : HN. &... f. I. &. f.#i Lemma 1 The number of local lnks that must be traversed n the route between any two nodes of NM0MN s ndependent of the order chosen to execute the r-cycles exstng between those nodes. Proof : Wthout loss of generalty, let the two nodes be. and. M. Let.N. N. be an r-cycle of, +,9,. We frst show that H does not depend on the sequence of lateral lnks chosen to execute M. If., there s only one such sequence (Eq. ). If. C, there are dfferent possble sequences (Eq. 3). However, due to the cyclc nature of these sequences, they all have the same cost (Eq. 8). By extenson, the total number of local lnks n the route,, must also be an nvarant. An mmedate consequence of Lemma 1 s that the number of local lnks between two nodes of NM0MN can be derved wthout further consderatons about routng. (Assumng, of course, that routng s accomplshed n adherence to Eqs. and 3, as s the case wth all routng algorthms presented n ths paper.) As an example, consder an r-cycle M C+ cycle can be executed wth a sequence of lateral lnks I+ requred n the executon of ths sequence s 'N + (8), and let. Such an r- +7. The number of local lnks

9 M Theorem 1 The NM0MP average number of par of nodes n s: local lnks that must be traversed n the route between a 9! Proof : The average number of local lnks that must be traversed between two adjacent lateral lnks s:. N +7 & M The average number of local lnks that must be traversed n the executon of an r-cycle..9. s: f. C 8 f. C Over all possble permutatons of symbols and for each nteger value, +9,,C, there s a total of <! r-cycles that nclude symbol 1 (.( ) and *> <! r-cycles that do not nclude symbol 1 (. / ). The average number of local lnks over all permutatons s therefore: 9! ! Average number of local lnks 9! &! 08 8 Recall that local lnks are needed to move between executon sequences of adjacent r-cycles ), to move nto the frst lateral lnk, and to move out of the last lateral lnk n a route (+,, NM0M between a par of nodes n. Theorem The average number of local lnks that must be traversed n the route between a par of nodes n NM0MP, under a random orderng of r-cycles, s: N & + 9 Proof : Over all possble permutatons of symbols and for each nteger value, +,,, there s a total of *> r-cycles. The total number of r-cycles of length at least n the possble permutatons of symbols s, therefore, # *>;N< 78;!. The average number of r-cycles, +,,, n a permutaton of symbols s 9 >!. The average number of local lnks that must be traversed between these r-cycles s: (9) (10) (11) (1) 8

10 & 5+7. C! 8 (13) 95+ Assumng that the source node s. and that the frst lateral lnk n the route to the destnaton node s., +,<.,, the average number of local lnks that must be traversed between. and. s:.n Note that. dffers from (Eq. 10), snce to compute. we must consder the case.n<.. Smlarly, the average number of local lnks that must be traversed between the last lateral lnk n the route and the destnaton node s 0.. Then, the average number of local lnks that must be traversed n the route between a par of nodes n the SCC graph, assumng a random orderng of r-cycles n the route, s #... The theorem follows. As descrbed n Secton 4, a properly desgned routng algorthm can optmze the orderng of the r-cycles and reduce the average number of local lnks further below the value provded by a random orderng of r-cycles (Eq. 1). The average number of local lnks, consderng that the shortest route between any two nodes of an SCC graph s determned by an optmal routng algorthm, s therefore bounded by: N, & (14) (15) 3.5 Average dstance n the SCC graph Theorem 3 The average dstance of NM0M s bounded by: NM0MN D,< +! + # (16) Proof : The theorem mmedately follows from Eqs. 5, 6, 9, 1 and Routng algorthms n the SCC graph 4.1 Orderng of r-cycles Routng between two nodes. and. n NM0MN s equvalent to routng from. to., where I & 8!, C +;G, and & s the nverse or recprocal of permutaton [, 1]. Let lateral lnks 0 denote a route from from. to. NM0MN n, along a sequence of C. The total cost of s gven wth: 9

11 ;. & ) Dependng on the order chosen to execute the r-cycles n, dfferent routes. (17). to. are produced. As explaned n Secton 3, a common feature to any of these routes s that they all have the same number of lateral lnks ( ) and local lnks ( ). Fndng the shortest route from. to. s therefore a matter of choosng an r-cycle orderng whch mnmzes the number of local lnks ( ). A routng algorthm whch acheves ths goal s gven n Subsecton 4.4. Non-optmal (but smpler) routng algorthms are presented n Subsectons 4. and 4.3. To llustrate the dfferent cost components n a route, and how they are affected by the order chosen to execute the r-cycles, assume routng from node G G + $ to node + +;G $ n NM0M. A route along the sequence 0+ G7NC+ + G7 contans four lateral lnks, four local lnks, and three local lnks (.e., + G77 G0C; ). However, f the sequence of lateral lnks 0G +7 #/G + +7 s used, a route wth four lateral lnks, four local lnks, and one local lnk results (.e., G +77 ). In some cases, the number of from local lnks n a route from. to. can be further reduced by nterleavng (rather than executng separately) the r-cycles n. For example, some possble sequences of lateral lnks from supernode +;G$ I J9+ G7 $7 to supernode +;G $ n NM0M are (, 3, 4, 5, 4), (, 3, 5, 4, 5), (4, 5, 4,, 3), (5, 4, 5,, 3), (, 4, 5, 4, 3) and (, 5, 4, 5, 3). The last two of these sequences nterleave r-cycles J#+ G7 and $7. All of the routng algorthms presented n ths paper account for the possblty of nterleavng r-cycles. 4. Random routng algorthm A smple routng algorthm for NM0M conssts of choosng a random order to execute the r-cycles n. Partcularly, a possble algorthm that can be used for ths purpose s the routng algorthm of the star graph [9]: Algorthm 1 (Non-determnstc routng n the star graph): Repeat untl N : 1. If the frst symbol n s 1, then exchange t wth any symbol not n ts correct poston.. If the frst symbol n s, then ether exchange t wth the symbol at poston, or exchange t wth any symbol n an r-cycle of length at least two, other than the r-cycle contanng. The above algorthm requres at most complexty s 0, or 0, snce,, EH>!+LK and,,<. steps of complexty 0J! each, and therefore ts 10

12 M M M M 4.3 Greedy routng algorthm A smple approach to mnmze the number of local lnks n the route between nodes. and. conssts of usng a greedy algorthm. Such an algorthm uses the followng data structures and varables: the set of r-cycles of length at least n. a subset of the symbols of, such that: If J#... s an r-cycle of, then. and... If.".0 =. s an r-cycle of that does not nclude symbol 1, then.(.... an nteger varable ntalzed to. <.. Algorthm (Greedy routng n the SCC graph): 1. If N, then route nsde the supernode and ext.. Identfy the r-cycles of length at least that exst n, and ntalze,, and.. 3. Choose a symbol. such that.. 7 s mnmal. Let M be the r-cycle that contans symbol.. Once. s chosen, make. <.. 4. If M has the form J (.e., ncludes symbol 1), then make J # J4. 0. # and B C!.!.. Otherwse, make - and - 7, where 7 denotes a functon that returns the set of symbols n r-cycle M. 5. Repeat Steps 3 and 4 untl. The greedy approach used by Algorthm conssts of choosng the r-cycle that has the mnmum dstance from. M as the next one to be executed. If the selected r-cycle ncludes symbol 1, then only M the frst lateral lnk of M s taken, whch allows for an nterleaved executon of that r-cycle. If does not nclude symbol 1, then M s executed completely. The complexty of the greedy routng algorthm s 0, or 0 snce,, EH>!+LK and,,/. The orderng of r-cycles chosen by ths algorthm, however, may not be optmal. 4.4 Optmal routng algorthm We now present an optmal routng algorthm whch provdes the shortest route between a par of nodes. and. n NM0MN. The goal of the algorthm s to fnd a sequence of lateral lnks 0, such that s mnmal (Eq. 17). We note that an earler verson of our optmal routng algorthm appeared n [1]. The algorthm we present here mproves that of [1] n 11

13 two ways: 1) t employs more selectve heurstcs to further constran the search space generated by the algorthm, and ) t accounts for the possblty of nterleavng r-cycles, whch s not possble wth the algorthm n [1]. The algorthm performs a depth-frst search on a weghted tree structure. The tree s bult by expandng at each step only those r-cycle orderngs that seem to result n a mnmal number of local lnks. Although the search tree can vrtually examne all possble r-cycle orderngs, ncludng nterleaved r-cycles, ts sze s sgnfcantly constraned n our algorthm. To guarantee that an optmal route s always found, backtrackng s used to enable expanson of prevous r-cycle orderngs that seem to be better than the most recently expanded orderngs. In the followng dscusson, we use the term vertex to refer to an element of the search tree. In addton, we use the term edge to refer to the logcal connecton between vertces n the search tree, whch s usually mplemented wth ponters or some form of ndexng. The followng data structures are stored wthn each vertex of the search tree and are used by our routng algorthm: the label of the node reached so far by the routng algorthm. a subset of the symbols of ", such that: If J#... s an r-cycle of 1, +, 0,<, then. and.. If s an r-cycle of 1, +<,,, such that , then The symbols n represent all possble lateral lnks that can be selected by the routng algorthm whle expandng the search tree from a gven vertex. For convenence, we defne a functon to generate from. Let ths functon be referred to as bsymbols, such that 7'. a subset of the symbols of ", such that: If J#... s an r-cycle of 1, +, 0,<, then. and..j&. If s an r-cycle of 1, +<,,, such that , then The symbols n represent all lateral lnks that can be possbly selected by the routng algorthm to enter supernode (.e., all possble r-cycle orderngs that can be selected by the routng algorthm from a gven vertex necessarly end wth a lateral lnk ). For convenence, we defne a functon to generate from. Let ths functon be referred to as fsymbols, such that 7H. the number of local lnks used so far by the routng algorthm n the route from. to. 1

14 M an estmate of the mnmum number of local lnks that may be needed by the routng algorthm to reach node. from node., usng the route already constructed by the algorthm up to the ntermedate node. For convenence, we use a functon dubbed as mnloc to compute from,,.. Such a functon s defned as: where.!,. N mn J 7', and!, = 7H. mn!. J (18) Note that mnloc s computed under the optmstc assumpton that the route from to. selects the best possble lateral lnks n the sets and. In addton, the summaton term used to compute the number of local lnks that are requred to execute all r-cycles M B (see Eq. 8) assumes that an optmal r-cycle orderng requrng no local lnks to move from one r-cycle to the next can be found by the routng algorthm. an enable/dsable bt whch ndcates whether the tree should contnue to be expanded from vertex or not. In addton, the tree structure generated by the optmal routng algorthm has the followng characterstcs: The number of levels n the search tree s at most +, wth beng gven by Eq. 4. We number these levels from 0 to, startng from the root level. Let be the parent of a vertex n the search tree. We refer to the data stored n and as and, respectvely. The weght of the edge connectng to corresponds to the number of local lnks that are requred to route from to n NM0MN and s gven by mn " 9< =. Hence,. Note that routng from to also requres one lateral lnk f " /, and zero lateral lnks otherwse. Snce the total number of lateral lnks that must be traversed n the route from. to. can be computed n advance (Eq. 4), the routng algorthm focuses on accountng for the local lnks only. The root vertex s ntalzed wth., I 7, < 7,,.!.. and ON. All vertces located at level n the tree have N., = and.!.. =. All vertces located at level n the tree have N used to enter supernode ),, and.! (wth beng the lateral lnk. N.. The backtrackng mechansm s trggered by comparng the estmated mnmum number of local lnks ( ) stored n the most recently generated chld vertces wth a global varable referred to as. Ths varable s updated whenever a backtrackng procedure occurs, meanng 13

15 that the mnmum number of local lnks that s requred n the route from. to. s actually greater than the prevous value of. As expected, the search for an optmal route becomes more selectve as ncreases, whch not only lmts the wdth of the search tree but also makes the backtrackng mechansm less lkely to be trggered agan. Gven the defntons above, the optmal routng algorthm for the SCC graph follows : Algorthm 3 (Optmal routng n the SCC graph): 1. If N, then route nsde the supernode and ext.. Create a root vertex wth., 7,,.!.. and ON. Also, ntalze wth the value.!... 7, 3. Generate chld vertces for all enabled vertces, such that the label for each chld corresponds to exactly one of the symbols stored n the set of each parent vertex. Set OFF at each recently expanded parent vertex. Also, obtan permutaton for each chld vertex by swappng the 1st and the th symbols of the correspondng permutaton stored n the parent vertex (1 ), and make 7, 7,,.!.. Enabled vertces located at level of the search tree must be expanded smlarly. However, they generate a sngle chld wth /., /,,,.,. In any case, a chld vertex s enabled wth ON f. Otherwse, we set OFF., 4. If one of the chld vertces has. and ON, then an optmal route has been found. The optmal sequence of lateral lnks 0 can be obtaned n reverse order by backng up towards the root of the tree and lstng the value of the symbol stored n each vertex located between the J'3 and the 1st levels. Once 0 has been obtaned, ext the algorthm. 5. If none of the enabled chld vertces has N., go to Step If there are no enabled chld vertces, do a backtrackng search n the tree. Among all exstng chld vertces, select those wth the smallest value of and set to ths value. Also, enable the selected nodes and go to Step 4. The heght of the search tree s 0, snce ts maxmum value s D + AEHG!J>!+LK +. A worst-case analyss of the wdth of the search tree can be done under the followng pessmstc assumpton: consderng that all possble orderngs of r-cycles n permutaton are examned by the optmal routng algorthm, the lowest level n the search tree would have at most vertces. Ths s justfed by the fact that there are at most possble ways to move the msplaced symbols n to ther correct postons, usng the mnmum number of lateral lnks gven by Eq. 4. In practce, the constrants placed on the number of expanded vertces by the heurstcs of the 14

16 l = 5 π = (153) (4) (6) B = {,4,5} F = {,3,4} L = 0 M = 6 e = OFF 1 0 l = π = (1453) (6) l = 4 π = (1453) (6) B = {4} F = {3} B = {} F = {3} L = M = 10 e = OFF L = 1 M = 8 e = OFF l = 5 π = (13) (4) (5) (6) B = {,3,4} F = {,3,4} L = 0 M = 5 e = OFF 1 l = π = (143) (5) (6) B = {4} F = {3} L = M = 7 e = OFF l = 3 π = (1) (4) (3) (5) (6) B = {,4} F = {,4} L = M = 8 e = OFF l = 4 π = (143) (5) (6) B = {} F = {3} L = 1 M = 6 e = OFF l = π = (143) () (5) (6) B = {4} F = {3} Dmensonalty of the SCC graph: n = 6 L = 3 M = 6 e = OFF Source node: < 5, (153) (4) (6) > Destnaton node: < 3, (1) () (3) (4) (5) (6) > Backtrackng threshold used: T = 6 l = 4 π = (13) () (4) (5) (6) Optmal sequence of lateral lnks found: (5,4,,4,3) B = {3} F = {3} Number of lateral lnks n the optmal path: 5 L = 5 M = 6 e = OFF Number of local lnks n the optmal path: 6 1 Total length of the optmal path: 11 lnks l = 3 π = (1) () (3) (4) (5) (6) B = { } F = { } L = 6 M = 6 e = ON Fgure 4: Example of search tree used for optmal routng n NM0M# 15

17 algorthm (.e., the estmated mnmum number of local lnks ) lmt the wdth of the search tree consderably. Smulatons carred out for 9,<, revealed that a very small number of vertces s enabled at each step, whch makes the maxmum wdth of the tree vrtually proportonal to. Fgure 4 llustrates an example of the search tree constructed by the algorthm. The man computatons that must be performed upon creaton of a vertex of the search tree refer to, and. Fortunately, each of these computatons can be accomplshed n 0J! tme by usng the correspondng values, and that are stored n the parent vertex, and takng nto account the dfferences n the r-cycle structures of permutatons = and. The reasonng above results n a worst-case complexty of 0. As explaned above, such computatonal requrements were not observed durng smulatons of the optmal algorthm. The potental need for backtrackng searches n the tree, added to fact that the maxmum wdth of the tree s n practce proportonal to, results n a complexty of 0, on the average (or 0, snce,,< ). 5 Smulaton results The performance of routng algorthms for the NM0MO graph was evaluated wth smulaton programs that compute the route of all <! nodes of the graph to the dentty. The routng algorthms that were tested are: 1) a random routng algorthm that generates all possble routes to the dentty wth equal probablty, whch s based on Algorthm 1, ) Algorthm, and 3) Algorthm 3. The smulatons were carred out for G, 5,. A log of worst-case routes that may result from the random routng algorthm was also made Graph sze J9! 8! * ,600 30,40 8,40,903,040 NM0MP Graph dam. D J Average number of lateral lnks J Average number of local lnks Average number of local lnks Average number of local lnks! NM0MP Average dst. D J Table 1: Average dstance of SCC graphs under optmal routng Table 1 and Fg. 5 show the smulaton results obtaned wth the optmal routng algorthm. The smulaton results obtaned for and match exactly the theoretcal values provded by 16

18 Eqs. 6 and 9. Also, the smulaton results obtaned for are closely bounded by Eq. 1. under an optmal routng algorthm Average dstance Average number of local lnks Average number of MI local lnks Average number of lateral lnks Average number of MB local lnks Dstances n Fgure 5: Average dstances on the SCC graph under optmal routng Average number of MB local lnks Random routng (worst-case) Random routng (average, smulaton) Random routng (average, theoretcal) Greedy routng Optmal routng n Fgure 6: Average number of local lnks for dfferent routng algorthms As expected, only the average number of local lnks vared among the dfferent routng algorthms that were tested. Fg. 6 compares smulaton results for. Note that the results 17

19 for the random routng algorthm are very close to the theoretcal values provded by Eq. 1. The model used to derve that equaton seems to result n an error proportonal to!>!, whch s neglgble consderng that Eq. 1 s stll a close upper bound for. As expected, both the greedy and the optmal routng algorthm outperform the random routng algorthm, as far as the average number of local lnks s concerned. Also observe that, for G,/<,, the greedy routng algorthm performs as well as the optmal routng algorthm. Besdes, our results ndcate that the performance of these algorthms s qute smlar for $5, I,, whch makes the less complex greedy routng algorthm partcularly attractve Optmal routng Greedy routng Random routng (theoretcal) Random routng (smulaton) Random routng (worst-case) Table : Average costs for dfferent routng algorthms Average costs of paths produced by the three routng algorthms are summarzed n Table. The random routng algorthm has a complexty of 0 and performs reasonably well on the average. Utlzaton of such an algorthm may, however, result n varatons n the average cost of routes up to the worst-case values shown n Table Optmal routng Greedy routng Random routng (average) Random routng (worst case) Dstance to the dentty Fgure 7: dstrbuton curves for a 9-SCC graph 18

20 Fgure 7 shows dstrbuton curves comparng the three routng algorthms n the case of an NM0M graph. A pont n one of these curves ndcates that the correspondng routng algorthm wll compute a route of cost to the dentty for nodes n the SCC graph. The average dstrbuton for the random routng algorthm s shown, but the results for that algorthm may actually vary from the optmal to the worst-case dstrbuton curves due to the non-determnstc nature of the algorthm. It s also nterestng to observe that the greedy routng algorthm provdes a dstrbuton curve whch s close to that of the optmal routng algorthm, presentng however a smaller complexty. 6 Consderatons on wormhole routng In ths secton, we brefly descrbe how the algorthms presented n the paper can be combned wth wormhole routng [8], whch s a popular swtchng technque used n parallel computers. All three algorthms can be used wth wormhole routng, when mplemented as source-based routng algorthms [13]. In source-based routng, the source node selects the entre path before sendng the packet. Because the processng delay for the routng algorthm s ncurred only at the source node, t adds only once to the communcaton latency, and can be vewed as part of the start-up latency. Source-based routng, however, has two dsadvantages: 1) each packet must carry complete nformaton about ts path n the header, whch ncreases the packet length, and ) the path cannot be changed whle the packet s beng routed, whch precludes ncorporatng adaptvty nto the routng algorthm. Dstrbuted routng elmnates the dsadvantages of source-based routng by nvokng the routng algorthm n each node to whch the packet s forwarded [13]. Thus, the decson on whether a packet should be delvered to the local processor or forwarded on an outgong lnk s done locally by the routng crcut of a node. Because the routng algorthm s nvoked multple tmes whle a packet s beng routed, the routng decson must be taken as fast as possble. From ths vewpont, t s mportant that the routng algorthm can be easly and effcently rendered n hardware, whch favors the random routng algorthm over the greedy and optmal routng algorthms. Besdes beng the most complex algorthm dscussed n ths paper, the optmal routng algorthm ncludes a feature whch precludes ts dstrbuted mplementaton n assocaton wth wormhole routng, namely ts backtrackng mechansm. Dstrbuted versons of the random and greedy algorthms, however, can be used n combnaton wth wormhole routng. A sub-optmal dstrbuted routng algorthm whch supports wormhole routng can be obtaned by removng the backtrackng mechansm from Algorthm 3. Such a sub-optmal algorthm s lkely to have computatonal complexty and average cost that le between those of the greedy and the optmal routng algorthm. Due to ts non-determnstc nature, the random routng algorthm also seems to be a good canddate for SCC networks employng dstrbuted adaptve routng [13]. Adaptvty s desrable, for example, f the routng algorthm must dynamcally respond to network condtons such as congeston and faults. Some degree of adaptvty s also possble n the greedy and optmal routng algorthms, whch n some cases can decde between paths of equal cost. 19

21 7 Concluson Ths paper compared the average cost and the complexty of three dfferent routng algorthms for the SCC graph. We dvded the route between a par of nodes n the graph nto three components (lateral lnks, local lnks and local lnks) and showed that only the number of local lnks may be affected by the routng algorthm beng consdered. Exact expressons for the average number of lateral lnks and the average number of local lnks were presented. Also, an upper bound for the average number of local lnks was derved, consderng a random routng algorthm. As a result, a tght upper bound on the average dstance of the SCC graph was obtaned. Smulaton results for a random, a greedy and an optmal routng algorthm were presented and compared wth theoretcal values. The complexty of the proposed algorthms s respectvely 0, 0, and 0, where s the dmensonalty of the NM0M graph. NM0M The results under optmal routng produce exact numercal values for the average dstance of, for G,<B,. Our results ndcate that the greedy algorthm performs as well as the optmal algorthm for G,,. The greedy algorthm also performs close to optmalty for $,,, and s an nterestng choce due to ts 0 complexty. The random routng algorthm has an 0 complexty and performs farly well on the average, but may ntroduce addtonal local lnks n the route under worst-case condtons. Fnally, we dscussed how the routng algorthms presented n ths paper can be used n assocaton wth the wormhole routng swtchng technque. Drectons for future research n ths area nclude an evaluaton of requrements for deadlock avodance (e.g., number of vrtual channels). References [1] S. B. Akers and B. Krshnamurthy, Group Graphs as Interconnecton Networks, Proc. 14th Int l Conf. on Fault-Tolerant Computng, 1984, pp [] S. B. Akers and B. Krshnamurthy, A Group-Theoretc Model for Symmetrc Interconnecton Networks, Proc. Int l Conf. on Parallel Processng, 1986, pp [3] S. B. Akers, D. Harel and B. Krshnamurthy, The Star Graph: An Attractve Alternatve to the -Cube, Proc. Int l Conf. on Parallel Processng, 1987, pp [4] M. M. Azevedo, N. Bagherzadeh and S. Latf, Broadcastng Algorthms for the Star-Connected Cycles Interconnecton Network, J. Par. Dst. Comp., 5, 09- (1995). [5] M. M. Azevedo, N. Bagherzadeh, and S. Latf, Embeddng Meshes n the Star-Connected Cycles Interconnecton Network, to appear n Math. Modellng and Scentfc Computng. [6] M. M. Azevedo, N. Bagherzadeh, and S. Latf, Fault-Dameter of the Star-Connected Cycles Interconnecton Network, Proc. 8th Annual Hawa Int l Conf. on System Scences, Vol. II, Mau, Hawa, January 3-6, 1995, pp [7] W.-K. Chen, M. F. M. Stallmann, and E. F. Gehrnger, Hypercube Embeddng Heurstcs: an Evaluaton," Int l Journal of Parallel Programmng, Vol. 18, No. 6, 1989, pp

22 [8] W. J. Dally and C. I. Setz, The Torus Routng Chp, Dstrbuted Computng, Vol. 1, No. 4, 1986, pp [9] K. Day and A. Trpath, A Comparatve Study of Topologcal Propertes of Hypercubes and Star Graphs, IEEE Trans. Par. Dst. Systems, Vol. 5, No. 1, January 1994, pp [10] D. E. Knuth, The Art of Computer Programmng, Vol. 1, Addson-Wesley, 1968, pp. 73, pp [11] S. Latf, Parallel Dmenson Permutatons on Star Graph, IFIP Transactons A: Computer Scence and Technology, 1993, A3, pp [1] S. Latf, M. M. Azevedo and N. Bagherzadeh, The Star-Connected Cycles: a Fxed-Degree Interconnecton Network for Parallel Processng, Proc. Int l Conf. Parallel Processng, 1993, Vol. 1, pp [13] L. M. N and P. K. McKnley, A Survey of Wormhole Routng Technques n Drect Routng Technques, Computer, February 1993, pp [14] F. P. Preparata and J. Vullemn, The Cube-Connected Cycles: A Versatle Network for Parallel Computaton, Comm. of the ACM, Vol. 4, No. 5, May 1981, pp [15] Y. Saad and M. H. Schultz, Topologcal Propertes of Hypercubes, IEEE Trans. Comp., Vol. 37, No. 7, July 1988, pp [16] S. Shoar and N. Bagherzadeh, Computaton of the Fast Fourer Transform on the Star-Connected Cycle Network, to appear n Computers & Electrcal Engneerng, [17] P. Vadapall and P. K. Srman, Two Dfferent Famles of Fxed Degree Regular Cayley Networks, Proc. Int l Phoenx Conf. on Computers and Communcatons, Scottsdale, AZ, March 8-31, 1995, pp