Reed, Daniel A.

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Purdue University Purdue e-Pubs Department of Computer Science Technical Reports Department of Computer Science 1983 Optimal Routing in the Cube-Connected Cycles Interconnection Network Daniel A. Reed Report Number: 81-412 Reed, Daniel A., "Optimal Routing in the Cube-Connected Cycles Interconnection Network" (1983). Department of Computer Science Technical Reports. Paper 336. https://docs.lib.purdue.edu/cstech/336 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information. Optimal Routing in the Cube-Connected CyclcsInlercollIlcction NetworkDaniel A. Reed tDepartment of Computer ScienceUniversity of North CarolinaChapel Hill, North Carolina 27514CSD-TR-412 (Revised)ABSTRACTAn optimal message routing algorithm for thc cube-connectedcycles processor interconnection network is described, and theaverage:- message path length is derived assuming a uniformmessage routing distribution. The optimal algorithm is comparetlto one preViously proposed and is shown to have significanllyshorter average path length.Key Words and Phrases:Parallel processing, message routing, mu] ticomputer netwodG::.performance evaluationt CSNET address: dar.unc@UDel-RelayThis work was performed at the Department o[ ComputerSciences, Purdue University.June 27, 19831IntroductionSeveral researr.hers have recently proposed large networks of computa-tion nodes whose communicaLion paradigm is message passing [1, 3, 11. I!:achnetwork node, implemented as one or two VLSI chips. would contain a process-ing clement wlLh some local memory, a communication p['ocessor capable ofrouting messages without delaying Lhe processing element, and a few connec-tions to other network nodes.Selecting an appropl"iate interconnection network is a particularly per-plexing design problem. Among Lhe proposed interconnections, the binary rnul-tidimcnslonal cube has been shown Lo cfficienLly support Lhe communicationpatterns of several important algorithms. Unfortunately, each of the 2D nodesin such a cube must be connected to D other nodes. As 1J increases lhisviolates the fanout limitations imposed by the VLSI implr~menlation of thenodes.However, Preparata and Vuillemin [2] have suggested a varialion of thecube, called the cube-connecled cycles, thaL eliminates the fanoul problem. Asthe name SlLgg~Sts, each node at a verLex of a D-dimension;ll cube is replacedwith a ring of D nodes, numbered from 0 through D-·l. Node addresses thentake the form (i ,j) where 0 ~ i < D and 0:.5 j < 2D . Each node is connected tothe Lwo neighboring ring nodes at ils verlex and the node with the same ringnumber i and. vertex number given by tOggli!lg Lhe i-th bit of the current ver-lex number j. (See F'igure 1.) Consequently, each node has Lhree neighborsregardless or the cube dimension.Allhough lhis approach solves the fanout pl'ublem, it complicales the a.lgo-rithrn for routing messages from source to destination nodes. In this paper, we2analyze a simple routing algorithm proposed by Wittie (4), outline all optimalrouting algorithm [or the cube-connecLed cycles interconnection. and discussthe relative messagt~ intensities on the communication link.s of 1:.he ordinarycube and the cube-COlmected cycles.Definilious and AssumptionsWe define a cube link as a linle connecting two nodes with dlHerenl verlexaddresses and a. ring link as one connecting LVro nodes with diffe~'ent ringaddresses. Eecause traversing a cube link moves a message to another nodewith the same ring address at another vertex, findlng a sharlusl paLrl from asource uode to a destination node can be reduced to the followIng optimizationproblem.(1) Consider a ring of D nodes.(2) Distinguish a .:::ource node (s), a destination node (d), and k inter-mediate nodes (0 ~ k < D-l).(3) Find a shortesl path from the source node to the deslinallOn nodethat passes through all inlermediale nodes.When analyzing routing algorithms [or solving this problem, we shallassume a uniform message routing dislribution. That is. all source arLd desti-nation addresses are presumed to be equally likely. Under thh: assumption.each ring node will be a distinguished node with probabiHly 0.5 since each bitor the source and destination vertex addresses differ with this probabiliLy.3Simple Routing AlgorithmThe routing algorithm proposed by Willie [4] begins at the source node andinvolves two steps:(A) Cross ring links in the clockwise direction until all dislmguisbedinlel"mediale nodes have been visited.(1:1) Find the shorlest path, clQckwi~e 01' countercllJckwise, !i-om thecurrent position to the final destination node.For step A, a message traverses i. ring links if and only if the node l linksaway from the source were distinguished and the last D -l - 1 nodes were notdistinguished. Under the uniform routing distribution, the [Jfobubility of·f·.,traversing l ring links is just a geometric random variate with valueThlis, the average number of ring links traversed in performing slep A isD.:;l lL: 2D-',~D + - 2.For step B, the number of links traversed is at mo:::t Ig]. On the average~l D2 I -D-' ( ),~2 DL: min I, D - I = D r;;venD 4·,~D~I2 I,~ D' - 1 DoddD 4Dlinks must be traversed. Hence, the average number of ring links traversed bythis algorithm is40 D even5D 1 - 2+ -4 2D - 11D odd.4D( 1)Optimal Routing AlgorithmWitlie's simple routing algorithm does not use the location of source anddestination address di1ferences to reduce the number of ring link traversals.One would expect any algorithm employing this information to performsignificantly better. Finkel and Solomon presented an optimal routing algo-rilbm for the lens [1], a shared bus interconnection, thaL can be applied to thecubc-connecLcd cycles tnlerconncctioll, and it docs exhibit shorler mean pathlength than the simple algorithmWr:. ucgin by observing that the t30urce and destination node::> dtvide thering into two arcs. Let A be the longer of these. and let a be its length. Let Bbe a maximal contiguous sequence of non- distinguished nodes of length b - 1in the interior of A. Finally. let C be a maximal contiguous sequence of non-distinguished nodes of length c - 1 outside A. Figure II shows one possiblearrangement of arcs for a ring or 11 nodes with distinguished nodes indicatedby·. Note that I~1~ a :=; D, 1:=; b ~ a, and 1:=; c :=; D - a. A shortest pathfrom source to destination is given by the minimum of cases I and II below.Case 1 (See Figure IIl)Move from the source (5) to one end of B and back to the other end of B.Finally, move to the final destination (d). Inspection shows the lenglh of thispath to be D + a - 2b.5Crl.Jfi if (Sp.e Figura IV).Move from the source (5) to one end of C and back to the other end of C.Fihally, move to Lhe final destination (d). The length of this path is2D -a -2c:.The optimality of this algorithm rests on the observaLion that any pathmust include either the links in A or those outside A. Having inclurlad theselinks, one need only Vi3it the distinguished nodes not encountered along thispath. Clearly, tlles:3 are best visited by short excursions from Lhe source ordesLino.tioll node. Thus. the minimum length ring path from source to destina-lion ismin [D + a - 2b . 2D - a - 20 ).Again. we would like to find the average number of ring link traversalsrequired to reach a destination under the uniform routing assumption. To dothis, we must tlrsl establish two apparently unrelated lemmas.The probability of arc A having length a is P(a. ,il) where1D D 2aa DT =DP(~,D) =2 otheT"uJise.D6Proof:Since all sources and destinations are equallr likely and A is, by definition,the longer of the two arcs between the source and destination, there are twocomplementary positions on the ring. each occurring with probability ~, suchthat A has length a;. The only exceptions occur if source and destination arccoincident. 01' a is half the ring circumference. In these cases, then: is onlyone possible ilrc of this length.Lemma 2The probability of a run of at least m successes in n trials is R(m,n)whereR(m,n) = pm + q"r.1piR(m,n-i-l);=0andProof:pqR(m,n) oprobability of a successpTobability of a failurem > n.Let peE) denote the probability of the desired run of m successes (i.e.,peE) =R(m,n». and let peS) denote the probability of an individual success(Le., P(S) =p). Applying the laws of conditional probability. we have7PtE) = P(S)P(E IS) + P(.5')P(E 1.5')= pP(E IS) + qP(E I.5').Now P(SIS) is just R(m,n-l). Employing conditional probabilities again. wehaveP(EIS) = P(SjS)P(EIS') + P(.5'IS)P(E!.5'S)= pP(E'IS') + qR(m,n-2)soP(E) = p'P(EIS') + pqR(m,n-2) + qR(m,n-l).By induction, one can show thati-'1P(E) = p'P(EIS') + qL.pJR(m,n-j-l).j:fJNote. however, that P(E Ism) = 1. Hence,m-'P(E) = R(m,n) = pm + qL.pJR(m,n-j-l).i =0IIFinally, Q(m,n) = R{rn.n) - R(m+1.n) is the probability of a run ofexa.ctly m suceesses in 'It trials. In the context of our discussion, Q(m,n)corresponds to the probability of i:l. contiguous group of non-distinguished ringnodes or length m in an arc of length n. Under the uniform message routingassumption, the probability of an individual success. p, is 0.5. We can nowstate the foll;)wing theOn:lffi.8TheoremThe average number of ring link traversals required to route a message toits destination using the optimal rouLing algorilhm is/)-1 a {)u (L: P(a.,f) L.;lf(ll-l,n-l) )~Q((~-I,J)-f1.-1)11Ii!-t 1J+r..u=I¥1 b=l c=]+P(D,D)t Q (b-l,D-l)min(2 I D - b),D),b =1Proof:-'J,b, 20 -(1 -2~}(2)Recall that P(a.D) is the probability of arc A having length a,Q(b-l,a-l) is the probability of a group o[ contiguous non-disLinguishednodes of lengLh b - 1 occurring in the interior of i1, and Q(c -! ,j) -a -1) is Lheprobability of a similar group of non~distinguishednodes occ.urring exterior LaA. Then the sum represents all posslble value:; of a, b. and c weigh led by theirprobability of occurretl.ce multiplied by th(~ minimum path 18ngth fN thosevalues. oAlthough (2) is unWieldy. is seems unlikely that a clo.3ed form solution canbe found. Fortunalely, Lwo relaLed factors make such a solution unnecessary.Fil'sl, Lhe compuLaUoIlill complexity of (2) is only O(D3) if recurrence valuesare precalculated. This c::ontrasts with the O(D2D ) operations needed to dctcr-mine the mean number of ring link traversals by exhaustive enumeration.Second, the number nelwork nodes rises exponentially with D. Thus, we needonly eonsidel· relatively smalt values or D Lo obtain networks with thousands ofnodes (e.g., D = 10 gives 10.240 nodes).9Comparisonsfigure V shows the average number 01 ring link traversals required forboth the simple routine algorithm and the optimal one. Over the range of Dsllown, the opLimal rouLing algorithm show~ a 20.l) percenl rcducLinn in themean number of ring link lraver::;als. As we shall see, only margin"l perfor-mance improvements i"esull if a reducLion of greater than 25 per'cent isobtained.The expected numoer of link traversals alone fails to capture the perfor-mance of an interconnection network. Jndeed, a single bus and a completelyconnecled network have the same performance by this metric. Another.pel'hap:'! man: Important. metric or inlcrCOlll:ccLion neLwork pedol'mdnee is thelink message inLensiLy, Lhe average number of link ll'i.\Vcr~als rcquit'cd by amessage diviUed by the number of links. l~or a network such as the cube-connecLed cycles. there are Lwo mC3sagc inLensities. one for the ring links andone for the cube links.lJnder uniform routing, the average number of cube link traversalsre':luired by a message is ~ because each of the D bits of the verLex addressesdiITcr wilh probability 0.5. Since: there are D2D - 1 cube links. the messageilll81lsity for these links is 2~·for the D2D ring links. equation (1) shows that the message intensiLy forthe simple routing algorithm is approximately 1~~5. Since Lhis is greater thanthe message intensity of the cube links. the ring Hnks will be the dominant fac-Lor in communication delays if this routing algorithm is used. Dy way of eon-LrasL, Lhe mc:nmge inlensily for Lhe ring links using Lhe oplimal rouLinp; algo-riLhm is only slightly greater than12D ·10This impHes that the- performance ofthe cube-connecled cyc.:lcs interconnection is ncar that of an ordinary cube,confirming Lhe cube-connecled cycles interconnection as a feasible alL8rnativeto an ol'dinary cube.ConclusionsWe have presented un optimal routing algorithm for the cube-connecLedcycles inlerconnection and analy:lcd iLs performance. The algoriLhHl is onlyslightly more complicated than the simple algorithm and significanLly reducesthe message intensity of the ring links. Because of this, the cube-connectedcycles interconnection performance should approach the performance of anordinary cube interconnection.11References[1J R. A. Finkel and M. H. Solomon, "The Lens lnterconnectlOn Stra-tegy," IEEE Trans. Comput. Vol. C-30. pp. 960-965, Dec. 1981.[2] [0'. P. Preparala and J. \'-ulliemin, "The Cube-Connected Cycles: AVel'satHe N~twork for Parallel Computation," Commun. Ass. Com-put. Mach, vol. 24, pp. 300-309. May 1981.[3] D. A. Reed Ilnd H. D. Schwetman. "Cost-Performance Bound~ forMulU-Microcornputer NeLworks," IEb'E 1'rans. Comput, Vol. C-32, pp.83-95. Jan. 1963.[4] L. D. Wittie, "Conununication Structures for Large Multimicrocom-puLer Sy~tem.s," IEEE Trans. Com:put, VoL C-30, pp. 264-273, Apr.1901.'.p- --/"J:l...0-" .....--r- ---<:r - /" /'?IIII/6.. ....../l'igUr" I3_dimensional cube-eonnecled cycles_.. -"c*FigureUArc lengths for a ring of eleven nodess----....**Figure IIIOptimal routing algorithm. case IPath length ;;; D + a + 2b ;:: 9D -.,-.--!'igureN....Optimal routing algorithm. case 11Path length :;: 2D - a - 2c = 1210o------ Simple Routing Algorithm.......... Optimal Routing Algorithm6 8Network Dimension D345 7 910Figo.re YHean ring path lengthfor the cube-connected cycles

Optimal Routing in the Cube-Connected Cycles Interconnection Network

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Optimal Routing in the Cube-Connected Cycles Interconnection Network

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