Acyclic orientations do not lead to optimal deadlock-free packet routing algorithms

Transcription

1 Acyclic orientations do not lead to optimal deadloc-ree pacet routing algorithms Daniel Šteanovič 1 Department o Computer Science, Comenius University, Bratislava, Slovaia Abstract In this paper e consider the problem o designing deadloc-ree shortest-path routing algorithms. A design technique based on acyclic orientations has proven to be useul or many important topologies, e.g., meshes, tori, trees and hypercubes. It as not non hether this technique alays leads to algorithms using an asymptotically optimal number o buers. We sho this is not the case by presenting a graph o size N hich has a deadloc-ree shortest-path routing algorithm using O(1) buers, but every deadloc-ree shortest-path routing algorithm based on acyclic orientations requires (log N / log log N ) buers Elsevier Science B.V. All rights reserved. Keyords: Deadloc-ree pacet routing; Acyclic orientations; Interconnection netors 1. Introduction Devising deadloc-ree routing algorithms is an important problem in the netor design. A ide range o dierent algorithms and design techniques has been proposed [1,4 11]. In this paper e concentrate on an important class o deadloc-ree routing algorithms called buer-reservation algorithms [2, 3]. Given source, destination and current buer o a pacet a buer-reservation algorithm speciies a set o buers to hich the pacet may move. The pacet can move to any buer rom the speciied set provided that it is empty. It is non that or each netor there exists a deadloc-ree routing algorithm using only 2 buers This research has been partially supported by VEGA 1/4315/ steano@zeus.dcs.mph.uniba.s. per node [8]. Hoever this algorithm suers rom congestion, because it uses only a spanning tree o the netor. It is reasonable to consider routing algorithms using only shortest paths. For such algorithms it as shon in [2] that there exists a netor o size N hich requires (log log N ) buers per node. It is also non that or every netor o diameter D there exists a shortest-path deadloc-ree routing algorithm using D + 1 buers per node [9]. A technique or constructing deadloc-ree routing algorithms based on acyclic orientations has proven to be useul or many important topologies [9]. Moreover only e additional bits o data besides routing inormation are required by the algorithms designed using this technique. Thus a natural question is ho good the technique is in general. We present a graph o size N hich has deadloc-ree shortest-path routing algorithm using O(1) buers per node, but every deadloc-ree shortest-path routing algorithm based

2 on acyclic orientations requires (log N / log log N ) buers per node. The proos presented in this paper can be adapted to obtain the same result or the cubeconnected-cycles graph. 2. Preliminaries A communication netor is modeled by an undirected graph G = (V, E), here vertices in V represent nodes and edges in E represent bidirectional communication lins. Every edge is considered to comprise to opositely oriented arcs. Given source, destination and the buer currently holding the pacet a buer-reservation algorithm speciies a set o buers to hich the pacet may move. The pacet can move to any o the speciied buers, provided it is empty. A deadloc is a situation in hich a set o pacets can never reach their destination, because speciied buers o each pacet are occupied by other pacets rom the set. A buer reservation algorithm is called deadloc-ree i it does not allo a deadloc to occur. A routing algorithm is called shortest path i the pacets are alays routed along shortest paths. Given a graph G an all-to-all path system P is a collection o paths connecting every to vertices in G. Anorientation DG o G is a directed graph obtained rom G by replacing each undirected edge in G by an arc (i.e., {u, v} is replaced by either (u, v) or (v, u)). An orientation is called acyclic i it doesn t contain a cycle. An orientation cover o a path system P is a sequence o orientations DG 1,...,DG s such that every path p P can be ritten as a concatenation o s paths p 1,...,p s here p i is a path in DG i or i [s]. Anacyclic orientation cover is an orientation cover consisting o acyclic orientations. Let P be a shortest-path all-to-all path system o a graph G. I an acyclic orientation cover or P o size s exists, then there exists a shortest-path deadloc-ree routing algorithm using only s buers per vertex [9]. A routing algorithm designed using this technique is said to be based on acyclic orientations. I there is no restriction on ho a routing algorithm as designed then it is said to use plain buers. 3. A gap beteen acyclic orientations and plain buers In this section e present a graph hich has shortest-path deadloc-ree routing algorithm using only O(1) plain buers per vertex, but every shortestpath deadloc-ree routing algorithm based on acyclic orientations requires (log N / log log N ) buers per vertex. Consider the olloing machine M n : It has a oring tape ith n cells and a head hich can be positioned above any cell. Each cell contains one binary digit. In one step the head can change the content o the scanned cell (shule) or it can move to the let or to the right neighboring cell. The state o the machine M n is the position o the head and the content o the tape. Deinition 1. Let G n be the graph ith vertices corresponding to states o the machine M n and edges connecting to vertices i the corresponding states are reachable in one step o M n. Remar 2. I the tape o the machine as raparound the resulting graph ould be the cube-connectedcycles graph. It is possible to sho the same results or this graph ith slightly more complicated proos and orse constants Upper bound or plain buers Theorem 3. There exists a shortest-path deadlocree routing algorithm o the graph G n using only 8 buers per vertex. Proo. Let u, v be any to vertices o the graph G n. Moving along a shortest path rom u to v corresponds to changing the state u to the state v using the minimal number o steps. Let h u and h v be positions o the head in u and v. Moreover let l u,v and r u,v be positions o the letmost and the rightmost cell that have to be visited (every cell in hich tapes o u and v dier as ell as cells at positions h u and h v must be visited).

3 In every shortest path the head maes a movement in one o the olloing to orms: l u,v h u h v r u,v CASE 1 (h u h v ) l u,v h v h u r u,v CASE 2 (h u h v ) Thus according to the head movement each path can be divided into three phases (the irst and/or third phase can be empty) in hich the head moves in one direction. No e sho that eight buers per vertex suice. Each vertex has buers ith names 1, 2, 3, 4, 1, 2, 3, 4. A pacet moving along a path ith the head movement CASE 1 is initially put in the buer 1 and then it moves beteen buers according to these rules: let-arc (phase 1) any buer buer 1 right-arc (phase 2) any buer buer 2 let-arc (phase 3) any buer buer 3 shule-arc (all phases) buer a buer a No shortest path contains to consecutive shuleedges and thus the rule or the shule-arc is correct (a pacet that ants to move along a shule-arc cannot be in a buer ith a prime). A pacet moving along a path ith the head movement CASE 2 is initially put in buer 2 and moves beteen buers according to these rules: right-arc (phase 1) any buer buer 2 let-arc (phase 2) any buer buer 3 right-arc (phase 3) any buer buer 4 shule-arc (all phases) buer a buer a For the sae o contradiction suppose that there is a deadloc. Then there must be a sequence such that (v 1,b 1 ) (v 2,b 2 ) (v m,b m ) (v 1,b 1 ), here (u, a) (v, b) means that a pacet rom the buer a in the vertex u aits or the buer b in the vertex v. Observethat a pacetnevermovestoa buer ith smaller number. Thereore all buers b 1,...,b m must have the same number some a [4]. Whena pacet moves to a buer a it uses shule-arc and hen it moves to a buer a it uses a let-arc i a is odd and a right-arc i a is even. Thereore all buers b 1,...,b m must be a. Hoever a pacet never moves rom a to a, a contradiction Loer bound or acyclic orientations The vertex set o an n-dimensional hypercube consists o all binary strings o length n. To vertices are connected by an edge i they dier in exactly one bit. For any to vertices u, v a movement along path rom u to v corresponds to a sequence o changes o some bits. I the bits are changed in order rom let to right then the path is called monotone. The loer bound or acyclic orientations depends on the olloing lemma: Lemma 4. Let P be the all-to-all path system o an n-dimensional hypercube ith monotone paths. Every orientation cover or P has size (n/log n). Proo. Let DG 1,...,DG s be an orientation cover or P. We sho that the size o this cover must be large by simulating movement o pacets in the netor. Each vertex generates a pacet or every other vertex. The pacets then move along their paths according to the orientation DG 1, then according to DG 2 andsoonand inally ater moving according to DG s every pacet must be in its destination. We observe the state o the netor (positions o pacets) ater each orientation. Let d be a number 0 d n and v beavertex. By Sv d e denote the set o all vertices that can dier rom v only in the irst d bits and by Dv d the set o all vertices that can dier rom v only in the last n d bits. Given a state o the netor e call the vertex v active ith respect to d i or each vertex u Dv d there exists a pacet hich has to move to u through v. Clearly i v has not yet received some pacet p rom some vertex Sv d then v is active ith respect to d, because pacets rom to every vertex in Dv d travel together ith p until they reach v (the paths are monotone). A vertex hich is not active ith respect to d is called passive ith respect to d. Clearly a vertex passive ith respect to d must have already received all pacets rom all vertices in Sv d.

4 Let = log 2 n. We ill construct a sequence o hypercubes Q 0,...,Q n/ such that ater moving according to the ith orientation the ratio o passive vertices in Q i ith respect to i is at most i/2. This ould imply that, because i some vertices are active then some pacets are not in their destination. For each i the hypercube Q i ill have dimension n i and it ill consist o vertices starting ith some string rom {0, 1} i. Initially all vertices are active and thus e can tae Q 0 to be the hole hypercube. No e sho ho to construct Q i+1 rom Q i.letq i be an (n i)-dimensional sub-hypercube consisting o vertices starting ith some string x {0, 1} i and suppose that the netor is in state ater moving according to the ith orientation. Let M denote the number o passive vertices (rom Q i ) ith respect to i. Forany {0, 1} n (i+1) let A denote the set o active vertices (rom Q i ) ith respect to i such that they end ith. No e move the pacets along their paths according to the orientation DG i+1. Ho many vertices rom Q i ending ith can be passive ith respect to (i + 1) ater the movement? Let N denote the set o these vertices. Let l [] be such that 2 l N > 2 l 1 (the case N 1 is treated separately). By induction it is easy to sho that there exist distinct z 1,...,z l [] such that or each z j,j [l], there exist to vertices in N o the orm: u 0 : u 1 : x a 0 i z j b x a 1 i z j c n (i+1) n (i+1) here a,b,c are some binary strings. I there ere to vertices in A o the orm: v 0 : v 1 : x d 0 i z j x e 1 i z j n (i+1) n (i+1) here d,e, are some binary strings, then either a pacet rom v 0 to u 1 or a pacet rom v 1 to u 0 cannot move beteen source and destination in the orientation DG i+1, because these to pacets need to cross the edge beteen vertices: x a i z j n (i+1) in the opposite direction. This is a contradiction to our assumption that u 0 and u 1 ill be passive ater the movement and thus there cannot be such v 0,v 1 pair in A. Thereore or each j [l] the (i + z j )th bit o each vertex rom A is determined by bits (i + z j + 1),..., (i + 1). Thisimpliesthat A 2 l. For any l []: N ( 2 A ) 2 l l 1. The inequality holds also in case N 1. Summing the inequalities or each {0, 1} n (i+1) e obtain: N M 2 n (i+1), here N is the total number o vertices rom Q i passive ith respect to (i + 1) ater the movement. Clearly there exists a string q {0, 1} such that at most (M + 2 n (i+1) )/2 vertices starting ith xq are passive ith respect to (i + 1) ater the movement. Let Q i+1 be the hypercube consisting o vertices starting ith xq. The ratio o passive vertices increased rom R i = M 2 n i to R i+1 = (M + 2n (i+1) )/2 2 n (i+1) = R i Thereore i in Q i the ratio as at most i/2 then in Q i+1 the ratio ill be at most (i + 1)/2. Theorem 5. Any shortest-path deadloc-ree pacet routing algorithm o the graph G n based on acyclic orientations requires (n/log n) buers. Proo. Let S be the set o all vertices rom G n having the head on the letmost bit and D be the set o all vertices having the head on the rightmost bit. For every u S and v D there is a unique shortest path rom

5 u to v in G n. It corresponds to changing the bits in hich u and v dier hile moving the head to the right. Let P be the set o the shortest paths rom every u S to every v D. LetC = DG 1,...,DG s be an acyclic orientation cover used to implement shortestpath deadloc-ree pacet routing algorithm on G n. Clearly C is also an acyclic orientation cover or P. By merging the vertices having the same tape content e obtain the n-dimensional hypercube and the path system P becomes all-to-all path system o the n-dimensional hypercube ith monotone paths. The acyclic orientation cover C becomes orientation cover o the ne path system. Using Lemma 4 e obtain that the size o C must be (n/log n). A consequence o Theorems 3 and 5 is: Theorem 6. Graph G n o size N = n2 n has a deadloc-ree shortest-path routing algorithm using O(1) buers per node, but each deadloc-ree shortestpath routing algorithm based on acyclic orientations requires (log N / log log N ) buers per node. 4. Conclusion There are many results on buer-reservation algorithms or speciic topologies, but or general graphs very little is non. It ould be interesting to have better upper and loer bounds on the number o buers required or shortest-path deadloc-ree routing on general graphs. Concerning acyclic orientations it ould be interesting to no ho large the gap beteen routing algorithms based on acyclic orientations and routing algorithms using only plain buers can be. Acnoledgement Thans are due to Ivona Bezáová, Rastislav Kráľovič, Branislav Rovan, Peter Ružiča and Richard Tan or helpul discussions and or careully reading the drat o this paper. Reerences [1] B. Aerbuch, S. Kutten, D. Peleg, Eicient deadloc-ree routing, in: Proc. 10th Annual ACM Symposium on Principles o Distributed Computing, 1991, pp [2] R. Cypher, Minimal, deadloc-ree routing in hypercubic and arbitrary netors, in: Proc. 7th IEEE Symposium on Parallel and Distributed Processing, [3] R. Cypher, L. Gravano, Requirements or deadloc-ree, adaptive pacet routing, in: Proc. 14th Annual ACM Symposium on Principles o Distributed Computing, 1992, pp [4] W.J. Dally, C. Seitz, Deadloc-ree message routing in multiprocessor interconnection netors, IEEE Trans. Comput. 36 (5) (1987) [5] J. Duato, Deadloc-ree adaptive routing algorithms or multicomputers: Evaluation o a ne algorithm, in: Proc. 3rd IEEE Symposium on Parallel and Distributed Processing, [6] I.S. Gopal, Prevention o store-and-orard deadloc in computer netors, IEEE Trans. Commun. 33 (12) (1985) [7] K.D. Günther, Prevention o deadlocs in pacet-sitched data transport systems, IEEE Trans. Commun. 29 (4) (1981) [8] P.M. Merlin, P.J. Scheitzer, Deadloc avoidance in store-andorard netors, IEEE Trans. Commun. 28 (3) (1980) [9] G. Tel, Deadloc-ree pacet routing, in: Introduction to Distributed Algorithms, Cambridge University Press, Cambridge, UK, 1994, Chapter 5. [10] S. Toueg, K. Steiglitz, Some complexity results in the design o deadloc-ree pacet sitching netors, SIAM J. Comput. 10 (4) (1981) [11] S. Toueg, J.D. Ullman, Deadloc-ree pacet sitching netors, SIAM J. Comput. 10 (3) (1981)