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Routing Permutations on Graphs via Matchings
 SIAM Journal on Discrete Mathematics
, 1994
"... We consider a class of routing problems on connected graphs G. Initially, each vertex v of G is occupied by a “pebble ” which has a unique destination π(v) in G (so that π is a permutation of the vertices of G). It is required to route all the pebbles to their respective destinations by performing a ..."
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Cited by 36 (2 self)
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We consider a class of routing problems on connected graphs G. Initially, each vertex v of G is occupied by a “pebble ” which has a unique destination π(v) in G (so that π is a permutation of the vertices of G). It is required to route all the pebbles to their respective destinations by performing a sequence of moves of the following type: A disjoint set of edges is selected and the pebbles at each edge’s endpoints are interchanged. The problem of interest is to minimize the number of steps required for any possible permutation π. In this paper we investigate this routing problem for a variety of graphs G, including trees, complete graphs, hypercubes, Cartesian products of graphs, expander graphs and Cayley graphs. In addition, we relate this routing problem to certain network flow problems, and to several graph invariants including diameter, eigenvalues and expansion coefficients. 2 1
ManytoMany Routing on Trees via Matchings
, 1996
"... In this paper we present an extensive study of manytomany routing on trees under the matching routing model. Our study includes online and offline algorithms. We present an asymptotically optimal online algorithm which routes k packets to their destination within d(k \Gamma 1) + d \Delta dist r ..."
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Cited by 10 (4 self)
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In this paper we present an extensive study of manytomany routing on trees under the matching routing model. Our study includes online and offline algorithms. We present an asymptotically optimal online algorithm which routes k packets to their destination within d(k \Gamma 1) + d \Delta dist routing steps, where d is the degree of tree T on which the routing takes place and dist is the maximum distance any packet has to travel. We also present an offline algorithm that solves the same problem within 2(k \Gamma 1)+dist steps. The analysis of our algorithms is based on the establishment of a close relationship between the matching and the hotpotato routing models that allows us to apply tools which were previously used exclusively in the analysis of hotpotato routing.
ManytoOne Packet Routing via Matchings
 In Proceedings of the Third Annual International Computing and Combinatorics Conference
, 1997
"... In this paper we study the packet routing problem under the matching model proposed by Alon, Chung and Graham [1]. We extend the model to allow more than one packet per origin and destination node. We give tight bounds for the manytoone routing number for complete graphs, complete bipartite graphs ..."
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Cited by 5 (2 self)
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In this paper we study the packet routing problem under the matching model proposed by Alon, Chung and Graham [1]. We extend the model to allow more than one packet per origin and destination node. We give tight bounds for the manytoone routing number for complete graphs, complete bipartite graphs and linear arrays. We also present an efficient algorithm for manytoone routing on an trees (and therefore any graph). Finally, we give bounds for routing arbitrary relations in this model. 1 Introduction Routing packets arises naturally in the design of largescale parallel computers and the study of data flow in parallel computing. Packet routing consists of moving packets of data from each node of a network to the other nodes in the network. The goal is to move all of the packets to their desired locations as quickly as possible. Various routing problems have been extensively studied under different models. We refer the reader to [4] for a survey of the topic. In this paper, we study ...
Permutation Routing via Cayley graphs with an Example for Bus Interconnection Networks
 IN PROCEEDINGS OF DIMACS WORKSHOP ON INTERCONNECTION NETWORKS AND MAPPING AND SCHEDULING PARALLEL COMPUTATIONS, DIMACS SERIES IN DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE
, 1994
"... Cayley graphs have been used extensively to design interconnection networks and provide a natural setting for studying pointtopoint routing [1, 2, 3, 5, 6, 7, 12]. The extension of these techniques to the more important problem of permutation routing on interconnection networks presents fundamen ..."
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Cited by 1 (1 self)
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Cayley graphs have been used extensively to design interconnection networks and provide a natural setting for studying pointtopoint routing [1, 2, 3, 5, 6, 7, 12]. The extension of these techniques to the more important problem of permutation routing on interconnection networks presents fundamental problems. This is due to the potentially explosive growth in both the size of the graph and the number of generating permutations, referred to as onestep permutation routes, used to define the underlying graph. This paper describes a technique for moderating that growth so that the techniques in [8] can be applied for finding optimal permutation routes. In a particularly striking example, a bus interconnection architecture involving 1.0×10 17 permutations (nodes of the Cayley graph) is reduced to a computation on a graph with only 3,950 nodes. Further, it is shown how many of the 58,624 generators (directed edges labelled by onestep permutation routes) at each node of the graph may be eliminated as locally redundant.
ROUTING PERMUTATIONS ON GRAPHS VIA MATCHINGS
, 1994
"... A class of routing problems on connected graphs G is considered. Initially, each vertex v of G is occupied by a "pebble " that has a unique destination (v) in G (so that r is a permutation of the vertices of G). It is required that all the pebbles be routed to their respective destinations ..."
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A class of routing problems on connected graphs G is considered. Initially, each vertex v of G is occupied by a "pebble " that has a unique destination (v) in G (so that r is a permutation of the vertices of G). It is required that all the pebbles be routed to their respective destinations by performing a sequence of moves of the following type: A disjoint set of edges is selected, and the pebbles at each edge’s endpoints are interchanged. The problem of interest is to minimize the number of steps required for any possible permutation r. This paper investigates this routing problem for a variety of graphs G, including trees, complete graphs, hypercubes, Cartesian products of graphs, expander graphs, and Cayley graphs. In addition, this routing problem is related to certain network flow problems, and to several graph invariants including diameter, eigenvalues, and expansion coefficients.
ROUTING NUMBERS OF CYCLES, COMPLETE BIPARTITE GRAPHS, AND HYPERCUBES
, 2010
"... The routing number rt(G) of a connected graph G is the minimum integer r so that every permutation of vertices can be routed in r steps by swapping the ends of disjoint edges. In this paper, we study the routing numbers of cycles, complete bipartite graphs, and hypercubes. We prove that rt(Cn) = n ..."
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The routing number rt(G) of a connected graph G is the minimum integer r so that every permutation of vertices can be routed in r steps by swapping the ends of disjoint edges. In this paper, we study the routing numbers of cycles, complete bipartite graphs, and hypercubes. We prove that rt(Cn) = n − 1(forn≥3) and for s ≥ t, rt(Ks,t) = ⌊ 3s ⌋ + O(1). We also prove 2t n +1 ≤ rt(Qn) ≤ 2n − 2forn≥3. The lower bound rt(Qn) ≥ n + 1 was previously conjectured
c ○ 1999 Society for Industrial and Applied Mathematics OPTIMAL BOUNDS FOR MATCHING ROUTING ON TREES ∗
"... Abstract. The permutation routing problem is studied for trees under the matching model. By introducing a novel and useful (socalled) caterpillar tree partition, we prove that any permutation on an nnode tree (and thus graph) can be routed in 3 n + O(log n) steps. This answers an open 2 ..."
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Abstract. The permutation routing problem is studied for trees under the matching model. By introducing a novel and useful (socalled) caterpillar tree partition, we prove that any permutation on an nnode tree (and thus graph) can be routed in 3 n + O(log n) steps. This answers an open 2
Optimal Permutation Routing for Lowdimensional Hypercubes
, 2006
"... We consider the offline problem of routing a permutation of tokens on the nodes of a ddimensional hypercube, under a queueless MIMD communication model (under the constraints that each hypercube edge may only communicate one token per communication step, and each node may only be occupied by a sing ..."
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We consider the offline problem of routing a permutation of tokens on the nodes of a ddimensional hypercube, under a queueless MIMD communication model (under the constraints that each hypercube edge may only communicate one token per communication step, and each node may only be occupied by a single token between communication steps). For a ddimensional hypercube, it is easy to see that d communication steps are necessary. We develop a theory of “separability ” which enables an analytical proof that d steps suffice for the case d = 3, and facilitates an experimental verification that d steps suffice for d = 4. This result improves the upper bound for the number of communication steps required to route an arbitrary permutation on arbitrarily large hypercubes to 2d − 4. We also find an interesting sideresult, that the number of possible communication steps in a ddimensional hypercube is the same as the number of perfect matchings in a (d + 1)dimensional hypercube, a combinatorial quantity for which there is no closedform expression. Finally we present some experimental observations which may lead to a proof of a more general result for arbitrarily large dimension d.
Cayley Graph Techniques for Permutation Routing on Bus Interconnection Networks
"... Cayley graphs and elementary group theory are used to efficiently find minimum length permutation routes in a bus interconnection network. Cayley graphs have been used extensively to design interconnection networks and provide a natural setting for studying pointtopoint routing [1, 2, 3, 4], bu ..."
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Cayley graphs and elementary group theory are used to efficiently find minimum length permutation routes in a bus interconnection network. Cayley graphs have been used extensively to design interconnection networks and provide a natural setting for studying pointtopoint routing [1, 2, 3, 4], but the technique has not been widely used for the more important problem of permutation routing. This is due to the potentially explosive growth in both the size of the graph and the number of generating permutations, referred to as onestep permutation routes, used to define the underlying graph. This paper describes a method for moderating that growth, using techniques from [6, 7], and applies that method to a model of bus interconnection networks. Some techniques from [6] are first reviewed. They show how the existence of a natural set of graph automorphisms that can be used to construct a homomorphism into a much smaller, reduced multigraph, from which shortest permutation routes c...