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44
On Cubical Graphs
 JOURNAL OF COMBINATORIAL THEORY (B) 18, 86 % (1975)
, 1975
"... It is frequently of interest to represent a given graph G as a subgraph of a graph H which has some special structure. A particularly useful class of graphs in which to embed G is the class of ndimensional cubes. This has found applications, for example, in coding theory, data transmission, and lin ..."
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Cited by 82 (5 self)
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It is frequently of interest to represent a given graph G as a subgraph of a graph H which has some special structure. A particularly useful class of graphs in which to embed G is the class of ndimensional cubes. This has found applications, for example, in coding theory, data transmission, and linguistics. In this note, we study the structure of those graphs 6, called cubical graphs (not to be confused with cubic graphs, those graphs for which all vertices have degree 3), which can be embedded into an ndimensional cube. A basic technique used is the investigation of graphs which are critically nonembeddable, i.e., which can not be embedded but all of whose subgrapbs can be embedded.
Efficient Communication Strategies for AdHoc Wireless Networks
, 2000
"... An adhoc wireless network is a collection of wireless mobile hosts forming a temporary network without the aid of any established infrastructure or centralized administration. This type of network is of great importance in situations where it is very difficult to provide the necessary infrastructur ..."
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Cited by 37 (3 self)
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An adhoc wireless network is a collection of wireless mobile hosts forming a temporary network without the aid of any established infrastructure or centralized administration. This type of network is of great importance in situations where it is very difficult to provide the necessary infrastructure, but it is a challenging task to enable fast and reliable communication within such a network. In this paper, we model and analyze the performance of socalled powercontrolled adhoc wireless networks: networks where the mobile hosts are able to change their transmission power. We concentrate on finding schemes for routing arbitrary permutations in these networks. In general, it is NPhard even to find a n 1 approximation for any constant to the fastest possible strategy for routing a given permutation problem on n mobile hosts. However, we here demonstrate that if we allow ourselves to consider slightly less general problems, efficient solutions can be found. We first demonstrate that there is a natural class of distributed schemes for handling nodetonode communication on top of which online route selection and scheduling strategies can be constructed such that the performance of this class of schemes can be exploited in a nearly optimal way for routing permutations in any static powercontrolled adhoc network. We then demonstrate
Packet Routing In FixedConnection Networks: A Survey
, 1998
"... We survey routing problems on fixedconnection networks. We consider many aspects of the routing problem and provide known theoretical results for various communication models. We focus on (partial) permutation, krelation routing, routing to random destinations, dynamic routing, isotonic routing ..."
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Cited by 35 (3 self)
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We survey routing problems on fixedconnection networks. We consider many aspects of the routing problem and provide known theoretical results for various communication models. We focus on (partial) permutation, krelation routing, routing to random destinations, dynamic routing, isotonic routing, fault tolerant routing, and related sorting results. We also provide a list of unsolved problems and numerous references.
Fast Gossiping by Short Messages
, 1995
"... Gossiping is the process of information diffusion in which each node of a network holds a packet that must be communicated to all other nodes in the network. We consider the problem of gossiping in communication networks under the restriction that communicating nodes can exchange up to a fixed numbe ..."
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Cited by 31 (12 self)
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Gossiping is the process of information diffusion in which each node of a network holds a packet that must be communicated to all other nodes in the network. We consider the problem of gossiping in communication networks under the restriction that communicating nodes can exchange up to a fixed number p of packets at each round. In the first part of the paper we study the extremal case p = 1 and we exactly determine the optimal number of communication rounds to perform gossiping for several classes of graphs, including Hamiltonian graphs and complete kary trees. For arbitrary graphs we give asymptotically matching upper and lower bounds. We also study the case of arbitrary p and we exactly determine the optimal number of communication rounds to perform gossiping under this hypothesis for complete graphs, hypercubes, rings, and paths. Finally, we investigate the problem of determining sparse networks in which gossiping can be performed in the minimum possible number of rounds.
Delayed path coupling and generating random permutations via distributed stochastic processes
, 1999
"... We analyze various stochastic processes for generating permutations almost uniformly at random in distributed and parallel systems. All our protocols are simple, elegant and are based on performing disjoint transpositions executed in parallel. The challenging problem of our concern is to prove that ..."
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Cited by 18 (3 self)
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We analyze various stochastic processes for generating permutations almost uniformly at random in distributed and parallel systems. All our protocols are simple, elegant and are based on performing disjoint transpositions executed in parallel. The challenging problem of our concern is to prove that the output configurations in our processes reach almost uniform probability distribution very rapidly, i.e. in a (low) polylogarithmic time. For the analysis of the aforementioned protocols we develop a novel technique, called delayed path coupling, for proving rapid mixing of Markov chains. Our approach is an extension of the path coupling method of Bubley and Dyer. We apply delayed path coupling to three stochastic processes for generating random permutations. For one
Optimal Bounds for Matching Routing on Trees
 In Proceedings of the 8th Annual ACMSIAM Symposium on Discrete Algorithms
, 1997
"... 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 2 n + O(log n) steps. This answers an open problem of Alon, ..."
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Cited by 18 (1 self)
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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 2 n + O(log n) steps. This answers an open problem of Alon, Chung and Graham. Key words. matching routing, offline algorithms, trees AMS subject classifications. 05C, 68M, 68R 1 Introduction Routing problems on networks arise in different fields such as communications, parallel architectures and VLSI theory, and have been extensively studied in recent years (see [9, 10] for a comprehensive survey). In this paper, we study permutation routing under the matching model, which was proposed by Alon, Chung and Graham[2]. The routing of this type is described as follows. Given a graph G = (V; E) with vertex set V and edge set E. Initially, each vertex v of G is occupied by a "packet" p. To each packet p is associated a destination ß(v) 2 V , so that di...
Optimal Sequential Gossiping by Short Messages
 DAMATH: DISCRETE APPLIED MATHEMATICS AND COMBINATORIAL OPERATIONS RESEARCH AND COMPUTER SCIENCE, VOL 86
, 1998
"... Gossiping is the process of information diffusion in which each node of a network holds a block that must be communicated to all the other nodes in the network. We consider the problem of gossiping in communication networks under the restriction that communicating nodes can exchange up to a fixed nu ..."
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Cited by 13 (3 self)
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Gossiping is the process of information diffusion in which each node of a network holds a block that must be communicated to all the other nodes in the network. We consider the problem of gossiping in communication networks under the restriction that communicating nodes can exchange up to a fixed number p of blocks during each call. We study the minimum numbers of call necessary to perform gossiping among n processor for any arbitrary fixed upper bound on the message size p.
Discrete isoperimetric inequalities
 Surveys in Differential Geometry IX, International Press, 53–82
, 2004
"... ..."
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 12 (5 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.
Routing on Trees
 INFORMATION PROCESSING LETTERS
, 1994
"... In this paper, we study the permutation packet routing problem on trees. We show that every permutation can be routed on a tree of n vertices in n  1 routing steps. We provide two algorithms that produce such routing schedules. The first algorithm builds in O(n²) time a schedule that needs O(n²) bi ..."
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Cited by 12 (3 self)
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In this paper, we study the permutation packet routing problem on trees. We show that every permutation can be routed on a tree of n vertices in n  1 routing steps. We provide two algorithms that produce such routing schedules. The first algorithm builds in O(n²) time a schedule that needs O(n²) bits for its description while the second one produces in O(n³) time a schedule that can be described with O(n log n) bits. Moreover, we describe an online algorithm that completes the routing of any permutation in n  1 routing steps by using at each vertex v buffering area of size at most d²(v) where d(v) is the degree of vertex v. Our results provide upper bounds on the number of routing steps required to route a permutation on an arbitrary connected graph G since the routing can be done by using only the edges of a spanning tree of G.