EMBEDDING VARIANTS OF HYPERCUBES WITH DILATION 2

The Computer Journal, 2003

Embeddings of various graph classes into hypercubes have been widely studied. Almost all these classes are regularly structured graphs such as meshes, complete trees or pyramids. In this paper, we present a general method for one-to-one embeddings of irregularly structured graphs into their optimal hypercubes, based on extended edge bisectors of graphs. An extended edge bisector is an edge bisector with the additional property that a certain subset of the vertices is distributed more or less evenly among the two halves of the bisected graph. The dilation and congestion of our embedding depends on the quality of the extended edge bisector. Moreover, if the extended bisection can be computed efficiently on the hypercube, then so can the embedding. Our embedding technique can also be applied to embeddings into hypercube-like topologies such as folded hypercubes, twisted cubes, crossed cubes, Möbius cubes, Fibonacci cubes or star graphs.

One important aspect of efficient use of a hypercube computer to solve a given problem is the assignment of subtasks to processors in such a way that the communication overhead is low. The subtasks and their inter-communication requirements can be modeled by a graph, and the assignment of subtasks to processors viewed as an embedding of the task graph into the graph of the hypercube network. We survey the known results concerning such embeddings, including expansion/dilation tradeoffs for general graphs, embeddings of meshes and trees, packings of multiple copies of a graph, the complexity of finding good embeddings, and critical graphs which are minimal with respect to some property. In addition, we describe several open problems. . the hypercube is used to simulate a network with graph G the nodes of G must be mapped to the nodes of Q n , and, in order to keep communication overhead down, adjacent nodes of G should map to adjacent nodes of Q n insofar as possible. In designing (or adapting) an algorithm that performs a task T on the hypercube network, T is modeled with a "task graph", G T , in which the nodes represent subtasks and the edges represent communication requirements between the corresponding subtasks. Once again, the efficiency of the implementation depends strongly on the nature of the mapping into Q n . To keep communication overhead low, the nodes of G T must be mapped to the nodes of Q n so that pairs of adjacent nodes of G T map to pairs of adjacent nodes of Q n .

Lecture Notes in Computer Science, 1996

Embeddings of several graph classes into hypercubes have been widely studied. Unfortunately, almost all investigated graph classes are regular graphs such as meshes, complete trees, pyramids. In this paper, we present a general method for one-to-one embedding irregular graphs into their optimal hypercubes based on extended-edge-bisectors of graphs. An extended-edge-bisector is an edge-bisector with the additional property that a subset of the vertices is distributed more or less evenly among the two halves of the bisected graph. The dilation and congestion of the embedding depends on the quality of the extended-edge-bisector. Moreover, if the extended bisection can be efficiently computed on the hypercube, so can the embedding.

IEEE Transactions on Computers, 1995

In this paper we consider the Supercube, a new interconnection network derived from the hypercube introduced by Sen in 10]. The Supercube has the same diameter and connectivity of a hypercube but can be realized for any number of nodes not only for powers of 2.

Information Processing Letters, 1987

The problem of embedding a graph into a fixed-size hypercube is shown to be NP-complete. This work complements recent work of the present authors showing that deciding whether a graph is embeddable into any size hypercube is NP-complete as well. The reduction is from 3-partition.

Electronic Notes in Discrete Mathematics, 2006

ABSTRACT Isometric subgraphs of hypercubes are known as partial cubes. These graphs have first been investigated by Graham and Pollak [R.L Graham, H.Pollak On the addressing problem for loop switching, Bell System Technol. J. 50 (1971) 2495–2519] and Djokovic̀ [D. Djokovic̀, Distance preserving subgraphs of the hypercubes, J. Combin. Theory, Ser B41 (1973), 263–267]. Several papers followed with various characterizations of partial cubes. In this paper, we determine all subdivisions of a given configuration which can be embedded isometrically in the hypercube. More specially, we deal with the case where this configuration is a connected graph of order 4 on one hand and the case where the configuration is a fan Fk(k⩾3) on the other hand. Finally, we conjecture that a subdivision of a complete graph of order n(n⩾5) is a partial cube if and only if this one is isomorphic to S(Kn) or there exists n−1 edges of Kn adjacent to a common vertex in the subdivision and the other edges of Kn contain odd added vertices. This proposition is true when the order n∈{4,5,6}.

Discrete Applied Mathematics, 2014

The aim of this paper is to generalize the Congestion Lemma, which has been considered an efficient tool to compute the minimum wirelength and thereby obtain the minimum wirelength of embedding hypercubes into sibling trees.

Mathematical and Computer Modelling, 1988

One important aspect of efficient use of a hypercube computer to solve a given problem is the assignment of subtasks to processors in such a way that the communication overhead is low. The subtasks and their inter-communication requirements can be modeled by a ...

Journal of Algorithms, 2002

In this paper, we present a one-to-one embedding of a graph with bounded treewidth into its optimal hypercube. This is the first time that embeddings of graphs with a highly irregular structure into hypercubes are investigated. The presented embedding achieves dilation of at most 3 log d + 1 t + 1 + 8 and nodecongestion of at most O d dt 3 , where t denotes the treewidth of the graph and d denotes the maximal degree of a vertex in the graph. Provided that the graph is given by its tree-decomposition the embedding can be computed efficiently on the hypercube itself. In particular, the embedding of a graph with constant treewidth and constant degree can be computed in time O log 2 n log log log n log * n . For graphs with constant treewidth, a minimal tree-decomposition can be computed efficiently in parallel due to a result of Bodlaender and Hagerup. In this case, the embedding can be computed on the hypercube in time O log 2 n d 2 + log n log log 2 n .  2002 Elsevier Science (USA)

SIAM Journal on Computing, 1992

The boolean hypercube is a particularly versatile network for parallel computing.

Journal of Computer and System Sciences, 1992

We consider efficient simulations of mesh connected networks (or good representations of array structures) by hypercube machines. In particular, we consider embedding a mesh or grid G into the smallest hypercube that at least as many points as G, called the optima/ hypercube for G. In order to minimize simulation time we derive embeddings which minimize dilation, i.e., the maximum distance in the hypercube between images of adjacent points of G. Our results are: (1) There is a dilation 2 embedding of the [m x k] grid into its optimal hypercube, provided that and (2) For any k < d, there is a dilation k + 1 embedding of a [a, x a2 x '. x a,] grid into its optimal hypercube, provided that x.:':: [log a,] + rlog Bkl < rc:'=, log a,], where ',

Discrete Mathematics, 2009

Isometric subgraphs of hypercubes are known as partial cubes. These graphs have first been investigated by Graham and Pollack [R.L. Graham, H. Pollack, On the addressing problem for loop switching, Bell System Technol. J. 50 (1971) 2495-2519; and D. Djoković, Distance preserving subgraphs of hypercubes, J. Combin. Theory Ser. B 14 (1973) 263-267]. Several papers followed with various characterizations of partial cubes. In this paper, we determine all subdivisions of a given configuration which can be embedded isometrically in the hypercube. More specifically, we deal with the case where this configuration is a connected graph of order 4, a complete graph of order 5 and the case of a k-fan F k (k ≥ 3).

Journal of Parallel and Distributed Computing, 1993

When each G j is a complete binary tree or a leap tree of 2 k-1 nodes, we describe an embedding achieving a dilation of 2 and a load of 5 and 6, respectively. For the cases when each G j is a linear array or a 2-dimensional mesh of 2 k nodes, we describe embeddings that achieve a dilation of 1 and an optimal load of 2 and 4, respectively. Using these embeddings, we also show that Tl complete binary trees, T2 leap trees, T3 linear arrays, and T 4 meshes can simultaneously be embedded into H with 1 constant dilation and load, L Tj :$ k.

Graph embedding or graph mapping is an important problem in interconnection networks. A good mapping is said to exist when adjacent processors in the guest network are mapped to reasonably close processors in the host network (i.e. small dilation) and when the paths between adjacent processors in the guest network are chosen in such a way that the congestion at each host node and across each host edge is moderately small (i.e. small nodeand edge-congestion). In the case of mapping guest networks onto smaller hosts, the processors of the host have to be assigned to about the same number of processes from the guest (i.e. small load-factor). In this paper an approach for embedding linear array onto Tree-Hypercubes networks is proposed.

Mathematical Methods in the Applied Sciences, 2016

We address various topologies (de Bruijn, chordal ring, generalized Petersen, meshes) in various ways (isometric embedding, embedding up to scale, embedding up to a distance) in a hypercube or a half-hypercube. Example of obtained embeddings: infinite series of hypercube embeddable Bubble Sort and Double Chordal Rings topologies, as well as of regular maps.