Showing papers on "Spectral graph theory published in 1995"
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TL;DR: A new domain mapping algorithm is presented that extends recent work in which ideas from spectral graph theory have been applied to this problem and provides better decompositions arrived at more economically and robustly than with previous spectral methods.
Abstract: Efficient use of a distributed memory parallel computer requires that the computational load be balanced across processors in a way that minimizes interprocessor communication. A new domain mapping algorithm is presented that extends recent work in which ideas from spectral graph theory have been applied to this problem. The generalization of spectral graph bisection involves a novel use of multiple eigenvectors to allow for division of a computation into four or eight parts at each stage of a recursive decomposition. The resulting method is suitable for scientific computations like irregular finite elements or differences performed on hypercube or mesh architecture machines. Experimental results confirm that the new method provides better decompositions arrived at more economically and robustly than with previous spectral methods. This algorithm allows for arbitrary nonnegative weights on both vertices and edges to model inhomogeneous computation and communication. A new spectral lower bound for graph bi...
550 citations
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22 Jan 1995TL;DR: This work considers two popular spectral separator algorithms, and provides counterexamples that show these algorithms perform poorly on certain graphs, and introduces some facts about the structure of eigenvectors of certain types of Laplacian and symmetric matrices.
Abstract: : Computing graph separators is an important step in many graph algorithms. A popular technique for finding separators involves spectral methods. However, there is not much theoretical analysis of the quality of the separators produced by this technique; instead it is usually claimed that spectral methods 'work well in practice.' We present an initial attempt at such analysis. In particular, we consider two popular spectral separator algorithms, and provide counterexamples that show these algorithms perform poorly on certain graphs. We also consider a generalized definition of spectral methods that allows the use of some specified number of the eigenvectors corresponding to the smallest eigenvalues of the Laplacian matrix of a graph, and show that if such algorithms use a constant number of eigenvectors, then there are graphs for which they do no better than using only the second smallest eigenvector. Further, when applied to these graphs the algorithm based on the second smallest eigenvector performs poorly with respect to theoretical bounds. Even if an algorithm meeting the generalized definition uses up to n% for 0 < E < % eigenvectors, there exist graphs for which the algorithm fails to find a separator with a cut quotient within %-4-e - 1 of the isoperimetric number. We also introduce some facts about the structure of eigenvectors of certain types of Laplacian and symmetric matrices; these facts provide the basis for the analysis of the counterexamples.
82 citations
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TL;DR: In this article, lower bounds for the toughness of a graph in terms of its eigenvalues were derived, and the best possible lower bounds were derived for each eigenvalue.
34 citations
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01 Jan 1995TL;DR: Spectral graph theory has a long history in representation theory and number theory and has been very useful for examining the spectra of strongly regular graphs with symmetries as mentioned in this paper, however, spectral graph theory is mostly algebraic.
Abstract: The study of eigenvalues of graphs has a long history. Since the early days, representation theory and number theory have been very useful for examining the spectra of strongly regular graphs with symmetries. In contrast, recent developments in spectral graph theory concern the effectiveness of eigenvalues in studying general (unstructured) graphs. The concepts and techniques, in large part, use essentially geometric methods.(Still, extremal and explicit constructions are mostly algebraic [20].) There has been a significant increase in the interaction between spectral graph theory and many areas of mathematics as well as other disciplines, such as physics, chemistry, communication theory, and computer sciences.
12 citations
01 Jan 1995
TL;DR: Anew domainmapping algorithm is presented that extends recent work in which ideas from spectral graph theory have been applied to this problem and provides better decompositions arrived at more economically and robustly than with previous spectral methods.
Abstract: Efficient use of a distributed memory parallel computer requires that the computational load be balanced across processors in away thatminimizes interprocessorcommunication. Anew domainmapping algorithm is presented that extends recent work in which ideas from spectral graph theory have been applied to this problem. The generalization of spectral graph bisection involves a novel use of multiple eigenvectors to allow for division of a computation into four or eight parts at each stage of a recursive decomposition. The resulting method is suitable for scientific computations like irregular finite elements or differences performed on hypercube or mesh architecture machines. Experimental results confirm that the new method provides better decompositions arrived at more economically and robustly than with previous spectral methods. This algorithm allows for arbitrary nonnegative weights on both vertices and edges to model inhomogeneous computation and communication. A new spectral lower bound for graph bisection is also presented. the computational load be balanced across processors in a way that minimizes interprocessor communication. This mapping requirement can be abstracted to a graph problem in which nodes represent computation, edges represent communication, and the objective is to assign an equal number of vertices to each processor in a way that, in some metric, minimizes the number ofedges crossing between processors. Extensive practical experience has shown that the quality ofthismapping has a substantialimpactonperformance, hence there is considerable interest in effective mapping algorithms. Finding a mapping that actually minimizes communication between balanced sets is an NP-hard problem (9), so it is unlikely that an efficient, general algorithm exists. The practical importance of this problem has, however, motivated a variety of heuristic approaches. A thorough review of these methods and the extensive literature associated with them is beyond the scope of this paper. We simply note that the established methods range from quick, linear time algorithms based on geometric assumptions (17), (19) or local graph information (5), (15) to very slow algorithms which approximate a global search for the minimum using genetic operators or simulated annealing 14), 16). The faster heuristics often do not provide mappings of adequate quality for bench-marking purposes or for performance-critical codes which will be used many times, while the more expensive mapping techniques are generally impractical for large problems. This paper describes amethoddesigned to provide high quality partitionings at moderate cost.