A Linear-Time Heuristic for Improving Network Partitions
01 Jan 1982-pp 241-247
TL;DR: An iterative mincut heuristic for partitioning networks is presented whose worst case computation time, per pass, grows linearly with the size of the network.
Abstract: An iterative mincut heuristic for partitioning networks is presented whose worst case computation time, per pass, grows linearly with the size of the network. In practice, only a very small number of passes are typically needed, leading to a fast approximation algorithm for mincut partitioning. To deal with cells of various sizes, the algorithm progresses by moving one cell at a time between the blocks of the partition while maintaining a desired balance based on the size of the blocks rather than the number of cells per block. Efficient data structures are used to avoid unnecessary searching for the best cell to move and to minimize unnecessary updating of cells affected by each move.
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TL;DR: This work presents a new coarsening heuristic (called heavy-edge heuristic) for which the size of the partition of the coarse graph is within a small factor of theSize of the final partition obtained after multilevel refinement, and presents a much faster variation of the Kernighan--Lin (KL) algorithm for refining during uncoarsening.
Abstract: Recently, a number of researchers have investigated a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph [Bui and Jones, Proc. of the 6th SIAM Conference on Parallel Processing for Scientific Computing, 1993, 445--452; Hendrickson and Leland, A Multilevel Algorithm for Partitioning Graphs, Tech. report SAND 93-1301, Sandia National Laboratories, Albuquerque, NM, 1993]. From the early work it was clear that multilevel techniques held great promise; however, it was not known if they can be made to consistently produce high quality partitions for graphs arising in a wide range of application domains. We investigate the effectiveness of many different choices for all three phases: coarsening, partition of the coarsest graph, and refinement. In particular, we present a new coarsening heuristic (called heavy-edge heuristic) for which the size of the partition of the coarse graph is within a small factor of the size of the final partition obtained after multilevel refinement. We also present a much faster variation of the Kernighan--Lin (KL) algorithm for refining during uncoarsening. We test our scheme on a large number of graphs arising in various domains including finite element methods, linear programming, VLSI, and transportation. Our experiments show that our scheme produces partitions that are consistently better than those produced by spectral partitioning schemes in substantially smaller time. Also, when our scheme is used to compute fill-reducing orderings for sparse matrices, it produces orderings that have substantially smaller fill than the widely used multiple minimum degree algorithm.
5,629 citations
26 Aug 2001
TL;DR: A new spectral co-clustering algorithm is used that uses the second left and right singular vectors of an appropriately scaled word-document matrix to yield good bipartitionings and it can be shown that the singular vectors solve a real relaxation to the NP-complete graph bipartitionsing problem.
Abstract: Both document clustering and word clustering are well studied problems. Most existing algorithms cluster documents and words separately but not simultaneously. In this paper we present the novel idea of modeling the document collection as a bipartite graph between documents and words, using which the simultaneous clustering problem can be posed as a bipartite graph partitioning problem. To solve the partitioning problem, we use a new spectral co-clustering algorithm that uses the second left and right singular vectors of an appropriately scaled word-document matrix to yield good bipartitionings. The spectral algorithm enjoys some optimality properties; it can be shown that the singular vectors solve a real relaxation to the NP-complete graph bipartitioning problem. We present experimental results to verify that the resulting co-clustering algorithm works well in practice.
1,836 citations
TL;DR: In this paper, it is shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph.
Abstract: The problem of computing a small vertex separator in a graph arises in the context of computing a good ordering for the parallel factorization of sparse, symmetric matrices. An algebraic approach for computing vertex separators is considered in this paper. It is, shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph. The Laplacian eigenvectors of grid graphs can be computed from Kronecker products involving the eigenvectors of path graphs, and these eigenvectors can be used to compute good separators in grid graphs. A heuristic algorithm is designed to compute a vertex separator in a general graph by first computing an edge separator in the graph from an eigenvector of the Laplacian matrix, and then using a maximum matching in a subgraph to compute the vertex separator. Results on the quality of the separators computed by the spectral algorithm are presented, and these are compared with separators obtained from other algorith...
1,762 citations
TL;DR: This paper presents and study a class of graph partitioning algorithms that reduces the size of the graph by collapsing vertices and edges, they find ak-way partitioning of the smaller graph, and then they uncoarsen and refine it to construct ak- way partitioning for the original graph.
1,715 citations
Cites methods from "A Linear-Time Heuristic for Improvi..."
...minima. One commonly used variation of the KL algorithm for bisection refinement is due to Fiduccia-Mattheyses [ 4 ]....
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...Kernighan-Lin (KL) partitioning algorithm [17] and their variants [ 4 , 12]....
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TL;DR: This paper employs approximation algorithms for the graph-partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities, and defines the network community profile plot, which characterizes the "best" possible community—according to the conductance measure—over a wide range of size scales.
Abstract: A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins with the premise that a community or a cluster should be thought of as a set of nodes that has more and/or better connections between its members than to the remainder of the network. In this paper, we explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions. Rather than defining a procedure to extract sets of nodes from a graph and then attempting to interpret these sets as "real" communities, we employ approximation algorithms for the graph-partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be i...
1,660 citations
Cites methods from "A Linear-Time Heuristic for Improvi..."
...This may be combined with local improvement methods like the Kernighan-Lin and Fiduccia-Mattheyses procedures [97, 65], which are fast and can climb out of some local minima....
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References
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TL;DR: A heuristic method for partitioning arbitrary graphs which is both effective in finding optimal partitions, and fast enough to be practical in solving large problems is presented.
Abstract: We consider the problem of partitioning the nodes of a graph with costs on its edges into subsets of given sizes so as to minimize the sum of the costs on all edges cut. This problem arises in several physical situations — for example, in assigning the components of electronic circuits to circuit boards to minimize the number of connections between boards. This paper presents a heuristic method for partitioning arbitrary graphs which is both effective in finding optimal partitions, and fast enough to be practical in solving large problems.
5,082 citations
26 Jun 1972
TL;DR: This paper expands on several aspects of the discrepancy: 1) its source, 2) the circumstances under which it is likely to be significant, and its magnitude for typical circuits, and 3) the comparative difficulty and expense of using a more appropriate representation.
Abstract: Partitioning algorithms for electrical circuits are often based on the heuristic manipulation of a simple element-to-element interconnection matrix. However, the element-to-element interconnection matrix does not properly represent an electrical interconnection, or “net”, among more than two elements. This paper expands on several aspects of the discrepancy: 1) its source, 2) the circumstances under which it is likely to be significant, and its magnitude for typical circuits, and 3) the comparative difficulty and expense of using a more appropriate representation.A physically correct “net-cut” model is presented. This model is computationally straightforward and is easily adapted to the typical heuristic solution strategies. The “net-cut” model is coupled with the Kernighan-Lin partitioning algorithm [3]; using the same algorithm, comparisons with the “edge-cut” model demonstrate that the correct model reduces net-cuts by 19 to 50% for four digital logic circuits.
199 citations
01 Jan 1977
TL;DR: A class of min-cut placement algorithms for solving some assignment problems related to the physical implementation of electrical circuits and the need for abandoning classical objective functions based upon distance, and introducing new objective functionsbased upon "signals cut".
Abstract: In this paper we present a class of min-cut placement algorithms for solving some assignment problems related to the physical implementation of electrical circuits. We discuss the need for abandoning classical objective functions based upon distance, and introduce new objective functions based upon "signals cut." The number of signals cut by a line c is a lower bound on the number of routing tracks which must cross c in routing the circuit. Three specific objective functions are introduced and the relationship between one of these and a classical distance measure based upon half-perimeter is presented. Two min-cut placement algorithms are presented. They are referred to as Ouadrature and Slice/Bisection. The concepts of a block and cut line are introduced. These two entities are the major constructs in developing any new min-cut placement algorithm.Most of the concepts presented have been implemented, and some experimental results are given.
194 citations
23 Jun 1980
TL;DR: This paper deals with placement and routing techniques for master slice LSIs to make wiring density on the chip more uniform.
Abstract: This paper deals with placement and routing techniques for master slice LSIs. The basic idea of both techniques is to make wiring density on the chip more uniform. Algorithms and some experimental results are described.
32 citations