Topic

# Bipartite graph

About: Bipartite graph is a research topic. Over the lifetime, 15108 publications have been published within this topic receiving 246712 citations. The topic is also known as: bigraph & bipartite network.

##### Papers published on a yearly basis

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TL;DR: A generic message-passing algorithm, the sum-product algorithm, that operates in a factor graph, that computes-either exactly or approximately-various marginal functions derived from the global function.

Abstract: Algorithms that must deal with complicated global functions of many variables often exploit the manner in which the given functions factor as a product of "local" functions, each of which depends on a subset of the variables. Such a factorization can be visualized with a bipartite graph that we call a factor graph, In this tutorial paper, we present a generic message-passing algorithm, the sum-product algorithm, that operates in a factor graph. Following a single, simple computational rule, the sum-product algorithm computes-either exactly or approximately-various marginal functions derived from the global function. A wide variety of algorithms developed in artificial intelligence, signal processing, and digital communications can be derived as specific instances of the sum-product algorithm, including the forward/backward algorithm, the Viterbi algorithm, the iterative "turbo" decoding algorithm, Pearl's (1988) belief propagation algorithm for Bayesian networks, the Kalman filter, and certain fast Fourier transform (FFT) algorithms.

6,637 citations

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TL;DR: This paper shows how to construct a maximum matching in a bipartite graph with n vertices and m edges in a number of computation steps proportional to $(m + n)\sqrt n $.

Abstract: The present paper shows how to construct a maximum matching in a bipartite graph with n vertices and m edges in a number of computation steps proportional to $(m + n)\sqrt n $.

2,785 citations

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Williams College

^{1}TL;DR: In this article, an exact formula for the entanglement of formation for all mixed states of two qubits having no more than two nonzero eigenvalues was given, and evidence suggests that the formula is valid for all states of this system.

Abstract: The ``entanglement of formation'' of a mixed state \ensuremath{\rho} of a bipartite quantum system can be defined as the minimum number of singlets needed to create an ensemble of pure states that represents \ensuremath{\rho}. We find an exact formula for the entanglement of formation for all mixed states of two qubits having no more than two nonzero eigenvalues, and we report evidence suggesting that the formula is valid for all states of this system.

2,386 citations

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26 Aug 2001TL;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

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TL;DR: An exact method is given which performs better than the Randall-Brown algorithm and is able to color larger graphs and the new heuristic methods, the classical methods, and the exact method are compared.

Abstract: This paper describes efficient new heuristic methods to color the vertices of a graph which rely upon the comparison of the degrees and structure of a graph. A method is developed which is exact for bipartite graphs and is an important part of heuristic procedures to find maximal cliques in general graphs. Finally an exact method is given which performs better than the Randall-Brown algorithm and is able to color larger graphs, and the new heuristic methods, the classical methods, and the exact method are compared.

1,510 citations