## Fast Estimation of Diameter and Shortest Paths (without Matrix Multiplication) (1996)

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Citations: | 68 - 2 self |

### BibTeX

@MISC{Aingworth96fastestimation,

author = {D. Aingworth and C. Chekuri and R. Motwani},

title = {Fast Estimation of Diameter and Shortest Paths (without Matrix Multiplication)},

year = {1996}

}

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### Abstract

this paper is organized as follows. We begin by presenting some definitions and useful observations in Section 2. In Section 3, we describe the algorithms for distinguishing between graphs of diameter 2 and 4, and the extension to obtaining a ratio 2=3 approximation to the diameter. Then, in Section 4, we apply the ideas developed in estimating the diameter to obtain the promised algorithm for an additive approximation for APSP. Finally, in Section 5 we present an empirical study of the performance of our algorithm for all-pairs shortest paths.

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(Show Context)
Citation Context ... bound of ~ O(n ! ) 1 where ! denotes the exponent in the running time of the matrix multiplication algorithm used. The current best matrix multiplication algorithm is due to Coppersmith and Winograd =-=[CW90]-=- and has ! = 2:376. In contrast, the naive algorithm for APSP performs breadthfirst searches from each vertex, and requires time O(nm). Given the fundamental nature of this problem, it is important to... |

712 |
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Citation Context ... greedy set cover algorithm repeatedly chooses the set that covers the most uncovered elements, and it is known to provide a set cover of size within a factor log n of the optimal fractional solution =-=[Jo74, Lo75]-=-. Since every vertex has degree at least s and therefore the corresponding set S v has cardinality at least s, assigning a weight of 1=s to every set in S gives a fractional set cover of total weight ... |

398 |
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Citation Context ...his distinction, we believe it is instructive to try and interpret the "algebraic" algorithms in purely graph-theoretic terms even with the use the simpler matrix multiplication algorithm of=-= Strassen [St69]-=-. Currently, the best known combinatorial algorithm is due to Feder and Motwani [FM91] which runs in time O(n 3 = log n), yielding only a marginal improvement over the naive algorithm. Supported by an... |

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Citation Context ...versity 1 Introduction Consider the problem of computing all-pairs shortest paths (APSP) in an unweighted, undirected graph G with n vertices and m edges. The recent work of Alon, Galil, and Margalit =-=[AGM91]-=-, Alon, Galil, Margalit, and Naor [AGMN92], and Seidel [Se92] has led to dramatic progress in devising fast algorithms for this problem. These algorithm are based on formulating the problem in terms o... |

73 | Clique partitions, graph compression and speeding-up algorithms
- Feder, Motwani
- 1995
(Show Context)
Citation Context ...orithms in purely graph-theoretic terms even with the use the simpler matrix multiplication algorithm of Strassen [St69]. Currently, the best known combinatorial algorithm is due to Feder and Motwani =-=[FM91]-=- which runs in time O(n 3 = log n), yielding only a marginal improvement over the naive algorithm. Supported by an NSF Graduate Fellowship and NSF NYI Award CCR-9357849. y Supported by an OTL grant an... |

41 | On the all-pairs-shortest-path problem
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(Show Context)
Citation Context ...airs shortest paths (APSP) in an unweighted, undirected graph G with n vertices and m edges. The recent work of Alon, Galil, and Margalit [AGM91], Alon, Galil, Margalit, and Naor [AGMN92], and Seidel =-=[Se92]-=- has led to dramatic progress in devising fast algorithms for this problem. These algorithm are based on formulating the problem in terms of matrices with small integer entries and using fast matrix m... |

34 | Polylog-time and near-linear work approximation scheme for undirected shortest paths - Cohen |

30 |
Witnesses for Boolean Matrix Multiplication and for Shortest Paths
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- 1992
(Show Context)
Citation Context ...em of computing all-pairs shortest paths (APSP) in an unweighted, undirected graph G with n vertices and m edges. The recent work of Alon, Galil, and Margalit [AGM91], Alon, Galil, Margalit, and Naor =-=[AGMN92]-=-, and Seidel [Se92] has led to dramatic progress in devising fast algorithms for this problem. These algorithm are based on formulating the problem in terms of matrices with small integer entries and ... |

20 | D.: Near-linear cost sequential and distributed constructions of sparse neighborhood covers - Awerbuch, Berger, et al. - 1993 |

18 |
Diameters of Graphs: Old problems and New Results. Congressus Numerantium 60
- Chung
(Show Context)
Citation Context ...es. The diameter can be determined by computing all-pairs shortest path (APSP) distances in the graph, and it appears that this is the only known way to solve the diameter problem. In fact, Fan Chung =-=[Chu87]-=- had earlier posed the question of whether there is an O(n 3\Gammaffl ) algorithm for finding the diameter without resorting to fast matrix multiplication. The situation with regard to combinatorial a... |

12 | On diameter verification and boolean matrix multiplication
- Basch, Khanna, et al.
- 1995
(Show Context)
Citation Context ...eter without resorting to fast matrix multiplication. The situation with regard to combinatorial algorithms for diameter is only marginally better than in the case of APSP. Basch, Khanna, and Motwani =-=[BKM95]-=- presented a combinatorial algorithm that verifies whether a graph has diameter 2 in time O i n 3 = log 2 n j . A slight adaptation of this algorithm yields a boolean matrix multiplication algorithm w... |

10 | Fast algorithms for t-spanners and stretch-t paths - Cohen - 1993 |