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## Faster Algorithms for All-Pairs Small Stretch Distances in Weighted Graphs

Citations: | 1 - 0 self |

### Citations

270 | Approximate distance oracles
- Thorup, Zwick
- 2001
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Citation Context ...nt, STRETCH2+ε(G) computes all-pairs stretch 2 + ε distances in expected time O(n 9/4 log n) since Õ(n 2.248 ) is o(n 9/4 ). Motivation. During the last 10-15 years, many new combinatorial algorithms =-=[2, 6, 1, 8, 7, 13, 3, 9]-=- were designed for the all-pairs approximate shortest paths problem in order to achieve faster running times in weighted and unweighted graphs. In weighted graphs, the current fastest randomized combi... |

90 | All-pairs almost shortest paths
- Dor, Halperin, et al.
- 2000
(Show Context)
Citation Context ...a): We split this further into two cases. Case (a): The vertex b /∈ ball2(v). We will show that s3(a) is our witness here. We have d[s3(a),u] + d[s3(a),v] ≤ δ(s3(a),a) + δ(a,u) + δ(s3(a),a) + δ(a,v). =-=(8)-=- 11 s2(v)sWe also have δ(s3(a),a) ≤ δ(a,s2(u)) (since s2(u) /∈ bunch3(a)) and δ(s3(a),a) ≤ w(a,b) (since the edge (a,b) is not present in ball3(a)). Substituting these bounds in Inequality (8) yields ... |

84 | 2002): All Pairs Shortest Paths using Bridging Sets and Rectangular Matrix Multiplication
- Zwick
- 1145
(Show Context)
Citation Context ...te distances between all pairs of vertices in G. While many efficient algorithms are known for this problem in unweighted graphs, not many results are known for this problem in weighted graphs. Zwick =-=[14]-=- showed that for any fixed ε > 0, stretch 1 1 + ε distances between all pairs of vertices in a weighted directed graph on n vertices can be computed in Õ(n ω ) time, where ω < 2.376 is the exponent of... |

82 |
Fast algorithms for constructing t-spanners and paths with stretch t
- Cohen
- 1998
(Show Context)
Citation Context ...nt, STRETCH2+ε(G) computes all-pairs stretch 2 + ε distances in expected time O(n 9/4 log n) since Õ(n 2.248 ) is o(n 9/4 ). Motivation. During the last 10-15 years, many new combinatorial algorithms =-=[2, 6, 1, 8, 7, 13, 3, 9]-=- were designed for the all-pairs approximate shortest paths problem in order to achieve faster running times in weighted and unweighted graphs. In weighted graphs, the current fastest randomized combi... |

76 | Computing almost shortest paths
- Elkin
(Show Context)
Citation Context ...nt, STRETCH2+ε(G) computes all-pairs stretch 2 + ε distances in expected time O(n 9/4 log n) since Õ(n 2.248 ) is o(n 9/4 ). Motivation. During the last 10-15 years, many new combinatorial algorithms =-=[2, 6, 1, 8, 7, 13, 3, 9]-=- were designed for the all-pairs approximate shortest paths problem in order to achieve faster running times in weighted and unweighted graphs. In weighted graphs, the current fastest randomized combi... |

48 | Nearlinear time construction of sparse neighborhood covers
- Awerbuch, Berger, et al.
- 1998
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40 | A new approach to all-pairs shortest paths on real-weighted graphs
- Pettie
(Show Context)
Citation Context ...ery important in several applications. The complexity of the fastest known algorithm for the APSP problem in a graph with m edges, n vertices and real non-negative edge weights is O(mn + n 2 loglogn) =-=[12]-=-. Thus this algorithm has a running time of Θ(n 3 ) when m = Θ(n 2 ). In fact, the best known upper bound on the worst case time complexity of this problem (in terms of n) is O(n 3 /logn) [5], which i... |

37 | All-pairs small-stretch paths
- Cohen, Zwick
- 1997
(Show Context)
Citation Context ...ication of two n × n matrices. Given an undirected weighted graph on n vertices, computing all-pairs stretch 3 distances in Õ(n 2 ) time and all-pairs stretch 7/3 distances in Õ(n 7/3 ) time is known =-=[7]-=- (these algorithms use only combinatorial techniques, i.e., fast matrix multiplication subroutines are not used). Researchers have been trying to explore the possible trade-off between stretch and run... |

30 | Faster algorithms for approximate distance oracles and all-pairs small stretch paths
- Baswana, Kavitha
- 2006
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Citation Context ...-pairs approximate shortest paths problem in order to achieve faster running times in weighted and unweighted graphs. In weighted graphs, the current fastest randomized combinatorial algorithms (from =-=[4]-=-) for computing all-pairs stretch t distances for t < 3 in G with m edges and n vertices are: computing all-pairs stretch 2 distances in expected Õ(m √ n + n 2 ) time and computing all-pairs stretch 7... |

13 | All-pairs nearly 2-approximate shortest paths in o(n2poly logn) time
- Baswana, Goyal, et al.
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11 |
Rajeev Motwani. Fast estimation of diameter and shortest paths (without matrix multiplication
- Aingworth, Chekuri, et al.
- 1999
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3 |
All-pairs shortest paths with real edge weights
- Chan
- 2005
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Citation Context ...oglogn) [12]. Thus this algorithm has a running time of Θ(n 3 ) when m = Θ(n 2 ). In fact, the best known upper bound on the worst case time complexity of this problem (in terms of n) is O(n 3 /logn) =-=[5]-=-, which is marginally subcubic. An almost cubic running time is inefficient for several applications, and this has motivated faster algorithms to compute approximate solutions for the APSP problem. Le... |

1 |
Fast rectangunlar matrix multiplication and applications
- Huang, Pan
- 1998
(Show Context)
Citation Context ...ss matrix Wi j for each pair (i, j) is Õ(C(n)), where C(n) is the time taken to multiply an n × n β matrix with an n β × n matrix. Here we will use the following result. Proposition 3 (Huang and Pan (=-=[10]-=- Section 8.2)). Multiplying an n × nβ matrix with an nβ × n matrix for 0.294 ≤ β ≤ 1 takes time O(nα ), where α = 2(1−β)+(β−0.294)ω 0.706 , and ω < 2.376 is the best exponent of multiplying two n × n ... |