## Flows in Undirected Unit Capacity Networks (1997)

Venue: | In Proceedings of the 30 th Annual Symposium on the Foundations of Computer Science |

Citations: | 15 - 1 self |

### BibTeX

@INPROCEEDINGS{Goldberg97flowsin,

author = {Andrew V. Goldberg and Satish Rao},

title = {Flows in Undirected Unit Capacity Networks},

booktitle = {In Proceedings of the 30 th Annual Symposium on the Foundations of Computer Science},

year = {1997},

pages = {32--35}

}

### OpenURL

### Abstract

We describe an O(min(m; n 3=2 )m 1=2 )-time algorithm for finding maximum flows in undirected networks with unit capacities and no parallel edges. This improves upon the previous bound of Karzanov and Even and Tarjan when m = !(n 3=2 ), and upon a randomized bound of Karger when v = \Omega\Gamma n 7=4 =m 1=2 ). (Here v is the maximum flow value.) 1 Introduction In this paper we consider the undirected maximum flow problem in a network with unit capacities and no parallel edges. Until recently, the fastest known way to solve this problem was using a reduction to the directed problem with unit capacities and no parallel arcs. Karzanov [7] and Even and Tarjan [2] have shown that Dinitz's blocking flow algorithm [1], applied to the directed problem, runs in O(min(m 1=2 ; n 2=3 )m) time. (Here n and m is the number of input vertices and edges, respectively.) Recently, Karger [6] developed two randomized algorithms for the undirected problem, with running times of O (m 5...

### Citations

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(Show Context)
Citation Context ...ay to solve this problem was using a reduction to the directed problem with unit capacities and no parallel arcs. Karzanov [7] and Even and Tarjan [2] have shown that Dinitz's blocking flow algorithm =-=[1]-=-, applied to the directed problem, runs in O(min(m 1=2 ; n 2=3 )m) time. (Here n and m is the number of input vertices and edges, respectively.) Recently, Karger [6] developed two randomized algorithm... |

93 |
Network Flow and Testing Graph Connectivity
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(Show Context)
Citation Context ...o parallel edges. Until recently, the fastest known way to solve this problem was using a reduction to the directed problem with unit capacities and no parallel arcs. Karzanov [7] and Even and Tarjan =-=[2]-=- have shown that Dinitz's blocking flow algorithm [1], applied to the directed problem, runs in O(min(m 1=2 ; n 2=3 )m) time. (Here n and m is the number of input vertices and edges, respectively.) Re... |

78 |
Computing edge-connectivity in multigraphs and capacitated graphs
- Nagamochi, Ibaraki
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(Show Context)
Citation Context ... lemma follows from [8]. Lemma 3.1 There is an O(m) implementation of Sparsify. Sparsify is defined so that the following lemma holds. The proof of the lemma is analogous to the proof of Lemma 2.1 in =-=[9]-=-. Lemma 3.2 The residual flow in the network output by Sparsify(v; G; f) is at least min(v; r), where r is the residual flow value in G f . Proof. We prove a stronger claim by induction. Let G i = (V;... |

67 |
A matroid approach to finding edge connectivity and packing arborescences
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Citation Context ...mbination with the augmenting path algorithm [3], this technique gives an O(nv 2 ) time bound. In combination with Karger's second algorithm, get an O (nv 5=3 ) expected time bound. See, for example, =-=[4]-=-. 2 Preliminaries For this paper, we consider computing a maximum flow in an undirected graph G = (V; E) with two distinguished vertices s and t. We consider only zero-one valued flows. Let jV j = n a... |

45 |
A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph
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Citation Context ...e previous deterministic bound for m = !(n 3=2 ) and Karger's randomized bound for v = \Omega\Gamma n 7=4 =m 1=2 ): Our improvements are based on the sparsification technique of Nagamochi and Ibaraki =-=[8]-=-. Their technique applies to undirected (e.g. symmetric) graphs. We use their technique in the context of residual graphs of flows in undirected graphs, which are not symmetric. We note that the Nagom... |

25 | Tarjan R E. Network flow and testing graph connectivity - Even |

13 |
0 nakhozhdenii maksimalâ€™nogo potoka v setyakh spetsialâ€™nogo vida i nekotorykh prilozheniyakh
- Karzanov
- 1973
(Show Context)
Citation Context ...th unit capacities and no parallel edges. Until recently, the fastest known way to solve this problem was using a reduction to the directed problem with unit capacities and no parallel arcs. Karzanov =-=[7]-=- and Even and Tarjan [2] have shown that Dinitz's blocking flow algorithm [1], applied to the directed problem, runs in O(min(m 1=2 ; n 2=3 )m) time. (Here n and m is the number of input vertices and ... |

12 | Global price updates help - Goldberg, Kennedy - 1997 |

12 | Better random sampling algorithms for flows in undirected graphs - Karger - 1998 |

10 | Using random sampling to find maximum flows in uncapacitated undirected graphs
- Karger
- 1997
(Show Context)
Citation Context ...t Dinitz's blocking flow algorithm [1], applied to the directed problem, runs in O(min(m 1=2 ; n 2=3 )m) time. (Here n and m is the number of input vertices and edges, respectively.) Recently, Karger =-=[6]-=- developed two randomized algorithms for the undirected problem, with running times of O (m 5=6 n 1=3 v 2=3 ) and O (m 2=3 n 1=3 v). (Here v is the maximum flow value.) We introduce an O(min(m; n 3=2 ... |

3 | On finding maximum flows in networks with special structure and some applications - Karzanov - 1973 |