## Planar Graphs, Negative Weight Edges, Shortest Paths, and Near Linear Time (2001)

Venue: | In Proc. 42nd IEEE Annual Symposium on Foundations of Computer Science |

Citations: | 53 - 0 self |

### BibTeX

@INPROCEEDINGS{Fakcharoenphol01planargraphs,,

author = {Jittat Fakcharoenphol and Satish Rao},

title = {Planar Graphs, Negative Weight Edges, Shortest Paths, and Near Linear Time},

booktitle = {In Proc. 42nd IEEE Annual Symposium on Foundations of Computer Science},

year = {2001},

pages = {232--241}

}

### Years of Citing Articles

### OpenURL

### Abstract

for finding shortest paths in a planar graph with real weights.

### Citations

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Flows in Networks
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(Show Context)
Citation Context ...our favorite applications for planar graphs include separator algorithms [17], multi-source multi-sink flow algorithms [15], or algorithms for finding minimum weighted cuts. In 1958, Bellman and Ford =-=[2, 7]-=- gave an O(mn) algorithm for finding shortest paths on an m-edge, nvertex graph with real edge weights. Gabow and Tarjan [10] showed that this problem could indeed be solved in e O(m p n): 1 Their alg... |

388 | Tarjan, A Separator Theorem for Planar Graphs
- Lipton, E
- 1979
(Show Context)
Citation Context ...edge weights, the problem is much easier. For example, Dijkstra's shortest path algorithm can be implemented in O(m + n log n) time. For planar graphs, upon the discovery of planar separator theorems =-=[14]-=-, an O(n 3=2 ) algorithm was given by Lipton, Rose, and Tarjan. [13]. Their algorithm is based on partitioning the graph into pieces, recursively computing distances on the borders of the pieces using... |

325 |
On a routing problem
- Bellman
- 1958
(Show Context)
Citation Context ...our favorite applications for planar graphs include separator algorithms [17], multi-source multi-sink flow algorithms [15], or algorithms for finding minimum weighted cuts. In 1958, Bellman and Ford =-=[2, 7]-=- gave an O(mn) algorithm for finding shortest paths on an m-edge, nvertex graph with real edge weights. Gabow and Tarjan [10] showed that this problem could indeed be solved in e O(m p n): 1 Their alg... |

183 |
Generalized nested dissection
- Lipton, Rose, et al.
- 1979
(Show Context)
Citation Context ...ortest path algorithm can be implemented in O(m + n log n) time. For planar graphs, upon the discovery of planar separator theorems [14], an O(n 3=2 ) algorithm was given by Lipton, Rose, and Tarjan. =-=[13]-=-. Their algorithm is based on partitioning the graph into pieces, recursively computing distances on the borders of the pieces using numerous invocations of Dijkstra's algorithm to build a dense graph... |

160 | Faster shortest-path algorithms for planar graphs
- Rauch, Klein, et al.
- 1997
(Show Context)
Citation Context ...rithms worked not only for planar graphs but for any p n-separable one. 2 Combining a similar approach with a (non-strongly) polynomial algorithm of Goldberg [11] for general graphs, Henzinger et al. =-=[12]-=- give an e O(n 4=3 ) algorithm for the shortest path problem in planar graphs (or any set of graphs with an O( p n) sized separator). In this paper, we present an O(n log 3 n) time algorithm for findi... |

152 |
The discrete geodesic problem
- Mitchell, Mount, et al.
- 1987
(Show Context)
Citation Context ...hose in [13]. The similar binary searching technique that we use in the Monge searching problem also appeared in the algorithm for finding shortest paths on a 3-dimensional polygon by Mitchell et al. =-=[16]-=-. 2 Preliminaries Given a directed graph, G = (V; E), and a weight function on the directed edges d : E ! R, a distance labelling for a source node s is a function d : V ! R, such that d(v) is the min... |

127 | Faster scaling algorithms for network problems
- Gabow, Tarjan
- 1989
(Show Context)
Citation Context ... algorithms for finding minimum weighted cuts. In 1958, Bellman and Ford [2, 7] gave an O(mn) algorithm for finding shortest paths on an m-edge, nvertex graph with real edge weights. Gabow and Tarjan =-=[10-=-] showed that this problem could indeed be solved in e O(m p n): 1 Their algorithm, however, depended on the Computer Science Division, University of California, Berkeley, CA 94720. E-mail: jittat@cs... |

118 |
Fast algorithms for shortest paths in planar graphs, with applications
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(Show Context)
Citation Context ...e novel and of independent interest. 1.3 More related work For planar graphs with positive edge weights, Henzinger et al. [12] gave an O(n) time algorithm. Their work improves on work of Frederickson =-=[9]-=- who had previously given O(n p log n) algorithms for this problem. Frederickson [8] gave an improved all-pairs shortest path algorithm for planar graphs with small hammock decompositions. Djidjev et ... |

56 | Scaling algorithms for the shortest paths problem
- Goldberg
- 1993
(Show Context)
Citation Context ...to construct a global solution. Their algorithms worked not only for planar graphs but for any p n-separable one. 2 Combining a similar approach with a (non-strongly) polynomial algorithm of Goldberg =-=[11]-=- for general graphs, Henzinger et al. [12] give an e O(n 4=3 ) algorithm for the shortest path problem in planar graphs (or any set of graphs with an O( p n) sized separator). In this paper, we presen... |

52 |
Ratio regions: A technique for image segmentation
- Cox, Rao, et al.
- 1996
(Show Context)
Citation Context ...ations in diverse areas. The problem has wide application even when the underlying graph is a grid graph. For example, there are recent image segmentation approaches that use negative cycle detection =-=[4, 5]-=-. Other of our favorite applications for planar graphs include separator algorithms [17], multi-source multi-sink flow algorithms [15], or algorithms for finding minimum weighted cuts. In 1958, Bellma... |

40 |
Flow in planar graphs with multiple sources and sinks
- Miller, Naor
- 1995
(Show Context)
Citation Context ...image segmentation approaches that use negative cycle detection [4, 5]. Other of our favorite applications for planar graphs include separator algorithms [17], multi-source multi-sink flow algorithms =-=[15]-=-, or algorithms for finding minimum weighted cuts. In 1958, Bellman and Ford [2, 7] gave an O(mn) algorithm for finding shortest paths on an m-edge, nvertex graph with real edge weights. Gabow and Tar... |

17 | Linear and O(n log n) time minimumcost matching algorithms for quasi-convex tours
- Buss, Yianilos
- 1998
(Show Context)
Citation Context ...aphs whose distance matrices obey a noncrossing condition (called the Monge condition). Efficient algorithms for searching for minima in Monge arrays have been developed previously. See, for example, =-=[1, 3]-=-. 2 Assuming that they were given a recursive decomposition of the graph. Our algorithm proceeds by combining Dijkstra's and the Bellman-Ford algorithms with methods for searching Monge matrices in su... |

12 | Efficient minimum cost matching and transportation using the quadrangle inequality
- Aggarwal, Bar-Noy, et al.
- 1995
(Show Context)
Citation Context ...aphs whose distance matrices obey a noncrossing condition (called the Monge condition). Efficient algorithms for searching for minima in Monge arrays have been developed previously. See, for example, =-=[1, 3]-=-. 2 Assuming that they were given a recursive decomposition of the graph. Our algorithm proceeds by combining Dijkstra's and the Bellman-Ford algorithms with methods for searching Monge matrices in su... |

8 |
Faster algorithms for finding small edge cuts in planar graphs
- Rao
- 1992
(Show Context)
Citation Context ...s a grid graph. For example, there are recent image segmentation approaches that use negative cycle detection [4, 5]. Other of our favorite applications for planar graphs include separator algorithms =-=[17]-=-, multi-source multi-sink flow algorithms [15], or algorithms for finding minimum weighted cuts. In 1958, Bellman and Ford [2, 7] gave an O(mn) algorithm for finding shortest paths on an m-edge, nvert... |

7 |
Dynamic programming for detecting, tracking and matching elastic contours
- Geiger, Gupta, et al.
- 1995
(Show Context)
Citation Context ...ations in diverse areas. The problem has wide application even when the underlying graph is a grid graph. For example, there are recent image segmentation approaches that use negative cycle detection =-=[4, 5]-=-. Other of our favorite applications for planar graphs include separator algorithms [17], multi-source multi-sink flow algorithms [15], or algorithms for finding minimum weighted cuts. In 1958, Bellma... |

2 |
A new approach to all pairs shortest paths in planar graphs
- Frederickson
- 1987
(Show Context)
Citation Context ...sitive edge weights, Henzinger et al. [12] gave an O(n) time algorithm. Their work improves on work of Frederickson [9] who had previously given O(n p log n) algorithms for this problem. Frederickson =-=[8]-=- gave an improved all-pairs shortest path algorithm for planar graphs with small hammock decompositions. Djidjev et al. [6] gave dynamic algorithms whose complexity are linear in the size of the hammo... |