## Finding the Hidden Path: Time Bounds for All-Pairs Shortest Paths (1993)

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Citations: | 63 - 0 self |

### BibTeX

@MISC{Karger93findingthe,

author = {David R. Karger and Daphne Koller and Steven J. Phillips},

title = {Finding the Hidden Path: Time Bounds for All-Pairs Shortest Paths},

year = {1993}

}

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### Abstract

We investigate the all-pairs shortest paths problem in weighted graphs. We present an algorithm---the Hidden Paths Algorithm---that finds these paths in time O(m* n+n² log n), where m is the number of edges participating in shortest paths. Our algorithm is a practical substitute for Dijkstra's algorithm. We argue that m* is likely to be small in practice, since m* = O(n log n) with high probability for many probability distributions on edge weights. We also prove an Ω(mn) lower bound on the running time of any path-comparison based algorithm for the all-pairs shortest paths problem. Path-comparison based algorithms form a natural class containing the Hidden Paths Algorithm, as well as the algorithms of Dijkstra and Floyd. Lastly, we consider generalized forms of the shortest paths problem, and show that many of the standard shortest paths algorithms are effective in this more general setting.

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Citation Context ...graph. The Hidden Paths Algorithm runs in time Supported by a National Science Foundation Graduate Fellowship. y Supported by NSF PYI grant CCR-8858030-03. O(m n+n 2 log n) if we use a Fibonacci heap =-=[8]-=- to implement a priority queue; the running time increases to O(m n log n) if a standard heap is used instead. The algorithm operates by running Dijkstra's singlesource shortest paths algorithm [3] in... |

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Citation Context ... all-pairs shortest path problem for all generalized path weight functions. Previous Work The most widely known algorithms for the all-pairs shortest paths problem are those of Dijkstra [3] and Floyd =-=[5]-=-. Dijkstra's algorithm for the single-source shortest path problem can be run from each vertex (as noted by Johnson [11]), resulting in a running time of \Theta(mn + n 2 log n) if Fibonacci heaps are ... |

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Citation Context ...hms for the all-pairs shortest paths problem are those of Dijkstra [3] and Floyd [5]. Dijkstra's algorithm for the single-source shortest path problem can be run from each vertex (as noted by Johnson =-=[11]-=-), resulting in a running time of \Theta(mn + n 2 log n) if Fibonacci heaps are used to implement priority queues. Floyd's algorithm, which can handle negative edge weights, works by dynamic programmi... |

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Citation Context ...ve a worst case running time of \Omega\Gamma mn). Fast algorithms exist for special cases of the all-pairs shortest paths problem, for instance when the graph is unweighted [4] or planar [6]. Fredman =-=[7]-=- shows that O(n 5=2 ) comparisons between sums of edge weights suffice to solve the allpairs shortest paths problem. He uses this fact to do preprocessing, producing an algorithm that runs in time O(n... |

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Citation Context ...revealed. The philosophy of the Hidden Paths Algorithm is thus similar to recent algorithms for connectivity, which work by first finding a sparse subgraph (or certificate) with the same connectivity =-=[2, 17]-=-. We have shown a lower bound of \Omega\Gamma mn) on the running time of comparison based algorithms for allpairs shortest paths. It is of particular interest that the construction and verification al... |

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Citation Context ...ch assigns to every path a weight equal to the weight of the maximal edge on the path. Solving single source shortest paths under this weight function is referred to as the bottleneck path problem in =-=[20]-=-. An algorithm for a generalized shortest paths problem receives as input the graph and a black box for the weight function. We assume that the black box takes constant time to compute the weight of a... |

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Citation Context ...s, works by dynamic programming and runs in time \Theta(n 3 ). If the edge weights are independently and identically distributed random variables, a variant of Dijkstra's algorithm developed by Spira =-=[18]-=- has an expected running time of O(n 2 log 2 n). Bloniarz [1] provided an algorithm with an expected running time of O(n 2 log n log n). Another algorithm, developed by Frieze and Grimmet [9], achieve... |

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Citation Context ...revealed. The philosophy of the Hidden Paths Algorithm is thus similar to recent algorithms for connectivity, which work by first finding a sparse subgraph (or certificate) with the same connectivity =-=[2, 17]-=-. We have shown a lower bound of \Omega\Gamma mn) on the running time of comparison based algorithms for allpairs shortest paths. It is of particular interest that the construction and verification al... |

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Citation Context ...the computational complexity of the all-pairs shortest paths problem have been proved in some other models. If the permissible operations are addition and minimum in a straight line computation, Kerr =-=[12]-=- shows that any algorithm requires \Omega\Gamma n 3 ) running time. Regarding algebraic decision tree complexity, Spira and Pan [19] show that\Omega\Gamma n 2 ) comparisons between sums of edge weight... |

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Citation Context ...robability when the input graph is the complete graph with edge weights chosen independently from any of a large class of probability distributions, including the uniform distribution on the interval =-=[0; 1]-=-. Our second contribution is a new lower bound, given in Section 3. Most algorithms for the allpairs shortest path problem use the edge weight function only in comparing the weights of two paths in th... |

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Citation Context ...lgorithm [3] in parallel from all nodes in the graph, using information gained at each node to reduce the work done at other nodes. Our algorithm is likely to be fast in practice, because it is known =-=[9, 15]-=- that m = O(n log n) with high probability when the input graph is the complete graph with edge weights chosen independently from any of a large class of probability distributions, including the unifo... |

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Citation Context ...n algorithm similar to the Hidden Paths Algorithm, with the same time bound, has been developed independently by McGeoch [16]. A variant of our algorithm has been developed independently by Jakobsson =-=[10]-=- as a transitive closure algorithm. Both these algorithms require data structures which are more complex than those used by the Hidden Paths Algorithm. Lower bounds on the computational complexity of ... |

6 |
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Citation Context ...tions are addition and minimum in a straight line computation, Kerr [12] shows that any algorithm requires \Omega\Gamma n 3 ) running time. Regarding algebraic decision tree complexity, Spira and Pan =-=[19]-=- show that\Omega\Gamma n 2 ) comparisons between sums of edge weights are necessary to solve the single-source shortest paths problem. 2 Finding Shortest Paths In this section, we describe an algorith... |

3 |
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Citation Context ...t allow such comparisons in order to improve on the \Omega\Gamma n 3 ) bound. An algorithm similar to the Hidden Paths Algorithm, with the same time bound, has been developed independently by McGeoch =-=[16]-=-. A variant of our algorithm has been developed independently by Jakobsson [10] as a transitive closure algorithm. Both these algorithms require data structures which are more complex than those used ... |

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