## A randomized parallel algorithm for single-source shortest paths (1997)

Venue: | Journal of Algorithms |

Citations: | 15 - 1 self |

### BibTeX

@ARTICLE{Klein97arandomized,

author = {Philip N. Klein and Sairam Subramanian},

title = {A randomized parallel algorithm for single-source shortest paths},

journal = {Journal of Algorithms},

year = {1997},

volume = {25},

pages = {205--220}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract We give a randomized parallel algorithm for computing single-source shortest paths in weighted digraphs. We show that the exact shortest path problem can be efficiently reduced to solving a series of approximate shortest-path subproblems. Our algorithm for the approximate shortest-path problem is based on a technique used by Ullman and Yannakakis in a parallel algorithm for breadth-first search. 1 Introduction One of the most fundamental and ubiquitous problems in combinatorial optimization is finding single-source shortest paths in a weighted graph. Aside from being important in its own right, the problem arises in algorithms for many other problems, especially those related to flow. In view of the importance of the single-source shortest paths problem, it is unfortunate that all known parallel algorithms for this problem are very inefficient on sparse graphs. This inability to make efficient use of parallelism in computing shortest paths is of both theoretical and practical significance. A fast and efficient parallel algorithm for this problem remains a major goal in the design of parallel graph algorithms.

### Citations

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(Show Context)
Citation Context ...rst search involves repeatedly squaring a matrix, where the element-wise operations are in the min--plus semiring. In fact, this algorithm can compute shortest paths. The technique is well-known; see =-=[1, 7]-=- for more details. The problem with this algorithm is that it requires too many processors; to achieve O(log n) time requires about n 3 processors. The processor bound has been improved somewhat [9] i... |

76 | Fast algorithms for constructing t-spanners and paths with stretch t - Cohen - 1999 |

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(Show Context)
Citation Context .... Additional support provided by ONR and DARPA contract N00014-83-K-0146 and ARPA Order No. 6320, Amendment 1. 1 to get exact shortest paths. The scaling technique is similar to the one used by Gabow =-=[8]-=- for sequential computation of shortest paths. In other related work, a parallel algorithm for shortest paths was discovered by Spencer [17]. Actually, his algorithm involves a tradeoff between time a... |

45 |
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Citation Context ...a 2 \Gamman ffi for some positive constant ffi . 2 Assuming the slightly weaker ARBITRARY CRCW PRAM, in which an arbitrary processor succeeds in writing, would entail using an O(log n)-time algorithm =-=[10]-=- for processor allocation. This change multiplies our time bounds by O(log n). 2.2 Parallel breadth-first search The fastest known parallel algorithm for breadth-first search involves repeatedly squar... |

34 | Polylog-time and near-linear work approximation scheme for undirected shortest paths
- Cohen
(Show Context)
Citation Context ...ors available, our algorithm would use the n processors to achieve a speed-up of about p n while Spencer's algorithm would achieve a speed-up of about n 1=4 . 1.1 Subsequent work In recent work Cohen =-=[5]-=- has given an algorithm to find (1 + ffl)-approximate shortest paths that works in poly-logarithmic time and does close to linear work. Her algorithm makes use of the limited-search algorithm presente... |

30 |
Efficient parallel shortest-paths in digraphs with a separator decomposition
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(Show Context)
Citation Context ... 000 and P 000 consists of at most 1 + p n log n edges. 2 Lemma 4.4 shows that to compute single-source distances in H [ G, it is sufficient to consider paths of size at most k = 1 + p n log n. Cohen =-=[3]-=- has observed that the following variant of Bellman-Ford correctly computes shortest paths in this case. Set d(v) := 1 for each node v, except set d(source) := 0. Repeat the following step k times. Fo... |

30 | Parallel Algorithmic Techniques for Combinatorial Computation
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(Show Context)
Citation Context ...rst search involves repeatedly squaring a matrix, where the element-wise operations are in the min--plus semiring. In fact, this algorithm can compute shortest paths. The technique is well-known; see =-=[1, 7]-=- for more details. The problem with this algorithm is that it requires too many processors; to achieve O(log n) time requires about n 3 processors. The processor bound has been improved somewhat [9] i... |

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Citation Context ...1, 7] for more details. The problem with this algorithm is that it requires too many processors; to achieve O(log n) time requires about n 3 processors. The processor bound has been improved somewhat =-=[9]-=- in the case of breadth-first search. The most elementary parallel search technique is parallel breadth-first search, in which the nodes are visited level by level as the search progresses. Level 0 co... |

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Citation Context ...le values to a single location, the value written is the bitwise OR of the values. This model enables us to do processor allocation using a randomized constant-time algorithm due to Hagerup and Raman =-=[11]-=- for approximate prefix summation. Their algorithm runs in constant time with probability 1 \Gamma 2 \Gamman ffi for some positive constant ffi . 2 Assuming the slightly weaker ARBITRARY CRCW PRAM, in... |

21 |
High-probability parallel transitive-closure algorithms
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(Show Context)
Citation Context ...gorithm for shortest paths, we see that our algorithm incurs an O( p n log 2 n log L) factor overhead. Our algorithm is based on a parallel breadth-first search algorithm due to Ullman and Yannakakis =-=[18]-=-. Ullman and Yannakakis show that a breadth-first search tree can be constructed in O( p n polylog n) time using a linear number of processors. Their algorithm is limited, however, in that it cannot h... |

16 | A Linear-processor Polylog-time Algorithm for Shortest-paths in Planar Graphs
- Klein, Subramanian
- 1993
(Show Context)
Citation Context ...ded then for undirected graphs the two algorithms require roughly the same amount of work. However like Cohen's algorithm, their algorithm does not generalize to directed graphs either. In other work =-=[14]-=- we have given a nearly work-optimal parallel algorithm for computing single-source shortest-paths if the underlying graph is planar. Combining the techniques of that paper with those of the present a... |

12 |
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Citation Context ...d, n and m denote, respectively the number of nodes and the number of edges in the input graph. 2.1 The model of parallel computation We assume a model of parallel computation called the OR CRCW PRAM =-=[2]-=-, in which multiple processors can simultaneously read and write to a shared memory. If multiple processors attempt to write multiple values to a single location, the value written is the bitwise OR o... |

11 |
A Parallel Randomized Approximation Scheme for Shortest Paths
- Klein, Sairam
- 1992
(Show Context)
Citation Context ... these distances in forming the auxiliary graph results in the final estimated distances being at least actual distances and at most (1 + ffl) times actual distances. An earlier version of this paper =-=[13]-=- used this limited search idea to compute approximte single-source shortest paths. Our second contribution is a reduction technique (similar to one due to Gabow [8]) that allows us to solve the exact ... |

9 |
Time-work tradeoffs of the single-source shortest paths problem
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- 1999
(Show Context)
Citation Context ...neralize to directed graphs; and there seems to be no way to use it to compute exact shortest paths by repeated approximation even if the underlying graph is undirected. More recently Shi and Spencer =-=[16]-=- have given a parallel shortest-path algorithm, for undirected graphs, with work-bounds that are similar to ours. Given log nstsn their algorithm runs in ~ O(t) time with ~ O((n 3 =t 2 ) +m) work, or ... |

8 |
Parallel algorithms for finding the maximum and the median almost surely in constant time
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(Show Context)
Citation Context ...he Bellman-Ford variant to compute single-source shortest paths requires O( p n log n) stages (as noted above). Each stage can be implemented in constant time with high probability by using Megiddo's =-=[15]-=- algorithm for computing minimum. Therefore the shortest paths in H [ G can be found in O( p n log n) time and O(m p n log n) work. We therefore get the bounds of Theorem 1.1. 5 Conclusion In this pap... |

3 |
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(Show Context)
Citation Context ...s. The scaling technique is similar to the one used by Gabow [8] for sequential computation of shortest paths. In other related work, a parallel algorithm for shortest paths was discovered by Spencer =-=[17]-=-. Actually, his algorithm involves a tradeoff between time and work. Spencer specifies his bounds in terms of a parameter ae and an upper bound L on edge-lengths (assuming the lengths are positive int... |