## Single-Source Shortest-Paths on Arbitrary Directed Graphs in Linear Average-Case Time (2001)

Venue: | In Proc. 12th ACM-SIAM Symposium on Discrete Algorithms |

Citations: | 28 - 5 self |

### BibTeX

@INPROCEEDINGS{Meyer01single-sourceshortest-paths,

author = {Ulrich Meyer},

title = {Single-Source Shortest-Paths on Arbitrary Directed Graphs in Linear Average-Case Time},

booktitle = {In Proc. 12th ACM-SIAM Symposium on Discrete Algorithms},

year = {2001},

pages = {797--806},

publisher = {ACM Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

The quest for a linear-time single-source shortest-path (SSSP) algorithm on directed graphs with positive edge weights is an ongoing hot research topic. While Thorup recently found an O(n + m) time RAM algorithm for undirected graphs with n nodes, m edges and integer edge weights in f0; : : : ; 2 w 1g where w denotes the word length, the currently best time bound for directed sparse graphs on a RAM is O(n + m log log n). In the present paper we study the average-case complexity of SSSP. We give a simple algorithm for arbitrary directed graphs with random edge weights uniformly distributed in [0; 1] and show that it needs linear time O(n + m) with high probability. 1 Introduction The single-source shortest-path problem (SSSP) is a fundamental and well-studied combinatorial optimization problem with many practical and theoretical applications [1]. Let G = (V; E) be a directed graph, jV j = n, jEj = m, let s be a distinguished vertex of the graph, and c be a function assigning a n...

### Citations

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Citation Context ...is the sum of the weights of its edges. Shortest-paths algorithms are typically classied into two groups: label-setting and label-correcting. The basic label-setting approach is Dijkstra's algorithm [=-=8]-=-. It maintains a partition of V into settled, queued, and unreached nodes, and for each node v a tentative distance tent(v); tent(v) is always the weight of some path from s to v and hence an upper bo... |

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Citation Context ... denser graphs have been proposed by Raman [22, 23]: they require O(m+n p log n log log n) and O(m+n (w logn) 1=3 ) time, respectively. The classic label-correcting algorithm of Bellman{ Ford [3, =-=10-=-] and all of its improved derivatives (see [4] for an overview) need nm) time in the worst case. Some of these algorithms, however, show very good practical behavior for SSSP on sparse graphs [4, 27].... |

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Citation Context ...) = tent(v)). In spite of poor worst-case running times, implementations of label-correcting approaches frequently outperform label-setting algorithms [4, 27]. 1.1 Previous Work Using Fibonacci heaps =-=[11]-=- Dijkstra's algorithm can be implemented to run in O(n log n +m) time for positive edge weights. A number of faster algorithms have been developed on the more powerful RAM (random access machine) mode... |

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Citation Context ... denser graphs have been proposed by Raman [22, 23]: they require O(m+n p log n log log n) and O(m+n (w logn) 1=3 ) time, respectively. The classic label-correcting algorithm of Bellman{ Ford [3, =-=10-=-] and all of its improved derivatives (see [4] for an overview) need nm) time in the worst case. Some of these algorithms, however, show very good practical behavior for SSSP on sparse graphs [4, 27].... |

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Citation Context ... those nodes until they aresnally settled (dist(v) = tent(v)). In spite of poor worst-case running times, implementations of label-correcting approaches frequently outperform label-setting algorithms =-=[4, 27]-=-. 1.1 Previous Work Using Fibonacci heaps [11] Dijkstra's algorithm can be implemented to run in O(n log n +m) time for positive edge weights. A number of faster algorithms have been developed on the ... |

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Citation Context ...model which basically re ects what one can use in a programming language such as C. Nearly all of these algorithms are based on Dijkstra's algorithm and improve the priority queue data structure (see =-=[25]-=- for an overview). Thorup [25] has given thesrst O(n +m) time RAM algorithm for undirected graphs with integer edge weights in f0; : : : ; 2 w 1g for word length w. His approach applies label-setting ... |

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Citation Context ...totic superiority of our new algorithm. 2 Adaptive Bucket-Splitting 2.1 Preliminaries Our algorithm is based on keeping nodes in buckets. This technique has already been used in Dial's implementation =-=[7]-=- of Dijkstra's algorithm for integer weights in f0; : : : ; Cg: a queued node v is stored in the bucket B i with index i = tent(v). In each iteration the algorithm removes a node v from thesrst nonemp... |

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Citation Context ... generalized Thorup's approach to directed graphs, the time complexity, however, remains super-linear O(n + m log w). The currently fastest RAM algorithm for sparse directed graphs is due to Thorup [26] and needs O(n + m log log n) time. Faster approaches for somewhat denser graphs have been proposed by Raman [22, 23]: they require O(m+n p log n log log n) and O(m+n (w logn) 1=3 ) time, re... |

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Citation Context ...on sparse graphs [4, 27]. Nevertheless, average-case analysis of shortest paths algorithms mainly focused on the All{Pairs Shortest Paths (APSP) problem on the complete graph with random edge weights =-=[5, 12, 15, 19, 24]-=-. Mehlhorn and Priebe [17] proved that for the complete graph with random edge weights, every SSSP algorithm has to check at least n log n) edges with high probability. Noshita [20] and Goldberg and T... |

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Citation Context ... those nodes until they aresnally settled (dist(v) = tent(v)). In spite of poor worst-case running times, implementations of label-correcting approaches frequently outperform label-setting algorithms =-=[4, 27]-=-. 1.1 Previous Work Using Fibonacci heaps [11] Dijkstra's algorithm can be implemented to run in O(n log n +m) time for positive edge weights. A number of faster algorithms have been developed on the ... |

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Citation Context ... w). The currently fastest RAM algorithm for sparse directed graphs is due to Thorup [26] and needs O(n + m log log n) time. Faster approaches for somewhat denser graphs have been proposed by Raman [=-=22,-=- 23]: they require O(m+n p log n log log n) and O(m+n (w logn) 1=3 ) time, respectively. The classic label-correcting algorithm of Bellman{ Ford [3, 10] and all of its improved derivatives (see [4... |

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Citation Context ... w). The currently fastest RAM algorithm for sparse directed graphs is due to Thorup [26] and needs O(n + m log log n) time. Faster approaches for somewhat denser graphs have been proposed by Raman [=-=22,-=- 23]: they require O(m+n p log n log log n) and O(m+n (w logn) 1=3 ) time, respectively. The classic label-correcting algorithm of Bellman{ Ford [3, 10] and all of its improved derivatives (see [4... |

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Citation Context ... applies label-setting but signicantly deviates from Dijkstra's algorithm in that it does not visit the nodes in order of increasing distance from s but traverses a so-called component tree. Hagerup [=-=14-=-] recently generalized Thorup's approach to directed graphs, the time complexity, however, remains super-linear O(n + m log w). The currently fastest RAM algorithm for sparse directed graphs is due t... |

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Citation Context ...pped to a single bucket: v is kept in the bucket with index btent(v)=c. The parameter is called the bucket width. Let denote the smallest edge weight in the graph. Dinitz [9] and Denardo and Fox [6] observed that taking , Dijkstra's algorithm correctly settles any node from B cur . Choosing > either requires to repeatedlysnd a node with smallest tentative distance in B cur or results in... |

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Citation Context ...on sparse graphs [4, 27]. Nevertheless, average-case analysis of shortest paths algorithms mainly focused on the All{Pairs Shortest Paths (APSP) problem on the complete graph with random edge weights =-=[5, 12, 15, 19, 24]-=-. Mehlhorn and Priebe [17] proved that for the complete graph with random edge weights, every SSSP algorithm has to check at least n log n) edges with high probability. Noshita [20] and Goldberg and T... |

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Citation Context ... is O(n + m log log n). In the present paper we study the average-case complexity of SSSP. We give a simple algorithm for arbitrary directed graphs with random edge weights uniformly distributed in [=-=0; 1]-=- and show that it needs linear time O(n + m) with high probability. 1 Introduction The single-source shortest-path problem (SSSP) is a fundamental and well-studied combinatorial optimization problem w... |

12 |
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Citation Context ...d. All remaining events are independent. We have E[Yu ] degree(u) 2 i+1 2 and by the elementary theory of branching processes, E[Z l ] (max-degree(G i ) 2 i+1 ) l 2 l . The tail bounds in [2] additionally show that Z l = O((max-deg.(G i ) 2 i+1 ) l log n) = O(2 l log n) whp. Furthermore, P jl Z j = O(2 l log n) whp. By Lemma 7 we have l l = O( log n log logn ) whp. Hence, jfu... |

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Citation Context ...hts [5, 12, 15, 19, 24]. Mehlhorn and Priebe [17] proved that for the complete graph with random edge weights, every SSSP algorithm has to check at least n log n) edges with high probability. Noshita =-=[20]-=- and Goldberg and Tarjan [13] analyzed the expected number of decreaseKey operations in Dijkstra's algorithm; the time bound, however, does not improve over the worst-case complexity of the algorithm.... |

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Citation Context ...verage-case analysis of shortest paths algorithms mainly focused on the All{Pairs Shortest Paths (APSP) problem on the complete graph with random edge weights [5, 12, 15, 19, 24]. Mehlhorn and Priebe =-=[17]-=- proved that for the complete graph with random edge weights, every SSSP algorithm has to check at least n log n) edges with high probability. Noshita [20] and Goldberg and Tarjan [13] analyzed the ex... |

11 |
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Citation Context ...with a FIFO queue requires (n 4=3 ) operations whp for any constant 0s1=3. The above input class also yields poor performance on other SSSP label-correcting approaches like Pallottino 's algorithm [2=-=1-=-] which has worst-case execution time O(n 2 m) but performs very well on many practical inputs [4, 27]. Pallottino's algorithm maintains two FIFO queues Q 1 and Q 2 . The next node to be removed is t... |

10 | Average-case complexity of shortest-paths problems in the vertex-potential model, Random Structures Algorithms 16
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Citation Context ...on sparse graphs [4, 27]. Nevertheless, average-case analysis of shortest paths algorithms mainly focused on the All{Pairs Shortest Paths (APSP) problem on the complete graph with random edge weights =-=[5, 12, 15, 19, 24]-=-. Mehlhorn and Priebe [17] proved that for the complete graph with random edge weights, every SSSP algorithm has to check at least n log n) edges with high probability. Noshita [20] and Goldberg and T... |

9 | Expected performance of Dijkstra’s shortest path algorithm
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Citation Context ...horn and Priebe [17] proved that for the complete graph with random edge weights, every SSSP algorithm has to check at least n log n) edges with high probability. Noshita [20] and Goldberg and Tarjan =-=[13]-=- analyzed the expected number of decreaseKey operations in Dijkstra's algorithm; the time bound, however, does not improve over the worst-case complexity of the algorithm. Meyer and Sanders [18] gave ... |

5 |
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Citation Context ...arjan [13] analyzed the expected number of decreaseKey operations in Dijkstra's algorithm; the time bound, however, does not improve over the worst-case complexity of the algorithm. Meyer and Sanders =-=[18-=-] gave a label-correcting algorithm for random edge weights in [0; 1] that runs in average-case time O(n + m + d d c ) where d denotes the maximum node degree in the graph and d c denotes the maximum... |

4 |
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Citation Context ...tive distances can be mapped to a single bucket: v is kept in the bucket with index btent(v)=c. The parameter is called the bucket width. Let denote the smallest edge weight in the graph. Dinitz [9] and Denardo and Fox [6] observed that taking , Dijkstra's algorithm correctly settles any node from B cur . Choosing > either requires to repeatedlysnd a node with smallest tentative distanc... |

3 |
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