Improved Algorithms for Dynamic Shortest Paths

Djidjev, Hristo; Pantziou, Grammati; Zaroliagis, Christos

doi:10.1007/s004530010043

Cited by 15 publications

(3 citation statements)

References 40 publications

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“…For the general case, when the girth is not bounded by a constant, Djidjev [9] presented and algorithm that computes the girth in O(n 5/4 log n) time. Djidjev's solution uses dynamic data structure for shortest paths [10], as well as a clever use of hammock decompositions [14]. Djidjev's algorithm is the fastest algorithm that solves this problem directly.…”

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confidence: 99%

Computing the Girth of a Planar Graph in O(n logn) Time

Weimann

Yuster

2009

Automata, Languages and Programming

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Abstract. We give an O(n log n) algorithm for computing the girth (shortest cycle) of an undirected n-vertex planar graph. Our solution extends to any graph of bounded genus. This improves upon the best previously known algorithms for this problem.Key words. Girth, shortest cycle, planar graph, graphs of bounded genus AMS subject classifications. 05C38, 68R101. Introduction. The girth of a graph is the length of its shortest cycle, or infinity if the graph does not contain any cycles. In addition to being a basic combinatorial characteristic of graphs, the girth has tight connections to many other graph properties. The connection between the girth of a graph and its chromatic number was studied by Erdős [13], Lovasz [19], Bollobás [4], and Cook [6]. Other important graph properties related to the girth include the minimum or average degree of the vertices, the diameter, the connectivity, the maximum genus, and the existence of certain type of minors (see Diestel's book [8] for a review of results).The problem of computing the girth of a graph is among the most natural and easily stated algorithmic graph problems. Itai and Rodeh [17] were the first to suggest an efficient algorithm to compute the girth. They presented an O(nm)-time algorithm for a graph of n vertices and m edges, and an O(n 2 )-time algorithm if an additive error of one is allowed. Monien [20] showed that finding the shortest cycle of even length is easier and can be done in O(n 2 α(n)) time, where α(n) is the inverse Ackermann function. For the case of planar graphs, Eppstein [11] proved that the girth can be found in O(n) time provided it is bounded by some constant. His result extended that of Itai and Rodeh [17] and of Papadimitriou and Yannakakis [21] who proved this for girth bounded by 3. For the general case, when the girth is not bounded by a constant, Djidjev [9] presented and algorithm that computes the girth in O(n 5/4 log n) time. Djidjev's solution uses dynamic data structure for shortest paths [10], as well as a clever use of hammock decompositions [14]. Djidjev's algorithm is the fastest algorithm that solves this problem directly. However, there is another, indirect approach to solve the girth problem in planar graphs. It is a known fact that cuts in an embedded planar graph correspond to cycles in the dual plane graph. Furthermore, minimum cuts correspond to shortest cycles in the dual plane graph. Chalermsook et al. [5] gave an O(n log 2 n) time algorithm for the minimum-cut problem in planar graphs. This algorithm can be used to solve the girth problem in planar graphs with positive edge weights in the same time, by reducing it (in linear time) to the min-cut problem in planar graphs. We note that this reduction introduces (not necessarily constant) weights in the dual graph even if the original graph was unweighted.

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confidence: 99%

Computing the Girth of a Planar Graph in O(n logn) Time

Weimann

Yuster

2009

Automata, Languages and Programming

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“…The algorithm in Ausiello et al [1992] is for graphs with nice topologies such as trees and outerplanar graphs. Djidjev et al [2000] achieves logarithmic query and update times for planar digraphs through graph decomposition. Feuerstein and Marchetti-Spaccamela [1993] and Klein et al [1994] also consider planar graphs.…”

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confidence: 99%

Incremental maintenance of shortest distance and transitive closure in first-order logic and SQL

Pang

Dong

Ramamohanarao

2005

ACM Trans. Database Syst.

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Given a database, the view maintenance problem is concerned with the efficient computation of the new contents of a given view when updates to the database happen. We consider the view maintenance problem for the situation when the database contains a weighted graph and the view is either the transitive closure or the answer to the all-pairs shortest-distance problem (APSD). We give incremental algorithms for APSD, which support both edge insertions and deletions. For transitive closure, the algorithm is applicable to a more general class of graphs than those previously explored. Our algorithms use first-order queries, along with addition (+) and less-than (<) operations (FO(+, <)); they store O(n 2 ) number of tuples, where n is the number of vertices, and have AC 0 data complexity for integer weights. Since FO(+, <) is a sublanguage of SQL and is supported by almost all current database systems, our maintenance algorithms are more appropriate for database applications than nondatabase query types of maintenance algorithms.Part of the results in this article appeared as Incremental FO(+; <) maintenance of all-pairs shortest paths for undirected graphs after insertions and deletions. In

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“…In general, the majority of work on dynamic algorithms for directed graphs has focused on shortest/longest paths and transitive closure (e.g., [King and Sagert 1999;Demetrescu and Italiano 2000;Djidjev et al 2000;Frigioni et al 1998;Baswana et al 2002;). For undirected graphs, there has been substantially more work and a survey of this area can be found in [Italiano et al 1999].…”

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confidence: 99%

A dynamic topological sort algorithm for directed acyclic graphs

Pearce

2007

ACM J. Exp. Algorithmics

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We consider the problem of maintaining the topological order of a directed acyclic graph (DAG) in the presence of edge insertions and deletions. We present a new algorithm and, although this has inferior time complexity compared with the best previously known result, we find that its simplicity leads to better performance in practice. In addition, we provide an empirical comparison against the three main alternatives over a large number of random DAGs. The results show our algorithm is the best for sparse digraphs and only a constant factor slower than the best on dense digraphs.

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Improved Algorithms for Dynamic Shortest Paths

Cited by 15 publications

References 40 publications

Computing the Girth of a Planar Graph in O(n logn) Time

Computing the Girth of a Planar Graph in O(n logn) Time

Incremental maintenance of shortest distance and transitive closure in first-order logic and SQL

A dynamic topological sort algorithm for directed acyclic graphs

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