## Improved Algorithms for Dynamic Shortest Paths (1996)

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Venue: | Algorithmica |

Citations: | 15 - 3 self |

### BibTeX

@ARTICLE{Djidjev96improvedalgorithms,

author = {Hristo N. Djidjev and Grammati E. Pantziou and Christos D. Zaroliagis},

title = {Improved Algorithms for Dynamic Shortest Paths},

journal = {Algorithmica},

year = {1996},

volume = {28},

pages = {2000}

}

### Years of Citing Articles

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### Abstract

We describe algorithms for finding shortest paths and distances in outerplanar and planar digraphs that exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. In the case of outerplanar digraphs, our data structures can be updated after any such change in only logarithmic time and a distance query is answered also in logarithmic time. In the case of planar digraphs, we give an interesting trade-off between preprocessing, query and update times depending on the value of a certain topological parameter of the graph. Our results can be extended to n-vertex digraphs of genus O(n 1\Gamma" ) for any " ? 0. Keywords: Shortest path, dynamic algorithm, planar digraph, outerplanar digraph. This work was partially supported by the NSF grant No. CCR-9409191 and by the EU ESPRIT LTR Project No. 20244 (ALCOM-IT). 1 Introduction 1.1 The prob...

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Citation Context ...ons of our results. Preliminary portions of this work have been appeared in [13]. 2 Preliminaries We assume that the reader is familiar with standard graph-theoretic terminology as contained e.g., in =-=[2, 23]-=-. A graph is called outerplanar if it can be embedded in the plane such that all of its vertices lie on one face. Let G = (V (G); E(G)) be a connected digraph. A separation pair is a pair (x; y) of ve... |

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Citation Context ...ons of our results. Preliminary portions of this work have been appeared in [13]. 2 Preliminaries We assume that the reader is familiar with standard graph-theoretic terminology as contained e.g., in =-=[2, 23]-=-. A graph is called outerplanar if it can be embedded in the plane such that all of its vertices lie on one face. Let G = (V (G); E(G)) be a connected digraph. A separation pair is a pair (x; y) of ve... |

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Citation Context ...ok for shortest paths between every pair of vertices in G; and the single-source shortest path in which we look for shortest paths between a specific vertex and all other vertices in G. Recent papers =-=[8, 12, 17, 18, 19, 20, 22, 27, 30, 32]-=- investigate the shortest path problem for different classes of input graphs and models of computation. All of the above mentioned results, however, relate to the static version of the problem, i.e., ... |

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Citation Context ...on, can be performed in O(log 3 n) time. This result, however, has been superseded by the O(n) time single-source shortest path algorithm for planar digraphs with nonnegative real edge costs given in =-=[29]-=-. Also in that paper [29] a dynamic algorithm is given for shortest paths in planar digraphs with integral edge costs (which may be negative). More precisely, the algorithm initializes an O(n)-size da... |

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Citation Context ... (G) whose removal leaves no connected component with more than ffjV (G)j vertices. It is well known that if G is outerplanar, then there exists a 2=3-separator of G which is a single separation pair =-=[7]-=-. Such a separation pair can be found by triangulating all internal faces of G and finding a separator edge in the dual graph of the resulting embedding, excluding the outer face, which is a tree. Thr... |

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Citation Context ... update time after an edge deletion or edge cost increase is equal to the time required to recompute all pairs shortest paths from scratch [22]. Improvements on these algorithms have been achieved in =-=[4]-=- with respect to the worstcase complexity of a sequence of edge insertions or edge cost decreases (thus providing a better bound per update in the amortized sense), in the special case where the edge ... |

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Citation Context ...thms for the dynamic shortest path problem. For general digraphs with real edge costs, the best previous algorithms, in the case of edge insertions, edge deletions and edge cost updates, are given in =-=[14, 34]-=-. The data structure in [14, 34] is updated in O(n 2 ) time after an edge insertion or edge cost decrease, and in O(nm + n 2 log n) time after an edge deletion or edge cost increase (m being the curre... |

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Citation Context ...ersion of the shortest path problem has several applications including dynamic maintenance of a maximum s-t flow in a planar network [24], computing a feasible flow between multiple sources and sinks =-=[31]-=-, as well as finding a perfect matching in bipartite planar graphs [31]. Dynamic algorithms for shortest paths appear also to be fundamental procedures in incremental computations for data flow analys... |

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Citation Context ...ok for shortest paths between every pair of vertices in G; and the single-source shortest path in which we look for shortest paths between a specific vertex and all other vertices in G. Recent papers =-=[8, 12, 17, 18, 19, 20, 22, 27, 30, 32]-=- investigate the shortest path problem for different classes of input graphs and models of computation. All of the above mentioned results, however, relate to the static version of the problem, i.e., ... |

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Citation Context ...blem domains which are of both theoretical and practical value, including (among others) communication networks, high level languages for incremental computations [38], incremental data flow analysis =-=[5]-=-, database and knowledge base systems [1, 37], and programming environments [25]. Finding shortest path information in graphs is an important and intensively studied problem with many applications. Gi... |

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Citation Context ...iate applications to a variety of problem domains which are of both theoretical and practical value, including (among others) communication networks, high level languages for incremental computations =-=[38]-=-, incremental data flow analysis [5], database and knowledge base systems [1, 37], and programming environments [25]. Finding shortest path information in graphs is an important and intensively studie... |

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Citation Context ...thms for the dynamic shortest path problem. For general digraphs with real edge costs, the best previous algorithms, in the case of edge insertions, edge deletions and edge cost updates, are given in =-=[14, 34]-=-. The data structure in [14, 34] is updated in O(n 2 ) time after an edge insertion or edge cost decrease, and in O(nm + n 2 log n) time after an edge deletion or edge cost increase (m being the curre... |

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Citation Context ...ear time and an embedding of the graph does not need to be provided by the input [9, 10]. If such an embedding is provided as an input, then one can find also an o(n)-decomposition for G in O(n) time =-=[3]; otherwis-=-e, this division is computed in O(n log n) time. Thus, we have the following two types of results for the class of digraphs of genus fl = O(n 1\Gamma" ), where " ? 0. If an embedding of G on... |

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Citation Context ...ching in bipartite planar graphs [31]. Dynamic algorithms for shortest paths appear also to be fundamental procedures in incremental computations for data flow analysis and interactive systems design =-=[33, 38]-=-. 1.2 Previous results There are a few previously known algorithms for the dynamic shortest path problem. For general digraphs with real edge costs, the best previous algorithms, in the case of edge i... |

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Citation Context ... as well as for edge cost modifications and edge deletions, while for edge insertions it is amortized. Other dynamic algorithms for the shortest path problem are known for special classes of digraphs =-=[6]-=-. On the other hand, efficient data structures for answering very fast on-line shortest path or distance queries in planar digraphs with real edge costs have been proposed in [12, 19], but they do not... |

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Citation Context ... plus a small number of non-tree edges and thus have a small value of q. Also, our algorithms seem to be very efficient for the class of all appropriately sparse graphs. As it has been established in =-=[15, 28]-=- random G n;p graphs with threshold function 1=n are planar with probability one and have expected value for q equal to O(1). Our solution is based on the following ideas: (a) The input planar digraph... |

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Citation Context ...s. Our results for planar digraphs are given in Section 4. Finally, in Section 5 we describe the extensions and generalizations of our results. Preliminary portions of this work have been appeared in =-=[13]-=-. 2 Preliminaries We assume that the reader is familiar with standard graph-theoretic terminology as contained e.g., in [2, 23]. A graph is called outerplanar if it can be embedded in the plane such t... |

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Citation Context ... any n-vertex digraph G of genus fl ? 0, a 2=3-separator of size O( p fln) exists. Such a separator can be found in linear time and an embedding of the graph does not need to be provided by the input =-=[9, 10]-=-. If such an embedding is provided as an input, then one can find also an o(n)-decomposition for G in O(n) time [3]; otherwise, this division is computed in O(n log n) time. Thus, we have the followin... |

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Citation Context ...hat any edge of the digraph belongs to O(log n) of those paths (n being the size of the outerplanar digraph). Our second result for planar digraphs is based on the above ideas and on the recent paper =-=[11]-=-, which studies tradeoffs between the preprocessing time and space and the query time for the shortest path problem in planar (non--dynamic) digraphs. Our result is as follows: Given an n-vertex plana... |

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Citation Context ... any n-vertex digraph G of genus fl ? 0, a 2=3-separator of size O( p fln) exists. Such a separator can be found in linear time and an embedding of the graph does not need to be provided by the input =-=[9, 10]-=-. If such an embedding is provided as an input, then one can find also an o(n)-decomposition for G in O(n) time [3]; otherwise, this division is computed in O(n log n) time. Thus, we have the followin... |

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Citation Context ... deletions or edge cost increases are not considered in [4].) For the important case of planar digraphs with nonnegative real edge costs, dynamic algorithms for the shortest path problem are given in =-=[16]-=-. The preprocessing time and space is O(n log n). (The space can be reduced to O(n), if the computation is restricted to finding distances only.) A shortest path or distance query can be answered in O... |

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Citation Context ...date) must be performed before the next one is known. The dynamic version of the shortest path problem has several applications including dynamic maintenance of a maximum s-t flow in a planar network =-=[24]-=-, computing a feasible flow between multiple sources and sinks [31], as well as finding a perfect matching in bipartite planar graphs [31]. Dynamic algorithms for shortest paths appear also to be fund... |

1 | Computing Shortest Paths and Distances - Djidjev, Pantziou, et al. - 1991 |