S. Even and H. Gazit, "Updating Distances in Dynamic Graphs," Methods of Operations Research 49 (1985), 371--387.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....worst case analysis. Furthermore we consider an intermediate model between worst case analysis and average case analysis: the semi random adversary introduced in [3] 1 Introduction Significant progress has been recently made in the design of algorithms and data structures for dynamic graphs [1, 5, 6, 8, 11, 12, 13, 16, 17, 18, 19, 20, 21, 24]. These data structures support insertions and deletions of edges and or nodes in a graph, in addition to several types of queries. The goal is to compute the new solution in the modified graph without having to recompute it from scratch. Usually, the sequence of insertions deletions of edges is ....

.... of vertices, the update amortized time for directed graphs is O(n) instead of O(ff(n; n) for undirected graphs [16, 19, 23] If we consider deletions of edges there are solutions for special classes of graphs such as directed acyclic graphs [17] The fully dynamic problem has also been studied [11, 19, 21] but, to the best of our knowledge, no fully dynamic data structure exists for general directed graphs that, in the worst case, achieves a bound of o(m) for reachability queries and update operations. Conversely, if we look to undirected graphs, the fully dynamic problem can be solved in O(n ) ....

S. Even and H. Gazit, Updating distances in dynamic graphs, Methods of Operations Research, 49, 1985.

....edge insertions are a special case of Decrease. Throughout the paper, we denote by n the number of vertices in G. Previous Work. The dynamic maintenance of shortest paths has been investigated for over three decades, as the first papers date back to 1967 [18, 19, 22] In 1985 Even and Gazit [5] and Rohnert [23] presented algorithms for maintaining shortest paths on directed graphs with arbitrary real weights. Their algorithms required O(n ) per edge insertion; however, the worst case bounds for edge deletions were comparable to recomputing APSP from scratch. Also Ramalingam and Reps ....

S. Even and H. Gazit. Updating distances in dynamic graphs. Methods of Operations Research, 49:371--387, 1985.

....the paper, we denote by n the number of vertices in the graph. Previous work. The dynamic maintenance of shortest paths has a remarkably long history, as the rst papers date back to 35 years ago [15, 16, 19] After that, many dynamic shortest paths algorithms have been proposed (see e.g. [6, 9, 10, 17, 18, 20]) but their running times in the worst case were comparable to recomputing APSP from scratch. The rst dynamic shortest path algorithms which are provably faster than recomputing APSP from scratch, only worked on graphs with small integer weights. In particular, Ausiello et al. 1] proposed a ....

S. Even and H. Gazit. Updating distances in dynamic graphs. Methods of Operations Research, 49:371-387, 1985.

....from x to y. Path(x, y) report a shortest path from x to y, if any. Throughout the paper, we denote by n the number of vertices in G. Previous Work. The dynamic maintenance of shortest paths has a long history, and the first papers date back to 1967 [26, 29, 32] In 1985 Even and Gazit [10] and Rohnert [33] presented algorithms for maintaining shortest paths on directed graphs with arbitrary real weights. Their algorithms required O(n per edge insertion; however, the worst case bounds for edge deletions were comparable to recomputing APSP from scratch. Also Ramalingam and Reps ....

S. Even and H. Gazit. Updating distances in dynamic graphs. Methods of Operations Research, 49:371--387, 1985.

....distance from x to y. Path(x; y) report a shortest path from x to y, if any. Throughout the paper, we denote by n the number of vertices in G. Previous Work. The dynamic maintenance of shortest paths has a long history, and the rst papers date back to 1967 [15, 17, 20] In 1985 Even and Gazit [6] and Rohnert [21] presented algorithms for maintaining shortest paths on directed graphs with arbitrary real weights. Their algorithms required O(n 2 ) per edge insertion; however, the worst case bounds for edge deletions were comparable to recomputing APSP from scratch. Also Ramalingam and Reps ....

S. Even and H. Gazit. Updating distances in dynamic graphs. Methods of Operations Research, 49:371-387, 1985.

....interest in dynamic problems on graphs. In particular, much attention has been devoted to the dynamic maintenance of connected components [14, 16, 43] and higher connectivity [8, 10, 18, 19, 20, 21, 32, 35, 53] transitive closure [29, 30, 31, 37, 47, 55] planarity [7, 8, 46] shortest paths [2, 5, 13, 39, 44], and minimum spanning trees [10, 11, 16] In these problems one would like to answer queries on graphs that are undergoing a sequence of updates, such as insertions and deletions of edges and vertices. The goal of a dynamic graph algorithm is to update efficiently the solution of a problem after ....

S. Even and H. Gazit. Updating distances in dynamic graphs. Methods of Operations Research, 49:371--387, 1985.

.... e Sistemistica, Universit a di Roma La Sapienza , Via Salaria 113 00198 Roma, Italy, ffrigioni,ioffreda,nanni,pasqualog dis.uniroma1.it 1 1 Introduction A lot of efforts have been done in the last years in order to devise efficient algorithms for dynamic graph problems (e.g. see [6, 9, 13, 14, 15, 16, 18, 20, 23, 24, 25, 26, 30, 31, 32]) motivated by theoretical as well as practical applications. In the literature, the most used dynamic model is the following: we are given a graph G and we want to answer queries on a property P of G, while the graph is changing due to insertions and deletions of edges. For instance, if the ....

....paper we provide the first experimental study of dynamic algorithms for the single source shortest paths problem. 1. 1 Previous theoretical results Many dynamic solutions have been proposed in the literature for the shortest paths problem, both for the single source and the all pairs versions [6, 9, 14, 15, 16, 18, 20, 26, 31, 32]. A fully dynamic solution for maintaining all pairs shortest paths on planar graphs with unrestricted edge weights is given in [26] but the algorithm is complex and far from being practical. In [9] efficient dynamic solutions are provided for graphs with bounded treewidth when the weights of ....

[Article contains additional citation context not shown here]

S. Even and H. Gazit. Updating distances in dynamic graphs. Methods of Operations Research, 49:371--387, 1985.

....allowed we refer to the fully dynamic problem; if we consider only insertions (deletions) of arcs then we refer to the incremental (decremental) problem. In the case of positive arc weights there is a number of papers that propose different solutions to deal with dynamic shortest paths problems [3, 4, 7, 8, 10, 11, 13, 17, 18]. However, in the general case, neither a fully dynamic solution nor a decremental solution for the single source shortest path problem is known in the literature that is asymptotically better than recomputing the new solution from scratch. Work partially supported by the ESPRIT Long Term ....

S. Even and H. Gazit. Updating distances in dynamic graphs. Methods of Operations Research, 49 (1985), 371--387.

....insertions and deletions of edges. An efficient solution for the incremental problem has been proposed in [3] assuming that edge weights are integers restricted in the range [1: C] Further results concerning the dynamic shortest paths problem for general graphs have been proposed, for example, in [7, 18, 19]. To the best of our knowledge, if insertions and deletions of edges are allowed and there is no restriction on the class of graphs then neither a fully dynamic solution nor a decremental solution for the single source shortest path problem is known that, in the worst case, is asymptotically ....

S. Even and H. Gazit, Updating distances in dynamic graphs, Meth. Oper. Res., 49 (1985), 371--387.

....in O(n p log n) time. Recently in joint work with Rao and Rauch [4] we have given an O(n) time algorithm for computing single source shortest paths. However, in the dynamic realm this problem is much less well understood. Though there are many algorithms for the dynamic problem (see for example [5, 6, 7], see also [8] none of them can simultaneously handle both updates and queries in time that is sublinear in the input size. Definition 1 Let G be an n node planar undirected graph with nonnegative integral edge lengths. Let D be the sum of lengths. The length of a path from u to v (denoted as ....

S. Even and H. Gazit, "Updating Distances in Dynamic Graphs," Methods of Operations Research 49 (1985), 371-387

....Interesting notions of completeness are sketched by Reif [14] He shows that some problems are unlikely to have efficient incremental solutions, but he does not develop a comprehensive theory or consider the necessary details of preprocessing. Some interesting techniques are explored in [2] [4] and [13] to derive lower bounds for incremental algorithms. Their main idea is to construct an algorithm for the batch version of the problem with input I by computing a dynamic data structure for some easily computed instance I 0 , whose Hamming distance from I is small, and then applying the ....

S. Even and H. Gazit, "Updating Distances in Dynamic Graphs," Methods of Operations Research 49 (1985), 371--387.

....Interesting notions of completeness are sketched by Reif [21] He shows that some problems are unlikely to have efficient incremental solutions, but he does not develop a comprehensive theory or consider the necessary details of preprocessing. Some interesting techniques are explored in [2] [6] and [20] to derive lower bounds for incremental algorithms. Their main idea is to construct an algorithm for the batch version of the problem with input I by computing a dynamic data structure for some easily computed instance I 0 , whose Hamming distance from I is small, and then applying the ....

S. Even and H. Gazit, "Updating Distances in Dynamic Graphs," Methods of Operations Research 49 (1985), 371--387.

....Grant CDA 9024735. z University of Venice Ca Foscari , Venice, Italy. Supported in part by the ESPRIT LTR Project no. 20244 (ALCOM IT) and by a Research Grant from University of Venice Ca Foscari . Most of the efficient data structures available for directed graphs are partially dynamic [2, 13, 29, 30, 31, 37, 39, 43, 53], and only preliminary results are available for fully dynamic problems [25] For this reason, an alternative viewpoint that has been proposed is to measure the complexity of a dynamic algorithm as a function of the output change [17, 40] The main dynamic problems considered on directed graphs ....

S. Even and H. Gazit. Updating distances in dynamic graphs. Methods of Operations Research, 49:371--387, 1985.

.... formatting, routing in communication systems, robotics (see e.g. 1] for a wide variety of application settings for the shortest paths problem) Many solutions have been proposed in the literature to deal with dynamic shortest paths problems both for the single source and the all pairs versions [4, 9, 12, 13, 15]. In the case of planar graphs a fully dynamic solution for maintaining all pairs shortest paths with unrestricted edge weights has been proposed in [13] but the proposed algorithm is complex and far from being practical. An efficient solution for the all pairs incremental problem has been ....

....no. 20244, and by Progetto Finalizzato Trasporti 2 (PFT 2) of the Italian National Research Council (CNR) of the problem on digraphs, with integer edge weights in [1: C] has been given in [10] Further results concerning the dynamic shortest paths problem have been proposed, for example, in [9, 12, 15]. All the mentioned solutions are characterized by a given setting, either for the kind of considered graph and edge weights, or for the set of allowed updates on the graph. However, to the best of our knowledge, no fully dynamic solution for the single source shortest path problem is known in the ....

S. Even, H. Gazit, Updating Distances in Dynamic Graphs, Methods of Operation Research 49 (1985), 371--387.

....with updates on the structure of the graph, while maintaining the possibility to answer queries on shortest paths without recomputing them from scratch. Various approaches have been considered in literature to deal with dynamic shortest path problems both for single source and all pairs versions [4, 7, 8, 13, 14, 18, 20], providing several noncomparable solutions, each characterized by a given setting for (a) the kind of considered graph and edge weights, b) the set of allowed updates on the structure of the graph, and (c) the adopted measure of performances. Some of the proposed solutions pursue a trade off ....

....These solutions use a topological partition of the graph based on a recursive application of the planar separator theorem [15] all these algorithms are complex and far from being practical. The explicit update of all pairs shortest paths for general graphs has been considered, for example, in [4, 7, 20]. In the particular case of edge insertions in a directed graph G = V; E) with jV j = n, and jEj = m, and integer edge weights in the range [0: C] an algorithm is provided in [4] requiring O(Cn log n) amortized time per edge insertion in any sequence of Omega Gamma m) insertions, while ....

S. Even, and H. Gazit, Updating distances in dynamic graphs, Methods of Operations Research 49 (1985), 371--387.

....updates on the structure of the graph, while maintaining the possibility to answer queries on shortest paths without recomputing them from scratch. Various approaches have been considered in the literature to deal with dynamic shortest path problems both for single source and all pairs versions [2, 4, 6, 8, 9, 11, 12, 17], providing several non comparable solutions, each characterized by a given setting for (a) the kind of considered graph and edge weights, b) the set of allowed updates on the structure of the graph, and (c) the adopted measure of performances. The most general repertoire of update operations ....

....from scratch after each deletion, which can be accomplished in O(m 2 ) total time. Note that our technique allows to decrementally maintain the all pairs shortest paths with unit edge weights in O(n 2 ) amortized time for each deletion, thus substantially improving the time bounds given in [4], where insertions and deletions are respectively performed in O(n 2 ) and O(mn n 2 log n) worst case time. Other results on the all pairs shortest paths can be found in [17] Our result for unit edge weights is extended to handle the case of integer edge weights in [1; C] allowing to ....

S. Even and H. Gazit, Updating distances in dynamic graphs, Methods of Operations Research 49 (1985), 371--387.

....Maximum flow and shortest path are fundamental problems in network optimization and have been extensively investigated [9,14] A long standing question has been the development of efficient dynamic algorithms for such problems. The dynamic data structures for all pairs shortest paths presented in [12,22] have O(n 2 ) space, O(1) query time, O(n 2 ) time for edge insertion, and O(mn n 2 log n) time for edge deletion, where n is the number of vertices and m is the number of edges. Such perfor1 mance bounds, especially for deletion, are quite unsatisfactory and indicate the difficulty of the ....

S. Even and H. Gazit, "Updating Distances in Dynamic Graphs," Methods of Operations Research 49 (1985), 371--387.

.... problems [11] Moreover, despite intensive research on dynamic problems on graphs (such as dynamic maintenance of connectivity [7, 8, 10, 11, 14, 20, 22, 29, 30] 2 and 3 connectivity [7, 12, 29, 30] transitive closure [3, 4, 15, 16, 17, 18, 19, 31] planar graphs [6, 7, 19, 25] shortest paths [2, 9, 21, 24, 31] and minimum spanning trees [5, 8, 11, 24] there are very few graphtheoretic problems for which a fully dynamic non trivial algorithm is known. As mentioned in [30] the fully dynamic maintenance of the connected components of a graph differs substantially from the fully dynamic maintenance of ....

S. Even, and H. Gazit, "Updating distances in dynamic graphs", Methods of Operations Research 49 (1985), 371--387.

....in response to changes in the input data. Domains for such algorithms have included: ffl graph theoretic algorithms: connectivity [ES81, Har83, Che84] spanning trees [SP73, CH78, FS84, Fre85] spanning forests [Wes89] shortest paths [Rod68, Che76, GSV78, Fuj81, CC82, Gaz83, EG85, AMSN89, AIMSN90, Ita91] biconnected components [Sac86, WT92, BT90] triconnected components [Ita91, BT90] transitive closure [IK83, Ita86, Ita88, LPv88, YS88, Yel91] planar graphs [Tam88, TP90, BT89, EIT 92, PT88] ffl computational geometry [Ov81, CBT 92] ffl data ....

....paucity of work in the area of lower bounds. In their paper describing an incremental algorithm for updating minimum paths, Even and Gazit prove that in the worst case no incremental algorithm that handles edge deletions can be faster than an algorithm that solves the problem from scratch [EG85] This work inspired the developments described in Chapter 4. Work begun by Alpern et al. AHR 90] and extended by Reps and Ramalingam [RR91] gives lower bounds based on ffi analysis, within a limited model of computation called local persistence; their work is described and extended further ....

[Article contains additional citation context not shown here]

S. Even and H. Gazit. Updating distances in dynamic graphs. Methods Oper. Research, 49:371--387, 1985.

....dynamic operations. 2.4 Shortest Path The shortest path query on a digraph G consists of answering queries of the type What is the length of a shortest path from v 1 to v 2 Sometimes the path itself is also requested. The fully dynamic data structures for the shortest path query presented in [40,95] have O(n 2 ) space, constant query time, O(n 2 ) time for edge insertion, and O(mn n 2 log n) time for edge deletion. Such performance bounds, especially for deletion, are quite unsatisfactory and indicate the difficulty of the query. The best known semidynamic data structures ....

S. Even and H. Gazit, "Updating Distances in Dynamic Graphs," Methods of Operations Research 49 (1985), 371--387.

....tree it belongs to. Dynamic problems on graphs have been extensively studied. Several algorithms have been proposed for maintaining fundamental structural information about dynamic graphs, such as connectivity [9, 10, 15, 24, 26] transitive closure [17, 18, 19, 20, 21, 34, 23] and shortest paths [1, 8, 25, 28, 34]. Dynamic planar graphs arise in communication networks, graphics, and VLSI design, and they occur in algorithms that build planar subdivisions such as Voronoi diagrams. Algorithms have been proposed for maintaining the embedding of a planar graph [29] and for incremental planarity testing [2, 3] ....

S. Even and H. Gazit. Updating distances in dynamic graphs. Methods of Operations Research, 49:371--387, 1985.

....structure quickly. In other words a small change to the input graph should not force us to recompute the entire data structure. Thus the challenge of constructing a dynamic data structure is to satisfy both these requirements simultaneously. Though there are many algorithms for the dynamic problem [7,27,33] (see also [24] none of them can simultaneously handle both updates and queries in time that is sublinear in the input size. 1.1.3 Finding shortest paths in parallel and dynamic settings In this thesis we give efficient exact and approximate algorithms for solving shortestpath problems in ....

....for fully dynamic data structures to various graph problems. See [81] for a complexity theoretic approach to dynamic computation. As we discussed in Chapter 1 the shortest path problem is not very well understood in the dynamic realm. Though there are many algorithms for the dynamic problem [7, 24,27,33], none of them can simultaneously handle both updates and queries in time that is sublinear in the input size. We say that a path is an ffl approximate shortest path if its length is at most 1 ffl times the distance between its endpoints. In this chapter we show that if we are willing to settle ....

S. Even and H. Gazit, "Updating distances in dynamic graphs," Methods of Operations Research 49 (1985), 371--387.

.... values of the edge lengths is D, then the time per operation is O(n 9=7 log D) worst case for queries, edge deletion, and length changes, and amortized for edge insertion) the space requirement is O(n) Several types of partially dynamic algorithms for shortest paths appear in [AuItMaNa90 ] [EvGa85], FrMaNa94] and [Ro85] Although it is one of the most important dynamic graph algorithms problems, there is less known about shortest paths than about many other problems, and this is an important topic for future study. In a recent breakthrough, Henzinger and King [HeKi95] obtained fully ....

S. Even and H. Gazit, Updating Distances in Dynamic Graphs, Methods of Operations Research, 49 (1985), pp. 371--387.

.... already been examined and discussed by various authors, such as Even Shiloach [4] and Ibaraki Katoh [8] transitive closures) La Poutr e van Leeuwen [11] transitive closures and reductions) Frederickson [6] minimum spanning trees) and Goto Sangiovanni Vincentelli [7] Rohnert [14] Even Gazit [3] and Ausiello et al. 2] least cost paths) Our problem can be formalized as follows: given a digraph G = V; E) and a cost function C : E IR, which does not imply negative cost cycles, we want to compute LeastCost(v; ffi ) the cost of the least cost path from a given origin r to the node ....

S.Even and H.Gazit. Updating Distances in Dynamic Graphs. Meth. of Oper. Res. 49 (1985) 371-387.

.... Several works have been published dealing with topological changes in which links may occasionally become unavailable (i.e. infinite weight) and others deal with quasi static models, that is, link weights that change from time to time but remain constant in between these (infrequent) changes[2,3]. Time dependent shortest path problems have been studied for the case of discrete delay functions whose domain and range are the positive integers. Such problems were addressed both directly [4,5] and indirectly in the context of maximal flow[6,7] In this paper we address the shortest path ....

S. Even and H. Gazit, "Updating Distances in Dynamic Graphs," Methods of Operations Research 49 pp. 371-387 (May 1985).

....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 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 current number of edges in the graph) Note that the update time after an edge deletion or edge ....

....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 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 current number of edges in the graph) Note that the update time after an edge deletion or edge cost increase is equal to the time ....

S. Even and H. Gazit, Updating distances in dynamic graphs, Methods of Operations Research, 49 (1985), 371-387.

No context found.

S. Even and H. Gazit, "Updating Distances in Dynamic Graphs," Methods of Operations Research 49 (1985), 371--387.

No context found.

Even, S. and H. Gazit, Updating distances in dynamic graphs, Methods of Operations Research 49 (1985), pp. 371--387.

No context found.

Even, S. and Gazit, H., "Updating distances in dynamic graphs," pp. 271-388 in IX Symposium on Operations Research, (Osnabrueck, W. Ger., Aug. 27-29, 1984), Methods of Operations Research, Vol. 49, ed. P. Brucker and R. Pauly,Verlag Anton Hain (1985).

No context found.

S. Even and H. Gazit. Updating distances in dynamic graphs. In P. Brucker and R. Pauly, editors, IX Symposium on Operations Research, volume 49 of Methods of Operations Research. Verlag Anton Hain, 1985.

No context found.

Even, S. and Gazit, H., "Updating distances in dynamic graphs," pp. 271-388 in IX Symposium on Operations Research, (Osnabrueck, W. Ger., Aug. 27-29, 1984), Methods of Operations Research, Vol. 49, ed. P. Brucker and R. Pauly,Verlag Anton Hain (1985).

No context found.

Even, S. and Gazit, H., "Updating distances in dynamic graphs," pp. 271-388 in IX Symposium on Operations Research, (Osnabrueck, W. Ger., Aug. 27-29, 1984), Methods of Operations Research, Vol. 49, ed. P. Brucker and R. Pauly,Verlag Anton Hain (1985). - 31 -

No context found.

S. Even and H. Gazit. Updating distances in dynamic graphs. Methods of Operations Research, 49:371--387, 1985.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC