Rohnert, H., A dynamization of the all pairs least cost path problem, in: Proc. Symp. Theoretical Aspects of Computer Science (STACS'85), LNCS 182 (1985), pp. 279--286.  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 ) ....

H.Rohnert, A dynamization of the all-pairs least cost path problem, Proc. of the 2nd Symp. on Theoretical Aspects of Computer Science, LNCS 182, Springer-Verlag, 1990.

....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 ....

H. Rohnert. A dynamization of the all pairs least cost path problem. In Proc. 2nd Annual Symp. on Theoretical Aspects of Computer Science, pages 279--286. Lecture Notes in Computer Science 182, Springer-Verlag, Berlin, 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 ....

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H. Rohnert. A dynamization of the all pairs least cost path problem. In Symposium on Theoretical Aspects of Computer Science, pages 279--286. Lect. Notes in Comp. Sci., 182, 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 ....

H. Rohnert. A dynamization of the all-pairs least cost path problem. Proc. 2nd Annual Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science 182, 279--286. 18

....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 ....

H. Rohnert, A dynamization of the all-pairs least cost path problem, Proc. Annual Symp. Th. Aspects Comp. Sci. (STACS '85), 1985; LNCS 182, 279--286.

.... d (v) then [15] AdjustHeap(Heap, v, min(rhs (v) d (v) 16] else [17] if v Heap then Remove v from Heap fi [18] fi [19] od [20] else u is underconsistent [21] d (u) 22] for v (Succ (u) u ) do [23] rhs (v) g v (d (x 1 ) d (x k ) 24] if rhs (v) d (v) then [25] AdjustHeap(Heap, v, min(rhs (v) d (v) 26] else [27] if v Heap then Remove v from Heap fi [28] fi [29] od [30] fi [31] od end postconditions Every vertex in V (G) is consistent ################################################################################################ Figure 1. ....

.... recomputeProductionValue(p) 19] od [20] while GlobalHeap do [21] Select and remove from GlobalHeap a non terminal X with minimum key value [22] if key (X) d (X) then X is overconsistent [23] d (X) key (X) 24] SP (X) p p is a production for X such that value (p) d (X) [25] Heap (X) 26] for every production p with X on the right hand side do [27] recomputeProductionValue(p) 28] od [29] else X is underconsistent [30] d (X) 31] SP (X) p p is a production for X [32] Heap (X) makeHeap( p p is a production for X with value (p) d (X) ....

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Rohnert, H., "A dynamization of the all pairs least cost path problem," pp. 279-286 in Proceedings of STACS 85: Second Annual Symposium on Theoretical Aspects of Computer Science, (Saarbruecken, W. Ger., Jan. 3-5, 1985), Lecture Notes in Computer Science, Vol. 182, ed. K. Mehlhorn,Springer-Verlag, New York, NY (1985).

....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 ....

H. Rohnert. A dynamization of the all pairs least cost path problem. In Proc. 2nd Symp. Theoretical Aspects of Computer Science, pages 279--286. Lecture Notes in Computer Science 182, Springer-Verlag, Berlin, 1985.

....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 ....

H. Rohnert, A dynamization of the all-pairs least cost path problem, Proceedings 2nd Annual Symposium on Theoretical Aspects of Computer Science, Saarbrucken, Germany, January 3--5, 1985; Lecture Notes in Computer Science 182, Springer-Verlag (1985), 279-- 286.

....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 ....

....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 maintain a sssp tree during a sequence of edge deletions in total time O(Cmn) thus obtaining a O(Cn) amortized time for each deletion. We also maintain a bfs tree of a graph G from ....

H. Rohnert, A dynamization of the all-pairs least cost path problem, Proceedings 2nd Annual Symposium on Theoretical Aspects of Computer Science, Saarbrucken, January 3--5, 1985; Lecture Notes in Computer Science 182, Springer-Verlag, Berlin, 279--286.

.... 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 ....

H. Rohnert, "A dynamization of the all pairs least cost path problem", Proc. 2nd Annual Symp. on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, vol. 182, Springer-Verlag, Berlin, 1985, 279--286.

....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] ....

H. Rohnert. A dynamization of the all pairs least cost path problem. In Proc. 2nd Annual Symp. on Theoretical Aspects of Computer Science, (STACS 85), Lecture Notes in Computer Science, vol. 182, pages 279-- 286. Springer-Verlag, Berlin, 1985.

.... 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 dynamic, randomized ....

H. Rohnert, A Dynamization of the All-Pairs Least-Cost Path Problem, in Proceedings of the 2nd Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, vol. 182, Springer, Berlin, 1985, pp. 279--286.

.... as well, and have 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 ....

H.Rohnert. A Dynamization of the All Pairs Least Cost Path Problem. Lecture Notes in Computer Science 182 (1985) 279-286.

....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 ....

H. Rohnert, A dynamization of the all pairs least cost path problem, in Proc. 2nd STACS'85, Lecture Notes in Computer Science, 182 (Springer-Verlag, 1985), 279-286.

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Rohnert, H., A dynamization of the all pairs least cost path problem, in: Proc. Symp. Theoretical Aspects of Computer Science (STACS'85), LNCS 182 (1985), pp. 279--286.

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Rohnert, H., "A dynamization of the all pairs least cost path problem," pp. 279-286 in Proceedings of STACS 85: Second Annual Symposium on Theoretical Aspects of Computer Science, (Saarbruecken, W. Ger., Jan. 3-5, 1985), Lecture Notes in Computer Science, Vol. 182, ed. K. Mehlhorn,Springer-Verlag, New York, NY (1985). - 37 -

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Rohnert, H., "A dynamization of the all pairs least cost path problem," pp. 279-286 in Proceedings of STACS 85: Second Annual Symposium on Theoretical Aspects of Computer Science, (Saarbruecken, W. Ger., Jan. 3-5, 1985), Lecture Notes in Computer Science, Vol. 182, ed. K. Mehlhorn,Springer-Verlag, New York, NY (1985).

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Rohnert, H., "A dynamization of the all pairs least cost path problem," pp. 279-286 in Proceedings of STACS 85: Second Annual Symposium on Theoretical Aspects of Computer Science, (Saarbruecken, W. Ger., Jan. 3-5, 1985), Lecture Notes in Computer Science, Vol. 182, ed. K. Mehlhorn,SpringerVerlag, New York, NY (1985).

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