J. A. La Poutre and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In Proc. Workshop on Graph-Theoretic Concepts in Computer Science, pages 106--120. Lecture Notes in Computer Science 314, Springer-Verlag, Berlin, 1988.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....O(IGReach(s)l) Dynamic Reachability A very efficient dynamic solution for this problem is possible in case of either sequences of edges insertion, or sequences of deletions in a day. The solution proposed here is similar to the one proposed in [7] and has the same time and space complexity of [3, 4, 9]. The performances of the algorithms are the following: O(m) time for any sequence of edge insertions, that is costant amortized time per edge insertion starting from an empty graph; O(m) time for any sequence of edge deletions, that is costant amortized time per edge deletion finishing ....

....to notice that procedure INS retains its validity while inserting edges in any directed graph, while DEL can be used only in acyclic graphs. Efficient deletion of edges in a cyclic directed graph is indeed a hard problem that still does not have a practical solution. The problem is addressed in [9]. More details about the proof of correctness and complexity of the proposed algorithms can be found in [7] 6 Dynamic Transitive Closure In this section we briefly describe the implementation of a simple and fast algorithm to maintain the transitive closure G of a graph G during insertion ....

J. A. La Poutr and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In Workshop on Graph-Theoretic Concepts in Computer 'cience, Lecture Notes in Computer Science, 314, pages 106 120. Springer-Verlag, 1988.

....0 otherwise. An edge insertion may result in the addition of more edges, and so on, recursively. The insert 19 operation returns a list of all edges that have been added, represented as node pairs. Initialization and maintenance of ITC can be done in O(n ) time, where n is the number of nodes [15, 21]. 2. PE: a data structure with potential edges. This data structure is computed in a preprocessing phase. It is a matrix of the same form as the adjacency matrix, but with each entry being a list of pairs of nodes. It is computed by, for each pair of L edges x 1 y 1 and x 2 y 2 , ....

J. A. La Poutr# and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In Proc. Workshop on Graph-Theoretic Concepts in Computer Science, pages 106120. Springer-Verlag (LNCS 314), 1988.

....For the incremental version of the problem, the first algorithm was proposed by Ibaraki and Katoh [11] in 1983: its running time was O(n 3 ) over any sequence of insertions. This bound was later improved to O(n) amortized time per insertion by Italiano [12] and also by La Poutr e and van Leeuwen [17]. Yellin [19] gave an O(m ffi max ) algorithm for m edge insertions, where m is the number of edges in the final transitive closure and ffi max is the maximum out degree of the final graph. All these algorithms maintain explicitly the transitive closure, and so their query time is O(1) The ....

....graph. All these algorithms maintain explicitly the transitive closure, and so their query time is O(1) The first decremental algorithm was again given by Ibaraki and Katoh [11] with a running time of O(n 2 ) per deletion. This was improved to O(m) per deletion by La Poutr e and van Leeuwen [17]. Italiano [13] presented an algorithm that achieves O(n) amortized time per deletion on directed acyclic graphs. Yellin [19] gave an O(m ffi max ) algorithm for m edge deletions, where m is the initial number of edges in the transitive closure and ffi max is the maximum out degree of the ....

J. A. La Poutr'e and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In Proc. Workshop on Graph-Theoretic Concepts in Computer Science, pages 106--120. Lecture Notes in Computer Science 314, Springer-Verlag, Berlin, 1988.

....Italy. 1 Introduction In the last decade there has been a growing 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 ....

J. A. La Poutr'e and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In Proc. Workshop on Graph-Theoretic Concepts in Computer Science, pages 106-- 120. Lecture Notes in Computer Science 314, Springer-Verlag, Berlin, 1988.

....and Katoh [Ibaraki and Katoh 1983] have described an incremental algorithm that maintains the transitive closure of a graph in O(nm ) time, where m is the number of edges of the transitive closure of G. This result was improved by Italiano [Italiano 1986] and by La Poutr e and van Leeuwen [La Poutr e and van Leeuwen 1988], their algorithms maintain the transitive closure of a graph in O(nm) time. This last result gives the best amortized update time (O(n) for maintaining transitive closure under edge additions [King 1999] Yellin [Yellin 1993] has presented an algorithm that solves the problem in O(dm ) time, ....

....have proposed an incremental algorithm which maintains the transitive closure of a graph in O(k 2 m m ) total time and O(kn) space. So it uses less space than the incremental algorithms that do not use the knowledge of the k paths (as Italiano [Italiano 1986] and La Poutr e and van Leeuwen [La Poutr e and van Leeuwen 1988] algorithms) but it could be slower than these algorithms when the number of paths is greater than the average length of these paths (k n=k) In [Abdedda m 1997] no experimental study was done in order to compare our algorithm real execution time with other incremental transitive closure ....

[Article contains additional citation context not shown here]

La Poutr e, J. A. and van Leeuwen, J. 1988. Maintenance of transitive closure and transitive reduction of graphs. In Proc. Workshop on Graph-Theoretic Concepts in Computer Science (1988), pp. 106-120. Lecture Notes in Computer Science 314, Springer-Verlag.

.... algorithms [13] On the other hand, the development of fully dynamic algorithms for maintenance of various properties on directed graphs (digraphs) turned out to be a much harder problem and much of the research so far was concentrated on the design of partially dynamic algorithms (see e.g. [4, 7, 8, 11, 26, 27, 28, 32]) Only recently, fully dynamic algorithms have started to appear for maintenance of shortest path trees [18, 19, 30] and transitive closure [9, 23, 24, 25] However, despite the number of interesting theoretical results achieved, very little has been done so far with respect to implementations ....

....G 0 = V 0 ; E 0 ) is a DAG. The idea is to use Italiano s algorithm to maintain the transitive closure of G 0 and additional information regarding the strongly connected components (SCCs) which is crucial for the decremental part of the algorithm. Note that a similar idea was used in [28]; however the 5 data structures and techniques used in that paper can answer only Boolean queries. To maintain path information in G 0 , we do not use the original data structures of Ital Opt, since whenever a SCC breaks (due to edge deletions) V 0 has to be updated and consequently its ....

[Article contains additional citation context not shown here]

J. A. La Poutre, and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In Proc. 14th Workshop on Graph-Theoretic Concepts in Computer Science, pp. 106-120. Lecture Notes in Computer Science, vol. 314, 1988.

....For the incremental version of the problem, the first algorithm was proposed by Ibaraki and Katoh [7] in 1983: its running time was O(n 3 ) over any sequence of insertions. This bound was later improved to O(n) amortized time per insertion by Italiano [8] and also by La Poutre and van Leeuwen [13]. Yellin [15] gave an O(m ffi max ) algorithm for m edge insertions, where m is the number of edges in the final transitive closure and ffi max is the maximum out degree of the final graph. All these algorithms maintain explicitly the transitive closure, and so their query time is O(1) The ....

....graph. All these algorithms maintain explicitly the transitive closure, and so their query time is O(1) The first decremental algorithm was again given by Ibaraki and Katoh [7] with a running time of O(n 2 ) per 1 deletion. This was improved to O(m) per deletion by La Poutre and van Leeuwen [13]. Italiano [9] presented an algorithm which achieves O(n) amortized time per deletion on directed acyclic graphs. Yellin [15] gave an O(m ffi max ) algorithm for m edge deletions, where m is the initial number of edges in the transitive closure and ffi max is the maximum out degree of the ....

J. A. La Poutre and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In Proc. Workshop on Graph-Theoretic Concepts in Computer Science, pages 106--120. Lecture Notes in Computer Science 314, Springer-Verlag, Berlin, 1988.

....quite different from the algorithm presented here. Other related work includes partially dynamic algorithms. The best result for updates allowing edge insertions only is O(n) amortized time per inserted edge, and O(1) time per query by Italiano (1986) 12] and by La Poutre and van Leeuwen (1987) [16]. This improved upon Ibaraki and Katoh s (1983) 11] algorithm with a total cost of O(n 3 ) for an arbitrary number of insertions. There is also Yellin s (1993) 19] algorithm, with a total cost of O(m Delta) for any number of insertions, where m is the number of edges in the transitive closure ....

....Yellin s (1993) 19] algorithm, with a total cost of O(m Delta) for any number of insertions, where m is the number of edges in the transitive closure and Delta is the out degree of the resulting graph. The best deletions only algorithm for general graphs is by La Poutre and van Leeuwen (1987)[16]. Their algorithm requires O(m) amortized time per edge deletion and O(1) per query. This improved upon the deletions only algorithm of Ibaraki and Katoh (1983) 11] which can delete any number of edges in O(n 2 (m n) total time. 4 Please write authorrunninghead Author Name(s) in file ....

H. La Poutr'e and J. van Leeuwen, "Maintenance of transitive closure and transitive reduction of graphs", Proc. Workshop on Graph-Theoretic Concepts in Computer Science, LNCS 314, Springer Verlag, Berlin, 1987, pp. 106--120.

....quite different from the algorithm presented here. Other related work includes partially dynamic algorithms. The best result for updates allowing only edge insertions is O(n) amortized time per inserted edge and O(1) time per query by Italiano (1986) 13] and by La Poutre and van Leeuwen (1987) [16]. This improved upon Ibaraki and Katoh s (1983) 11] algorithm with running time O(n 3 ) for an arbitrary number of insertions. Also there is Yellin s (1993) 18] algorithm with cost O(m Delta) for m insertions, where m is the number of edges in the transitive closure and Delta is the ....

....Also there is Yellin s (1993) 18] algorithm with cost O(m Delta) for m insertions, where m is the number of edges in the transitive closure and Delta is the out degree of the final graph. The best deletions only algorithm for general graphs is by La Poutre and van Leeuwen 3 (1987)[16] and uses O(m) amortized time per edge deletion and O(1) per query. This improved upon the deletions only algorithm of Ibaraki and Katoh (1983) 11] which has an update time of O(n 2 ) For acyclic graphs, Italiano (1988) 14] has a deletions only algorithm with amortized time O(n) per edge ....

H. La Poutr'e and J. van Leeuwen, "Maintenance of transitive closure and transitive reduction of graphs", Proc. Workshop on Graph-Theoretic Concepts in Computer Science, LNCS 314, Springer Verlag, Berlin, 1987, pp. 106--120.

....of O(n 2 2 d ) Here, we incorporate the King Sagert idea of keeping a count with the forest data structure of Henzinger King. The best amortized update times for partially dynamic problems are as follows: for maintaining transitive closure with insertions, O(n) 8; 11] with deletions, O(m) [11]; with deletions in acyclic graphs, O(n) 9] for maintaining shortest paths with insertions and positive integer weights no greater than b, O(nb log n) 1] Klein et.al. give a fully dynamic algorithm for the allpairs shortest path problem on planar graphs. If the sum of the absolute values of ....

H. L. Poutr e and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In Proc. Workshop on Graph-Theoretic Concepts in Computer Science, pages 106--120. LNCS 314, Springer Verlag, 1988.

....scratch after each update. Using a dynamic transitive closure algorithm, it is possible to update the reachability of vertices more efficiently. A dynamic transitive closure algorithm developed by Cicerone [39] which is a generalization of another algorithm proposed by La Poutr e and van Leeuwen [37] is used. Other similar algorithms include Italiano s [28, 29] algorithm and Yellin s algorithm [44] The algorithm proposed by Cicerone uses a counting technique to solve the problem. Information on edges existing in the graph is maintained explicitly in an adjacency matrix. As resource edges are ....

Poutr' e, J. A. L., and van Leeuwen, J. Maintenance of transitive closure and transitive reduction of graphs. In Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science 314 (1988), Springer-Verlag, pp. 106--120.

....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 ....

J. A. La Poutr'e and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In Proc. Workshop on Graph-Theoretic Concepts in Computer Science, pages 106-- 120. Lecture Notes in Computer Science 314, Springer-Verlag, Berlin, 1988.

....iterations, but at the price of using recursive algorithms. Incremental evaluation of arbitrary Datalog [16] An algorithm is given in [16] for transforming an arbitrary Datalog query into an incremental query for arbitrary updates, but which is not in general nonrecursive. Graph algorithms [21, 20, 26, 24]. Graph algorithms for on line evaluation of transitive closure of graphs are given in [21, 20] and a method to optimize transitive queries by using subtrees in graphs constructed in previous evaluations is presented in [24] The main difference is that they use more elaborate data structures and ....

J. La Poutre and J. van Leeuwen. Maintenance of transitive closures and transitive reductions of graphs. Technical Report RUU-CS-87-25, Department of Computer Science, University of Utrecht, The Netherlands, 1987. An extended abstract of this paper appeared in LNCS 314, pp. 106-120.

....for two classes of graphs (including the acyclic graphs) Incremental evaluation of arbitrary Datalog [15] An algorithm is given in [15] for transforming an arbitrary Datalog query into an incremental query for arbitrary updates, but which is not in general nonrecursive. Graph algorithms [20, 19, 25, 23]. Graph algorithms for on line evaluation of transitive closure of graphs are given in [20, 19] and a method to optimize transitive queries by using subtrees in graphs constructed in previous evaluations is presented in [23] The main difference is that they use more elaborate data structures and ....

J. La Poutre and J. van Leeuwen. Maintenance of transitive closures and transitive reductions of graphs. Technical Report RUU-CS-87-25, Department of Computer Science, University of Utrecht, The Netherlands, 1987. An extended abstract of this paper appeared in LNCS 314, pp. 106-120.

....j. We have implemented three algorithms for dynamic transitive closure in this release of the library, namely Italiano s algorithm [15, 16] Yellin s algorithm [25] and the algorithm of Cicerone et.al. 3] which is a generalization of another algorithm proposed by La Poutr e and van Leeuwen [18]) All of these algorithms are partially dynamic: the incremental versions apply to any digraph, while the decremental ones apply only on directed acyclic graphs (DAGs) Query operations come in two forms: a Boolean path query (which simply returns yes or no) and a find path query (which returns ....

J. A. La Poutr'e and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In Workshop on Graph-Theoretic Concepts in Computer Science, Lect. Notes in Comp. Sci. 314, pages 106--120, 1988.

.... no better bound than O( p m ) is known for the corresponding fully dynamic 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 ....

J. A. La Poutr'e, and J. van Leeuwen, "Maintenance of transitive closure and transitive reduction of graphs", Proc. Workshop on Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, vol. 314, Springer-Verlag, Berlin, 1988, 106--120.

.... in [64] An extension of this result to finding regular paths for a regular language is given in [14] A semi dynamic algorithm for acyclic digraphs that support edge deletions in in amortized O(n) time, queries in O(1) time, and use O(n 2 ) space is presented in [65] La Poutr e and van Leeuwen [85] present two semi dynamic data structures supporting transitive closure on general digraphs. The first supports edge insertions in amortized O(n) time. The second supports edge deletions in amortized O(n 2 ) time. Both data structures use O(n 2 ) space and can be used to answer transitive ....

J.A. La Poutr'e and J. van Leeuwen, "Maintenance of Transitive Closures and Transitive Reductions of Graphs," Proc. WG '87, LNCS 314(1988), 106--120.

....e is currently a spanning edge, and if so, which 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 ....

J. A. La Poutr'e and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In Proc. International Workshop on Graph-Theoretic Concepts in Computer Science, (WG 87), Lecture Notes in Computer Science, vol. 314, pages 106--120. Springer-Verlag, Berlin, 1988.

....path from u to v) However, in the dynamic realm the problem seems much harder. The best previously known dynamic results for general digraphs are semi dynamic data structures with O(n 2 ) space, constant query time, and O(n) update time. Data structures that support insertions only are given in [12,50,84]. Data structures that are restricted to acyclic digraphs and support only deletions can be found in [12,51] We can obtain better results if we restrict ourselves to specific classes of graphs. For series parallel digraphs, Italiano, Marchetti Spaccamela, and Nanni [52] gave an O(n) space data ....

J.A. La Poutr'e and J. van Leeuwen, "Maintenance of transitive closures and transitive reductions of graphs," Proc. WG '87, LNCS 314 (1988), 106--120.

....recomputing everything from scratch. Dynamization problems in graph theory are not only of practical interest, but of theoretical 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, ....

J.A. La Poutr' e and J. van Leeuwen. Maintenance of Transitive Closures and Transitive Reductions of Graphs. Lecture Notes in Computer Science 314 (1987) 106-120.

....= 0 otherwise. An edge insertion may result in the addition of more edges, and so on, recursively. The insert operation returns a list of all edges that have been added, represented as node pairs. Initialization and maintenance of ITC can be done in O(n 3 ) time, where n is the number of nodes [15, 21]. 2. PE: a data structure with potential edges. This data structure is computed in a preprocessing phase. It is a matrix of the same form as the adjacency matrix, but with each entry being a list of pairs of nodes. It is computed by, for each pair of L edges x 1 l 1 y 1 and x 2 l 2 y 2 ....

J. A. La Poutr'e and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In Proc. Workshop on Graph-Theoretic Concepts in Computer Science, pages 106--120. Springer-Verlag (LNCS 314), 1988.

....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 O(ff) for undirected graphs Then, for an arbitrary sequence of insertions and connectivity (for undirected graphs) or reachability (for directed graphs) queries between a pair 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 ....

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J.A.La Poutr'e, J.van Leeuwen, Maintenance of transitive closure and transitive reduction of graphs, Proc Work. on Graph Theoretic concepts in Comp. Sci., LNCS 314, Springer Verlag, Berlin, 106120, 1985.

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J. A. La Poutre and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In Proc. Workshop on Graph-Theoretic Concepts in Computer Science, pages 106--120. Lecture Notes in Computer Science 314, Springer-Verlag, Berlin, 1988.

No context found.

J. A. La Poutr'e and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 106--120. Lect. Notes in Comp. Sci., 314, 1988.

No context found.

J. La Poutre and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In Proceedings of the 13th International Workshop on Graph-Theoretic Concepts in Computer Science, Amsterdam, The Netherlands, volume 314, 1987.

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J. A. La Poutr e and J. van Leeuwen. Maintenance of transitive closure and transitive reduction of graphs. In Workshop on Graph-Theoretic Concepts in Computer Science, number 314 in Lecture Notes in Computer Science, pages 106--120. Springer-Verlag, Heidelberg, Germany, 1988.

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