T. Ibaraki and N. Katoh. On-line computation of transitive closure for graphs. Information Processing Letters, 16:95--97, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....closure and G = V, E its transitive reduction (cf. 1] Let E and E be represented by incidence matrices. Suppose edges are inserted in and deleted from G one at a time. We consider the problem of efficiently updating G and G each time an edge is inserted or deleted. Ibaraki and Katoh [2] presented two algorithms that update G when edge insertions and deletions are considered separately. Their insertion algorithm takes O(IVI : time for q consecutive insertions and their deletion algorithm takes O(IWl; Iroldl IW] time for q consecutive deletions (where the subscript old ....

....[3] in combination with the use of auxiliary information about the number of ways one can arrive at nodes coming from other nodes. The information is updated by both the insertion and deletion algorithm. The algorithms especially yield better time complexities (compared with those presented in [2]) for graphs with Eol d ( E [ e.g. planar graphs) and for graphs with relatively small components (i.e. graphs with mcyc eold = o( E[ For example, for planar graphs our algorithms both take O( V[ 2) time for any q consecutive applications, whereas the algorithm presented in [2] take O(IVI 3) ....

[Article contains additional citation context not shown here]

T. Ibaraki and N. Katoh, On-line computation of transitive closures of graphs, Infor- mation Processing Letters 16 (1983) 9597.

....18] this approach yields O(1) time per query and O(n ) time per update in the worst case, where is the best known exponent for matrix multiplication (currently 2:38 [2] Previous Work. 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 ....

....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 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 ....

T. Ibaraki and N. Katoh. On-line computation of transitive closure for graphs. Information Processing Letters, 16:95--97, 1983. 51

....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 ....

T. Ibaraki and N. Katoh. On-line computation of transitive closure for graphs. Inform. Process. Lett., 16:95--97, 1983.

....incremental algorithm, when it allows only additions of edges. In the following, the eciency of an incremental algorithm is measured by the global number of operations required for maintaining the transitive closure in a graph G that has n vertices and m edges in the nal state. Ibaraki 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] ....

Ibaraki, T. and Katoh, N. 1983. On-line computation of transitive closure for graphs. Inform. Proc. Lett. 16, 95-97.

....received. Similarly, each partial solution (as well as the final one) may need to be returned as soon as it is available [32, 40, 56] It is helpful to note here that, when no deadlines are imposed, computations for which inputs arrive while the algorithm is in progress are referred to as on line [27, 33, 35, 36], incremental [21, 22, 49, 59] dynamic [11, 12, 67] and updating [20, 23, 28, 38, 53, 54, 62, 66] It is also important to note that our definition, while striving to be as general as possible, is particularly suitable for our purposes in this paper. Many other definitions exist; see, for ....

T. Ibaraki and N. Katoh, On-line computation of transitive closure graphs, Information Processing Letters , 16, 1983, 95--97.

....14] this approach yields O(1) time per query and O(n ) time per update in the worst case, where is the best known exponent for matrix multiplication (currently 2:736 [2] Previous Work. 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 ....

....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 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 ....

T. Ibaraki and N. Katoh. On-line computation of transitive closure for graphs. Information Processing Letters, 16:95--97, 1983.

....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 and Delta is the out degree of the resulting ....

....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 For acyclic graphs, Italiano (1988) 13] has a deletions only algorithm which requires O(n) amortized time per edge deletion, and O(1) per query. There is also ....

T. Ibaraki and N. Katoh, "On-line computation of transitive closure of graphs, Information Processing Letters, 1983, pp. 95-97.

....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 out degree of the final graph. The best deletions only ....

....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 deletion and O(1) per query. There is also Yellin s (1993) 18] deletionsonly algorithm with cost O(m Delta) for m deletions, where m is the number ....

T. Ibaraki and N. Katoh, "On-line computation of transitive closure of graphs, Information Processing Letters, 1983, pp. 95-97.

....received. Similarly, each partial solution (as well as the final one) may need to be returned as soon as it is available [31, 39, 55] It is helpful to note here that, when no deadlines are imposed, computations for which inputs arrive while the algorithm is in progress are referred to as on line [26, 32, 34, 35], incremental [20, 21, 48, 58] dynamic [10, 11, 66] and updating [19, 22, 27, 37, 52, 53, 61, 65] It is also important to note that our definition, while striving to be as general as possible, is particularly suitable for our purposes in this paper. Many other definitions exist; see, for ....

T. Ibaraki and N. Katoh, On-line computation of transitive closure graphs, Information Processing Letters, 16, 1983, 95--97.

.... Similarly, each partial solution (as well as the final one) may need to be returned as soon as it is available [21, 28, 37] It is helpful to note here that, when no deadlines are imposed, computations for which inputs arrive while the algorithm is in progress are referred to as on line [18, 22, 23, 24], incremental [14, 15, 32, 39] dynamic [7, 8, 44] and updating [13, 16, 19, 26, 34, 35, 41, 43] 2.2 Real time optimization The first example of a computation for which a parallel solution is consistently better than a sequential one was provided by real time optimization. The realtime ....

T. Ibaraki and N. Katoh, On-line computation of transitive closure graphs, Information Processing Letters , 16, 1983, 95--97.

....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 ....

T. Ibaraki and N. Katoh. On-line computation of transitive closure for graphs. Inform. Proc. Lett., 16:95--97, 1983.

....materialized views upon updates. It is also closely related to the problem of partially evaluating definite logic programs [27] Finally, when restricted to standard transitive closure programs, our task can be viewed as solving the incremental transitive closure computation problem for graphs [10, 14, 20, 21]. More detailed comparison will be given in Section 6. In general, all these optimization approaches store extra information to reduce the time required for subsequent computations. In our case, we store the answer to the query in one database state (and possibly additional derived facts) to ....

....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 ....

[Article contains additional citation context not shown here]

T. Ibaraki and N. Katoh. On-line computation of transitive closure of graphs. Information Processing Letters, 16:95-97, 1983.

....materialized views upon updates. It is also closely related to the problem of partially evaluating definite logic programs [26] Finally, when restricted to standard transitive closure programs, our task can be viewed as solving the incremental transitive closure computation problem for graphs [10, 14, 19, 20]. More detailed comparison will be given in Section 6. In general, all these optimization approaches store extra information to reduce the time required for subsequent computations. In our case, we store the answer to the query in one database state (and possibly additional derived facts) to ....

....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 ....

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T. Ibaraki and N. Katoh. On-line computation of transitive closure of graphs. Information Processing Letters, 16:95-97, 1983.

....as well. Hence, it is the insertion or removal of edges that causes changes to the transitive closure. A number of algorithms have been proposed for solving the problem of on line maintenance of transitive closure for graphs. The method proposed in [Ita88] an improvement on the one proposed in [IK83] is applicable to acyclic graphs only. It is readily applicable to the condensed form of the dependence graph provided an appropriate indexing of the graph vertices is performed. It is not applicable however for general, possibly cyclic, graphs. In this section we propose a method for computing ....

T. Ibaraki and N. Katoh. On-Line Computation of Transitive Closures of Graphs. Information Processing Letters, 16(3):95--97, 1983.

....of computing a solution. The newly arrived data must be incorporated in the solution at hand. The final solution is to be returned by a certain deadline. Real time computations form a subclass of a larger class of problems known variably as on line, incremental , dynamic, and updating computations [7, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 30, 31, 32, 36, 38, 40, 41]. What distinguishes a real time problem from problems in the larger class is the presence of deadlines by which the input is to be processed, by which the output is to be produced, and so on. 3 Modern Cryptography The purpose of contemporary cryptography is the protection of digital ....

T. Ibaraki and N. Katoh, On-line computation of transitive closure graphs, Information Processing Letters , 16, 1983, 95--97.

.... 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 ....

T. Ibaraki, and N. Katoh, "On-line computation of transitive closure for graphs", Inform. Process. Lett. 16 (1983), 95--97.

.... 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 bases [ABJ89] ffl syntax directed editors and grammars [Rep82, RTD83, Rep88, ACR 87] ffl data flow analysis [Ryd83, RP88, RC86, CR88, Mar89, ....

....so the DAG algorithm requires O(n 3 ) as a practical matter it is much faster than the standard multiplicative algorithm for 51 many graphs. 6. 2 Incremental Upper Bounds There has been quite a bit of work in developing incremental algorithms for this problem, beginning with Ibaraki and Katoh [IK83] 1 Ibaraki and Katoh give algorithms for updating the transitive closure for a graph with edge additions and edge deletions. After each change in the graph, the transitive closure information can be accessed in constant time. That is, the cost of a query pathexists(a,b) which returns true ....

T. Ibaraki and N. Katoh. On-line computation of transitive closure of graphs. Information Processing Letters, 16:95--97, 1983.

....on the weight biased binary trees [7] 2.2 Transitive Closure Given a directed graph G and two vertices u and v of G, a transitive closure query determines if there is a directed path in G from u to v. This is also referred to as a reachability query. A basic data structure is presented in [63]. Each transitive closure query takes O(1) time, while a sequence of additions takes O(n 3 ) time, which amortizes to O(n) time per edge addition in dense digraphs. A sequence of deletions takes O(n 2 (n m) time, which amortizes to O(n 2 ) time per edge deletion in dense digraphs. The data ....

T. Ibaraki and N. Katoh, "On-Line Computation of Transitive Closure of Graphs," Information Processing Letters 16 (1983), 95--97.

....is computed. Subsequently, more data arrive at regular intervals. Each new datum received must be incorporated into the solution. Real time computation is sometimes known as on line computation, by contrast with off line computation in which all the required data are available at the outset [14, 16, 17, 21, 28]. The adjectives updating , incremental , and dynamic are also often used to refer to algorithms that receive and process new data [4, 9, 10, 11] Examples of real time computations include sorting a sequence of numbers, computing the convex hull of a set of points in the plane, and finding the ....

T. Ibaraki and N. Katoh, On-line computation of transitive closure graphs, Information Processing Letters , 16, 1983, 95--97.

....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 ....

T. Ibaraki and N. Katoh. On-line computation of transitive closure for graphs. Inf. Process. Lett., 16:95--97, 1983.

....of updates of rules and constraints can be found elsewhere (Plexousakis, 1994a) In addition to updating the dependence graph, we also need to incrementally compute its transitive closure. Incremental transitive closure algorithms available in literature can deal only with directed acyclic graphs (Ibaraki and Katoh, 1983; Italiano, 1988) In our research we have developed an algorithm that incrementally computes transitive closure for general graphs (Plexousakis, 1994a) Our preliminary experiments have shown that this algorithm can efficiently update the transitive closure of a dependence graph. In the ....

Ibaraki, T. and Katoh, N. (1983). On-Line Computation of Transitive Closures of Graphs. Information Processing Letters, 16(3):95--97.

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T. Ibaraki and N. Katoh. On-line computation of transitive closure for graphs. Information Processing Letters, 16:95--97, 1983.

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T. Ibaraki and N. Katoh. On-line computation of transitive closures of graphs. Information Processing Letters, 16:95--97, 1983.

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T. Ibaraki and N. Katoh. On-line computation of transitive closures of graphs. Information Processing Letters, 16:95--97, 1983.

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T. Ibaraki and N. Katoh, On-line computation of transitive closure graphs, Information Processing Letters , 16, 1983, 95--97.

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