F. P. Preparata and R. Tamassia, Fully Dynamic Techniques for Point Location and Transitive Closure in Planar Structures, Proceedings of the 29th IEEE Symposium on Foundations of Computer Science, 1988, pp. 558-567.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....v W is the only source and v E is the only sink of G 2 . 2 Both G 1 and G 2 are the so called s t planar graphs. An s t planar graph is a directed acyclic planar graph with exactly one source s and exactly one sink t on its exterior face. The properties of these graphs have been studied in [7, 10, 13]. Using these properties, the structure of G 1 can be summarized as follows: a) For each vertex v other than v S and v N , the edges entering v appear consecutively around v in G 1 . The edges leaving v appear consecutively around v in G 1 . Let e 1 and e 2 be the left most and the right most ....

F. P. Preparata and R. Tamassia, Fully Dynamic Techniques for Point Location and Transitive Closure in Planar Structures, Proceedings of the 29th IEEE Symposium on Foundations of Computer Science, 1988, pp. 558-567.

....the order defined by lef t . We call lef t the leftmost canonical ordering of G derived from . Note that lef t is different from the leftmost canonical ordering defined in [12, 13] Our algorithm will use lef t , which can be computed in linear time from by using the method in [15]. According to lef t , if there are more than one candidate vertex for G k , the leftmost one is added first. Fig 3 shows the leftmost canonical ordering of the graph shown in Fig 2. 1 2 3 4 5 6 7 8 9 10 12 13 14 18 11 17 16 19 unstable vertices: 3,4,5,6,7,8,12,15 stable vertices: ....

F. P. Preparata and R. Tamassia, Fully dynamic techniques for point location and transitive closure in planar structures, in Proc. 29th IEEE FOCS, 1988, pp. 558-567.

.... 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 ....

.... 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 graph differs substantially from ....

F. P. Preparata, and R. Tamassia, "Fully dynamic techniques for point location and transitive closure in planar structures", Proc. 29th Annual Symp. on Foundations of Computer Science, 1988, 558--567.

.... 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, Bur90, PS89, Zad84, Gho83, KRvM88] ffl code generation and optimization [Pol86] 2 ffl ....

F. P. Preparata and R. Tamassia. Fully dynamic techniques for point location and transitive closure in planar structures. In Proceedings of the Twenty-Ninth Annual IEEE Symposium on the Foundations of Computer Science, pages 558--567. Institute of Electrical and Electronics Engineers -- Computer Society, 1988.

....O(n 2 ) time. Both data structures use O(n 2 ) space and can be used to answer transitive closure queries in O(1) time. Better results can be achieved for specific classes of digraphs. In particular, there are fully dynamic data structures for dynamic reachability in planar st graphs [87, 108], spherical st graphs [112] and series parallel digraphs [66] Each of these require O(log m) time per query and update using O(m) space. 2.3 Connectivity Connectivity is a fundamental property of graphs. Dynamic algorithms have been presented for both vertex and edge connectivity. Given a ....

F.P. Preparata and R. Tamassia, "Fully Dynamic Techniques for Point Location and Transitive Closure in Planar Structures," Proc. 29th IEEE Symp. on Foundations of Computer Science (1988), 558--567.

....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 ....

F. P. Preparata and R. Tamassia. Fully dynamic techniques for point location and transitive closure in planar structures. In Proc. 29th IEEE Symp. on Foundations of Computer Science, pages 558--567, 1988.

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