Z. Galil & G. F. Italiano, "Maintaining biconnected components of dynamic planar graphs," Proc. 18th Int. Colloquium on Automata, Languages, and Programming. (1991).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of the rst edge on the path to the root, from which we immediately get a parent pointer. Unfortunately, the above axiomatic interface has been found too limited for many application of dynamic trees, and instead authors have worked directly with the Sleator and Tarjan s underlying representation [30, 5, 21, 24, 23, 14, 4, 1, 16, 9, 8, 7, 22]. In particular, this is the case for the previous solutions to the dynamic center [6] and median problems [3] and we believe part of the reason for their worse bounds and more complex solutions is diculties in working directly with Sleator and Tarjan s underlying representation. Of course, one ....

Z. Galil and G. Italiano. Maintaining biconnected components of dynamic planar graphs. In ICALP'91, 1991.

....T 2 is a spanning forest of G n E(T 1 ) Thus, their algorithm achieves O( p n log(m=n) time per update of general graphs. Hershberger et al. 10] improved Frederickson s algorithm for planar graphs down to O(log 2 n) time per update. 5 3. 2 Results for biconnectivity Galil and Italiano [7] showed an algorithm for planar graphs that takes O(n 2=3 ) per operation. Rauch [12] presented an algorithm that achieves, for general graphs, O(m 2=3 ) time per update and constant time per biconnectivity query. She used Frederickson s partition of graphs [2] to devise algorithm that works ....

Z. Galil and G.F Italiano. Maintaining Biconnected Components of Dynamic Planar Graphs. ICALP, LNCS, S--V, 18:331--350, 1991.

....techniques appears instead to be lacking in the area of dynamic graph algorithms. The goal of this research is exactly to provide such generalized techniques in the realm of dynamic tree and graph problems. Our approach is motivated by the observation that a number of dynamic graph algorithms [10,11,13,14,15,16,19,23,31], developed mostly for connectivity problems, appear to be based on the following fundamental idea: Decompose a graph into subgraphs with limited overlap, and represent such a decomposition by means of a tree so that dynamic operations on the graph are reflected into corresponding dynamic tree ....

Z. Galil and G.F. Italiano, "Maintaining Biconnected Components of Dynamic Planar Graphs," Automata, Languages and Programming (Proc. 18th ICALP), Lecture Notes in Computer Science (1991). 34

....higher by a logarithmic factor for max flow. To obtain a dynamic algorithm, we use an approach 2 used previously for dynamically approximating shortest paths in planar undirected graphs [20] an approach based in turn on that used in dynamic algorithms for a variety of problems in planar graphs [6, 10,11,27]. To compute the shortest path from a given source to a given sink, one operates on the union of complete graphs with two of the complete graphs replaced by the regions they represent, one for the source and one for the sink. To update the cost of an edge, for example, one recomputes the complete ....

....n) with O(n 4=3 log 3 n) work. Dynamic maintenance of shortest paths These partitioning techniques can also be used to derive efficient dynamic algorithms for shortest paths and other related problems. The ideas are similar to a number of other fully dynamic algorithms for graph problems [6, 10,11,20,27]. The basic idea is to divide G into suitable sized pieces and precompute all boundary pair shortest paths in each piece. These precomputed answers are used to answer any given query quickly. At the same time when edges are added or removed we need only recompute the shortest paths in a few ....

Z. Galil and G. F. Italiano, "Maintaining biconnected components of dynamic planar graphs," Proc. 18th Int. Colloquium on Automata, Languages, and Programming. (1991), 339--350.

....higher by a logarithmic factor for max flow. To obtain a dynamic algorithm, we use an approach used previously for dynamically approximating shortest paths in planar undirected graphs [KlS] an approach based in turn on that used in dynamic algorithms for a variety of problems in planar graphs [Frea, GaI, GIS, Sub]. To compute the shortest path from a given source to a given sink, one operates on the union of complete graphs with two of the complete graphs replaced by the regions they represent, one for the source and one for the sink. To update the cost of an edge, for example, one recomputes the complete ....

....paths in a planar directed graph that can have both negative and nonnegative length edges. As with the sequential algorithm our dynamic algorithm also uses the division based of Frederickson. We use the cluster partitioning approach previously used by Frederickson, Galil and Italiano, and others [Frea, GaI, GIS, Sub]. The basic idea is to divide G into suitable sized pieces and precompute all boundary pair shortest paths in each piece. These precomputed answers are used to answer any given query quickly. When edges are aded or removed we need only recompute the shortest paths in a few pieces. Throughout ....

Z. Galil & G. F. Italiano, "Maintaining biconnected components of dynamic planar graphs," Proc. 18th Int. Colloquium on Automata, Languages, and Programming. (1991).

....minimum spanning trees in general graphs. In the context of planar graphs [3] he used a separator based cluster decomposition (obtained by repeated division of the graph using separators) to derive improved sequential algorithms for single source shortest paths. Galil, Italiano, and Sarnak [10, 11] used the separator algorithm due to Lipton and Tarjan [12] to repeatedly divide the underlying planar graph into clusters. Galil and Italiano [10] used such a decomposition to derive a fully dynamic data structure for maintaining two and three vertex connectivity information in planar graphs. ....

....division of the graph using separators) to derive improved sequential algorithms for single source shortest paths. Galil, Italiano, and Sarnak [10, 11] used the separator algorithm due to Lipton and Tarjan [12] to repeatedly divide the underlying planar graph into clusters. Galil and Italiano [10] used such a decomposition to derive a fully dynamic data structure for maintaining two and three vertex connectivity information in planar graphs. Galil, Italiano, and Sarnak [11] used cluster decompositions to develop a fully dynamic planarity testing algorithm. We borrow this technique and ....

Z. Galil and G. F. Italiano, "Maintaining Biconnected Components of Dynamic Planar Graphs," Proceedings of the 18th International Colloqium on Automata, Languages, and Programming (1991), 339-350.

....the problem of maintaining the transitive closure in a directed graph under both edge insertions and deletions is studied and analyzed in the average case. Connectivity. Several dynamic edge connectivity problems have been efficiently solved using novel data structures and algorithmic techniques [272, 248, 249, 245, 244, 292]. Another interesting result in this area is the discovery of a reduction from vertex connectivity to edge connectivity [247] Also, for what concerns vertex connectivity, the on line maintenance of a triconnected graph is investigated in [297] The problem consists of performing the following ....

Z. Galil and G. F. Italiano. Maintaining biconnected components of dynamic planar graphs. In Proc. 18th Internat. Colloq. Automata Lang. Program., volume 510 of Lecture Notes in Computer Science, pages 339--350. Springer-Verlag, 1991.

....presented in this paper, since it only speeds up very specific 2 vertex connectivity algorithms. For a plane graph we achieve O( p n log n) time per update and O(log 2 n) time per query operation. The best previously known solution requires time O(n 2=3 ) per update or query operation [10]. Our algorithm uses linear space and preprocessing time. Eppstein et al. 14] improve the running time for this problem to O( p n) The new techniques for fully dynamic graph algorithms introduced in this paper involve precomputation and lazy updating. We split the graph into connected ....

Z. Galil, G. F. Italiano, "Maintaining Biconnected Components of Dynamic Planar Graphs" Proc. 18th ICALP, Lecture Notes in Computer Science, Vol., Springer-Verlag, Berlin, 1991, 339--350.

....complexity, design of algorithms, graph theory, dynamic algorithms, line graphs. 1 Introduction 1.1 Motivation In recent years, the dynamization of graph algorithms has become a current research field. In particular if we think of testing predicates on graphs, see for example Galil and Italiano[1] or Rauch[9] Accordingly, we consider line graphs which are a classical topic in the theory of special graphs 1 . Line graphs have some interesting algorithmic aspects. In order to illustrate this, let L be a line graph and G its root graph. Then a matching in G is an independent set in L and ....

Z. Galil and G.F. Italiano. Maintaining biconnected components of dynamic planar graphs. 18th Int. Coll. Automata, Languages and Programming. Springer-Verlag LNCS 510 (1991) 339--350.

....Least Common Ancestor return the least common ancestor of two tree nodes. ffl Find Minimum Find the minimum weight node on a path. Dynamic trees were introduced as internal data structures in sequential maximum flow algorithms [24,55,100] Since then, a large number of dynamic algorithms [27,28, 48,49,50,51,69,83,122] have used dynamic trees as part of their data structures. Initial data structures [24,55] based on balanced binary trees (e.g. AVL trees [1] or Red Black trees [57] take O(log 2 n) time per operation. Sleator and Tarjan improve this to O(log n) time per operation by basing their data ....

....The availability of such general techniques appears instead to be lacking in the area of dynamic graph algorithms. This chapter provides such generalized techniques in the realm of dynamic graph problems. Our approach is motivated by the observation that a number of dynamic graph algorithms [27,28,48,49,50,51,69,83,122], developed mostly for connectivity problems, appear to be based on the following fundamental idea: Decompose a graph into subgraphs with limited overlap, and represent such a decomposition by means of a tree so that dynamic operations on the graph are reflected into corresponding dynamic tree ....

Z. Galil and G.F. Italiano, "Maintaining Biconnected Components of Dynamic Planar Graphs," Automata, Languages and Programming (Proc. 18th ICALP), Lecture Notes in Computer Science (1991).

....a face restricted k cluster partition can be obtained in O(n log n) time. Frederickson [28,30] used the idea of partitioning a graph into clusters to develop dynamic data structures for maintaining minimum spanning trees, connected components, and two edge connected components. Galil and Italiano [36] used a k cluster partition to develop a fully dynamic data structure for two and three connectivity in planar graphs. Later Galil, Italiano, and Sarnak [37] used such a decomposition for designing a dynamic planarity testing algorithm. The basic idea behind these dynamic data structures is the ....

....both additions and deletions of edges while it is said to be semi dynamic if it supports only one of them. Unfortunately designing fully dynamic algorithms seems to be considerably harder than designing their sequential counterparts, and very few graph problems have fully dynamic solutions. See [28,30,36,37] 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, ....

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Z. Galil and G. F. Italiano, "Maintaining biconnected components of dynamic planar graphs," Proc. 18th Int. Colloquium on Automata, Languages, and Programming. (1991), 339--350.

....2=3 ) bound per operation also for the fully dynamic maintenance of 3 edgeconnected components [7] Efficient fully dynamic algorithms for 2 vertex connectivity seem harder to obtain. Using different techniques, we have been able to achieve an O(n 2=3 ) bound per operation for planar graphs [8]. Acknowledgments We would like to thank Dany Breslauer, Greg Frederickson, Kurt Mehlhorn, Neil Sarnak, and Moti Yung for useful comments. ....

Z. Galil, and G. F. Italiano, "Maintaining biconnected components of dynamic planar graphs", Proc. 18th ICALP, 1991.

....CCR 9014605. z On leave from Universit a di Roma, 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 ....

....in a planar graph are either biconnected or triconnected, in O(n 2=3 ) time per operation. The times for edge deletions and queries are worst case, while the time for edge insertion is amortized. The same bound for biconnectivity was already known (and with worst case time also for insertions) [21]; however, the new algorithm is simpler. On the other hand, this is the first fully dynamic algorithm known for maintaining information about triconnectivity in a planar graph. A fully dynamic setting is different and often much more complex than a partially dynamic setting. In the incremental ....

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Z. Galil and G. F. Italiano. Maintaining biconnected components of dynamic planar graphs. In Proc. 18th Int. Colloquium on Automata, Languages and Programming, pages 339--350. Lecture Notes in Computer Science 510, Springer-Verlag, Berlin, 1991. 53

....intermixed sequence of edge deletions, edge insertions that keep the graph planar, and connectivity queries in O(n 1 2 ) amortized time per operation. All our algorithms improve previous bounds: for 2 and 3 vertex and 3 edge connectivity, the best previous time bound was O(n 2 3 ) amortized [8, 19, 21], while for 4 edge connectivity nothing better than testing the graph from scratch after each update was known. These bounds apply to problems in which insertions need not respect a fixed embedding of the graph; a number of other papers have worked on dynamic graph problems such as minimum ....

....each update was known. These bounds apply to problems in which insertions need not respect a fixed embedding of the graph; a number of other papers have worked on dynamic graph problems such as minimum spanning forests, connectivity, and planarity testing for graphs with a fixed planar embedding [12, 14, 15, 18, 19, 22, 21, 24, 32, 33]. Finally, our methods apply to static as well as dynamic graph problems. A general certificate construction method from our companion paper, together with the certificates defined here, gives a unified method of testing 3 and 4 edge , and 2 and SPARSIFICATION II: EDGE AND VERTEX CONNECTIVITY ....

Z. Galil and G. F. Italiano, Maintaining biconnected components of dynamic planar graphs, in Proc. 18th Int. Colloq. Automata, Languages, and Programming, Lecture Notes in Computer Science 510, Springer-Verlag, New York, 1991, pp. 339--350.

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Z. Galil & G. F. Italiano, "Maintaining biconnected components of dynamic planar graphs," Proc. 18th Int. Colloquium on Automata, Languages, and Programming. (1991).

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