Reif J.H. A topological approach to dynamic graph connectivity Information Processing Letters, 25(1):65-70, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....two vertices x and y. As mentioned in [18] in such a fully dynamic setting there has been no known solution better than a repeated application of an off line algorithm, which yields a single operation worst case time complexity of O(m) where m is the current number of edges in the graph. Reif [14] introduced the notion of complete dynamic problems, as a class of problems with the property that if one problem can be solved in o(m) time per operation, then all the problems in the class can be solved in o(m) time per operation (examples of complete dynamic problems include the acceptance of a ....

J. H. Reif, "A topological approach to dynamic graph connectivity", Inform. Process. Lett. 25 (1987), 65--70.

....y Work partially supported by NSF Grant 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 ....

J. H. Reif. A topological approach to dynamic graph connectivity. Inform. Process. Lett., 25:65--70, 1987.

....are the stable [4] and the well founded [14] For propositional logic programs, computing inferences under the stable semantics is known to be co NP complete [7] computing inferences under the well founded is known to be quadratic time (folklore) and no faster algorithm is known. Work in [8] (on questions of updating accessibility relations in graphs) suggests that it may be quite difficult, using standard techniques, to break the quadratic time bound on the well founded semantics. We are concerned here with speeding up the computation of the well founded semantics; in particular, we ....

....some property P of interest which, for all interesting EDB s, the properties of Proposition 7 hold, may be very difficult for some properties P . We would like a far more general speedup technique. Van Gelder s algorithm repeatedly deletes edges from H and rechecks for accessibility from s. Reif [8] has studied algorithms for updating vertex accessibility information in directed graphs during dynamic edge deletion. His work suggests it may in general be very difficult to speed up in the worst case past the size of the graph times the number of cycles of deletions. Thus his result seems to ....

J. Reif. A topological approach to dynamic graph connectivity. Information Processing Letters 25(1), pages 65-70.

....preprocessing. Though their results are interesting, they are somewhat weak in that the issues of data structures and preprocessing are overlooked. They conjecture that the incremental versions of all P complete problems are P complete. Interesting notions of completeness are sketched by Reif [14]. He shows that some problems are unlikely to have efficient incremental solutions, but he does not develop a comprehensive theory or consider the necessary details of preprocessing. Some interesting techniques are explored in [2] 4] and [13] to derive lower bounds for incremental algorithms. ....

J.H. Reif, "A Topological Approach to Dynamic Graph Connectivity," Information Processing Letters 25 (1987), 65--70.

....that supports operations on past versions. There are many problems for which no efficient dynamic data structures are known. It has been observed that there are strong similarities among the classes of problems that are difficult to parallelize and those that are difficult to dynamize (see, e.g. [31]) Further investigations are needed to study the relationship between parallel and incremental complexity [25] Implicit vs. Explicit Two fundamental data organization mechanisms are used in data structures. In an explicit data structure, pointers (i.e. memory addresses) are used to link the ....

J. H. Reif. A topological approach to dynamic graph connectivity. Inform. Process. Lett., 25:65--70, 1987.

....of preprocessing. Though their results are interesting, they are somewhat weak in that the issues of data structures and preprocessing are overlooked. They conjecture that the incremental versions of all P complete problems are P complete. Interesting notions of completeness are sketched by Reif [21]. He shows that some problems are unlikely to have efficient incremental solutions, but he does not develop a comprehensive theory or consider the necessary details of preprocessing. Some interesting techniques are explored in [2] 6] and [20] to derive lower bounds for incremental algorithms. ....

J.H. Reif, "A Topological Approach to Dynamic Graph Connectivity," Information Processing Letters 25 (1987), 65--70.

.... operation [11] using the dynamic trees of Sleator and Tarjan [22, 23] However, in both cases 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 ....

J. H. Reif, "A topological approach to dynamic graph connectivity", Inform. Process. Lett. 25 (1987), 65--70.

....requiring 43 the second to process the entire graph even though kffik is still bounded. This method is used to show that the Single Source Reachability problem is non ffi incremental for locally persistent algorithms. The second method is a problem reduction approach similar to that in [Rei87] This method is used to extend the reachability result to Single Source (or Single Sink) Closed Semiring Path problems and Meet Semilattice Data Flow Analysis problems. 5.4 New ffi Analysis Lower Bounds for Undirected Graphs In this section we apply the techniques discussed above to several ....

....from the two methods may give a better understanding of the difficulty of finding incremental algorithms for certain problems. Thus we see the methods as complementary. 104 7. 6 Another Approach Complete Dynamic Problems Reif has proposed another approach to analyzing incremental algorithms [Rei87] By using Turing Machine reductions, he is able to categorize a group of problems as equally difficult to solve incrementally. While this does not yield a lower bound result, it does give some insight into the difficulty of developing incremental algorithms for a class of problems. Reif begins ....

J. H. Reif. A topological approach to dynamic graph connectivity. Information Processing Letters, 25:65--70, April 1987.

....of [48] takes O( p m) time per operation. This is the best result handling edge deletions. Other semi dynamic techniques supporting only edge deletions are shown in [41] where a data structure is presented supporting constant time queries and O(q mn) time to perform q edge deletions; and in [89] where q edge deletions are performed in O(mg m log m) time for a graph of genus g. These results have been extended to the maintenance of 2, 3, and 4 connectivity. Each of the following is implemented with a semi dynamic data structure, supporting only insertions. Suppose we perform a sequence ....

J.H. Reif, "A Topological Approach to Dynamic Graph Connectivity," Information Processing Letters 25 (1987), 65--70.

....weight of each tree, whether an edge 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 ....

J. H. Reif. A topological approach to dynamic graph connectivity. Inf. Process. Lett., 25:65--70, 1987.

No context found.

J. Reif, "A topological approach to dynamic graph connectivity", Inform. Process. Lett. , 25, pp. 65--70, 1987.

No context found.

J. Reif, "A topological approach to dynamic graph connectivity", Inform. Process. Lett. , 25, pp. 65--70, 1987.

No context found.

Reif J.H. A topological approach to dynamic graph connectivity Information Processing Letters, 25(1):65-70, 1987.

No context found.

Reif, J.H., "A topological approach to dynamic graph connectivity," Information Processing Letters 25(1) pp. 65-70 (1987).

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