J. A. La Poutre. Dynamic graph algorithms and data structures. Phd thesis, Department of Computer Science, Utrecht University, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....randomized time bounds for general graphs are polylogarithmic per update (amortized) and per query (worst case) HK95, Hen97] in graphs with polylogarithmic maximum degree. For the incremental problem the running time for general and planar graphs is O(a(q;n) amortized for each of q operations [Pou91] WT92] where a( Delta; Delta) is the inverse of Ackermann s function. For the decremental problem in planar graphs Giammarresi and Italiano [GI96] For the fully dynamic case it is not enough to give the assertion that the graph is planar, i.e. that there exists a combinatorial embedding. ....

J. A. La Poutre. Dynamic graph algorithms and data structures. Phd thesis, Department of Computer Science, Utrecht University, 1991.

....) a spanning tree T such that D(T ) is the smallest possible value over all spanning trees of G or at most one larger than this optimal value. That is D(T ) D(T ) Gamma 1, for all spanning trees T in G. a( Delta; Delta) denotes the inverse of Ackermann s function (cf. Tar83] or [La 91] 1.1 Previous Work It is well known that for any fixed D T 2 the problem of testing whether a graph has a spanning tree of maximum degree bounded by D T is N P complete [GJ79] On the other hand, Furer and Raghavachari [FR94] showed that one can find in polynomial time a spanning tree whose ....

.... tree T start has maximal degree D(T start ) we conclude that there are O(nlog(D(T start ) D(T end ) phases, In each phase we try to find improvements which propagate to vertices of S D(T) Each phase can be implemented in nearly linear time using a fast union find data structure [Tar83] La 91] for maintaining the connected components. Therefore the entire algorithm runs in time O(n m a(m;n) log(D(T start ) D(T end ) t build ) where m is the number of edges, t build is the running time to build the initial spanning tree T start , T end is the FR spanning tree, and a the inverse ....

J. A. La Poutre. Dynamic Graph Algorithms and Data Structures. Phdtheses, Utrecht University, Department of Computer Science, 1991.

....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 ....

J. A. La Poutre. Dynamic graph algorithms and data structures. PhD thesis, Dep. Comp. Sci., Utrecht Uni., 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 ....

J. A. La Poutr'e. Dynamic graph algorithms and data structures. PhD thesis, Department of Computer Science, Utrecht University, September 1991.

.... not based on cactus trees Finally, can we exploit known partially dynamic algorithms to speed up the insertion times in our bounds This would be particularly appealing, since there are very fast, partially dynamic algorithms already available in the literature for edge and vertex connectivity [6, 20, 27, 29, 30, 36]. Acknowledgments. We are deeply indebted to the anonymous referees, whose thorough reading of the paper lead to a substantial improvement in its presentation. ....

J. A. La Poutr e, Dynamic Graph Algorithms and Data Structures, Ph.D. Thesis, Utrecht University, the Netherlands, September 1991.

....is the structure of the graph with respect to its vertex connectivity. Knowing the structure of a graph can lead to the solution of important graph theoretical problems such as the augmentation problem that is studied here (see the survey chapter in [Hsu93] and dynamic graph algorithms [LP91]. The structure of an undirected graph that is not biconnected (i.e. 2 vertex connected) is well known [Har69] and is represented as a 2 block graph. The structure of a biconnected graph that is not triconnected (i.e. 3 vertex connected) is also well known and is represented as a 3 block graph ....

J. A. La Poutr'e. Dynamic Graph Algorithms and Data Structures. PhD thesis, Utrecht University, 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 ....

J.A. La Poutre, "Dynamic Graph Algorithms and Data Structures," Dept. of Computer Science, University of Utrechet, Utrechet, Ph.D. Thesis, 1991.

....this as one I O. Function symbols and conventions: The symbol IL will denote the iterated log function. IL (n) is the number of times we must apply the log function to n before the result becomes 2. It is also the 3rd row inverse of the Ackerman function. IL (n) is the inverse of A(3; n) [26] where A denotes the ackerman function. Unless otherwise stated all logarithms will be with respect to base 2. Also, throughout this article we will use the terms in core and main memory to mean the same thing. The efficiency of our algorithms will be measured in terms of the number of I O ....

J. A. La Poutr'e, "Dynamic Graph Algorithms and Data Structures," Department of Computer Science, University of Utrecht, Ph. D. Thesis.

....research in recent years. Currently, several results exist for incremental and fully dynamic graph problems, like for maintaining spanning trees, the 2 edge or the 2 vertex connected components of a graph, or the planarity of a graph under the insertions and or deletions of edges and vertices [3, 4, 5, 7, 8, 9, 10, 11, 12, 14]. In [4, 5, 12] algorithms for maintaining minimum spanning trees and the connectivity, 2 edge connectivity and the 2 vertex connectivity relations in fullydynamic graphs were presented that run in O( # m) or O( # m log n) time per operation (amortized time for 2 vertex connectivity) In this ....

....as follows. First, a breadth first search forest B of G T b is made and the edges are enumerated in some way. A (green) edge of B is converted into a red edge only if it changes the number of components of T b # G r . This can be done by maintaining the biconnected components dynamically [8, 14], thus yielding an add on sequence indeed. As was shown in [1] the graph T b # B is a certificate for biconnectivity, hence, so is T b # G r . We do not really need the breadth first forest, but can do this with all the existing edges once they are ordered as well. The edges in G r are ....

J.A. La Poutre, "Dynamic Graph Algorithms and Data Structures" Ph.D. Thesis, Utrecht University, 1991.

....arbitrary sequences of Union and Find operations in the best time complexity possible. Here a Union works on two disjoint subsets fusing them into one; a Find identifies the subset a certain element belongs to. For an introduction and overview to Union Find see e.g. Meh84] for recent results see [LP91]. In the following we will only assume a straight forward implementation of Union Find that could easily be implemented on an arbitrary pointer machine. In fact there exist versions of Union Find that are much more sophisticated, see e.g. Tar75] or [LP89b] that perform in time O(ff(n; m)m) and ....

....on pointer machines for the general case where no restrictions to the Union s or Find s apply. There is also a version that performs in linear time on a special case, first shown to work well when implemented on a random access machine, see [GT84] and then generalized to pointer machines in [LP91]. For these algorithms it is necessary to determine a tree of the elements in advance such that all subsets form connected subtrees of that tree at any time of the algorithm. For the application considered here this is not adequate because e.g. the number of pairs of elements that may form two ....

J. A. La Poutr'e, Dynamic graph algorithms and data structures, Ph.D. thesis, Rijksuniversiteit te Utrecht, 1991.

....The tree T c is the condensation of TH IN . Hence, each vertex in T c represents the set of vertices of a 2 edge connected component of TH IN . We shall describe a method for maintaining T c that runs in time O(p q) Our technique is similar to methods discovered independently by [23] and [12]. One result in [23] shows that the 2 edge connected components of an initially connected graph on n vertices can be maintained under m edge insertions in time O(mff(m; n) n) This bound is improved to O(m n) by [12] Although the bound of [12] matches our bound, we choose to present our ....

.... q) Our technique is similar to methods discovered independently by [23] and [12] One result in [23] shows that the 2 edge connected components of an initially connected graph on n vertices can be maintained under m edge insertions in time O(mff(m; n) n) This bound is improved to O(m n) by [12]. Although the bound of [12] matches our bound, we choose to present our method because it was discovered independently of [12] Moreover, the presentation of our technique will be helpful in understanding the more complicated solution for the biconnected case. The tree T c is represented by a ....

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J.A. La Poutr'e. Dynamic Graph Algorithms and Data Structures. PhD thesis, Department of Computer Sciences, University of Utrecht, The Netherlands, September 1991.

....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 ....

....insertion. In the case of 3 connectivity, Di Battista and Tamassia [28] give an algorithm which requires a total of O(q n log n) time in general. If we start with an initially biconnected graph, then the algorithm runs optimally in amortized O(ff(q; n) time per dynamic operation. La Poutr e [83,84] extends this to an optimal algorithm with amortized O(qff(q; n) time per edge insertion for general graphs. For 4 connectivity, the algorithm of [69] takes O(q n log n) time. If we start with an initially triconnected graph, then the algorithm runs optimally in amortized O(ff(q; n) time per ....

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J.A. La Poutre, "Dynamic Graph Algorithms and Data Structures," Dept. of Computer Science, University of Utrechet, Utrechet, Ph.D. Thesis, 1991.

....nodes. The structure utilizes the micro set approach of Gabow and Tarjan [8] but is based on a different tree representation. We note that several groups of researchers have developed fast data structures for maintaining 3 edge connected and 3 vertex connected component tree decompositions [2, 14, 16], but they did not give algorithms to find nearest common ancestors. It is straightforward to reduce the union find problem to the problem of maintaining 3 connected components, and lower bounds of Omega Gamma mff(m; n) are known for the union find problem in both pointer machine [15] and cell ....

J. A. L. Poutr'e. Dynamic Graph Algorithms and Data Structures. PhD thesis, University of Utrecht, Netherlands, 1991.

....is O(1) for general graphs. The randomized bounds for general graphs are currently O(log 4 n) amortized time per update and O(log 2 n) worst case time per query [8] For the incremental problem the running time for general and planar graphs is O(ff(q; n) amortized for each of q operations [10] [12] Partially supported by EU ESPRIT Long Term Research Project 20244 (ALCOM IT) DFG Graduiertenkolleg Parallele Rechnernetzwerke in der Produktionstechnik Me872 4 1, DFG Leibniz Grant Me872 6 1, and DFG Research Cluster Efficient Algorithms for Discrete Problems and their Applications ....

J.A. La Poutr'e. Dynamic Graph Algorithms and Data Structures. PhD. Thesis, Dep. Comp. Sci. Utrecht Univ., 1991.

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