M. Rauch. Improved data structures for fully dynamic biconnectivity. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 686--695, 1994. Submitted to SIAM J. Computing.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....deletion are allowed, incremental if only insertion is allowed, and decremental if only deletion is allowed. In this chapter we consider the biconnectivity problem, where the (main) query is to decide if two given vertices are biconnected. The fully dynamic problem has been extensively studied [Rau94] HL95] HK95] In [Rau94] the amortized update time is O( m) in this chapter n is the number of vertices and m is the number of edges) and the worst case query time is O(1) for general graphs. For plane graphs these time bounds are O(log n) for an update and O(logn) for a query. In ....

....if only insertion is allowed, and decremental if only deletion is allowed. In this chapter we consider the biconnectivity problem, where the (main) query is to decide if two given vertices are biconnected. The fully dynamic problem has been extensively studied [Rau94] HL95] HK95] In [Rau94] the amortized update time is O( m) in this chapter n is the number of vertices and m is the number of edges) and the worst case query time is O(1) for general graphs. For plane graphs these time bounds are O(log n) for an update and O(logn) for a query. In [HL95] the amortized update ....

[Article contains additional citation context not shown here]

M. Rauch. Improved data structures for fully dynamic biconnectivity. Revised manuscript, ftp://ftp.cs.cornell.edu/pub/mhr/papers/2-vertex2. ps.gz. Extended Abstract appeared on STOC`94, 1994.

....deletion are allowed, incremental if only insertion is allowed, and decremental if only deletion is allowed. In this chapter we consider the biconnectivity problem, where the (main) query is to decide if two given vertices are biconnected. The fully dynamic problem has been extensively studied [Rau94] [HL95] HK95] In [Rau94] the amortized update time is O( p m) in this chapter n is the number of vertices and m is the number of edges) and the worst case query time is O(1) for general graphs. For plane graphs 1 these time bounds are O(log 2 n) for an update and O(logn) for a query. In ....

....if only insertion is allowed, and decremental if only deletion is allowed. In this chapter we consider the biconnectivity problem, where the (main) query is to decide if two given vertices are biconnected. The fully dynamic problem has been extensively studied [Rau94] HL95] HK95] In [Rau94] the amortized update time is O( p m) in this chapter n is the number of vertices and m is the number of edges) and the worst case query time is O(1) for general graphs. For plane graphs 1 these time bounds are O(log 2 n) for an update and O(logn) for a query. In [HL95] the amortized update ....

[Article contains additional citation context not shown here]

M. Rauch. Improved data structures for fully dynamic biconnectivity. Revised manuscript, ftp://ftp.cs.cornell.edu/pub/mhr/papers/2-vertex2. ps.gz. Extended Abstract appeared on STOC94, 1994.

....in a variety of contexts including operating systems, information systems, network management, and graphical applications. A number of important theoretical results have been obtained for both fully and partially dynamic maintenance of several properties on undirected graphs (see e.g. [12, 13, 14, 15, 22, 31]) Recently, an equally important e ort has started on implementing these techniques and showing their practical merits [1, 2] These were the rst implementations concerning fully dynamic maintenance of certain properties (connectivity, minimum spanning tree) in undirected graphs, as well as the ....

M. Rauch. Improved data structures for fully dynamic biconnectivity. In Proc. 26th ACM Symposium on Theory of Computing, pp. 686-695, 1994.

....version of Dyn FO. In [TY79] Tarjan and Yao propose a dynamic model whose complexity measure is the number of probes into a data structure and any other computation is free. A log n= log log n lower bound on a dynamic prefix multiplication problem was proved in [FS] Other lower bounds [M2] [R94] have been proved using these methods. Other work on dynamic complexity for databases includes the theory of maintaining materialized views upon updates ( J92] GMS93] Io85] and in integrity constraint simplification ( LST87] N82] The design of dynamic algorithms is an active field. See, ....

....methods. Other work on dynamic complexity for databases includes the theory of maintaining materialized views upon updates ( J92] GMS93] Io85] and in integrity constraint simplification ( LST87] N82] The design of dynamic algorithms is an active field. See, for example, E 92] E2 92] [R94], CT91] F85] F91] among many others. There is also a large amount of work in the programming language community on incremental computation, see for example [RR93, LT94] 2 This paper is organized as follows. In Section 2, we begin with some background on Descriptive Complexity. In Section ....

M. Rauch. " Improved Data Structures for Fully Dynamic Biconnectivity," ACM Symp. Theory Of Comput. (1994), 686--695.

....mentioned improvement for connectivity of Henzinger and Thorup [27] does not affect the O(log 5 n) bound for 2 edge 45 connectivity. For biconnectivity, the previous results are a lot worse. The first nontrivial result was a deterministic bound of O(m 2=3 ) from 1992 by Rauch [20] In 1994 [33], Rauch improved this bound to O(minf p m log n; ng) In 1995, Rauch) Henzinger and Poutre further improved the deterministic bound to O( p n log n logdm=ne) 26] In 1995 [21] Henzinger and King generalized their randomized algorithm from [22] to the biconnectivity problem to achieve an ....

M. Rauch. Improved data structures for fully dynamic biconnectivity. In Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, may 1994.

....all j = 1; n: Flip(j) Negate the value of x j . Query(j) Return L j i=1 x i , the parity of the first j elements. We reduce this problem to the Dynamic Transitive Closure Problem introduced above; similar reductions have recently also been used by Miltersen et al. 10] and Rauch [13] for other graph problems. We give the full proof to gain more familiarity with the topology and the update operations. Note that there is no obvious way to transform the proof to the case of plane st graphs or to the update repertory of [18] Theorem 2.1. The Dynamic Transitive Closure Problem ....

Monika Rauch, Improved data structures for fully dynamic biconnectivity, 26th Ann. Symp. on Theory of Computing (STOC), ACM, 1994, pp. 686--695. 26

....only an amortized constant number of compressed certificates need major updates after an update in G (see Lemma 2.33) Second (Section 3) we study the dynamic biconnectivity problem for plane graphs. We use a topology tree approach based on [5] An earlier version of this paper appeared in [20]. 2 General graphs 2.1 The graph G # and the relaxed partition of order k Let G be an undirected graph with n vertices and m edges. The size G of a graph is the total number of its nodes and edges. We assume in the paper that G is connected, which implies m # n 1. If G is not ....

M. H. Rauch, "Improved Data Structures for Fully Dynamic Biconnectivity", Proc. 26 Annual Symposium on Theory of Computing, 1994, 686-695.

....their sparsification technique on the static algorithm to achieve, for general graphs, O(n log(m=n) time per update and O(1) time per query. As certificate they used T 1 [ T 2 , where T 1 is a breadth first forest of the given graph, G, and T 2 is a breadth first forest of G n E(T 1 ) Rauch [13] improved her results from [12] and achieved O( p m log n) amortized time per update and O(1) time per query. It is done by using ambivalent data structure [3, 4] and sparsification to enable recomputing the high level spanning tree in just O( z m=z) log n) amortized time, which yields the ....

....It is done by using ambivalent data structure [3, 4] and sparsification to enable recomputing the high level spanning tree in just O( z m=z) log n) amortized time, which yields the desired update time. The algorithm s variant for planar graph works in O(log 2 n) time per operation. Rauch [13] also showed a lower bound for k edge and k vertex connectivity testing: In fully dynamic (planar) graphs the amortized time per operation has a lower bound of Omega Gamma 17 n = log log n) in the cell probe model. Her technique for bi 2 edge connectivity can be described as two steps reduction. ....

M. Rauch. Improved Data Structure for Fully Dynamic Biconnectivity. STOC 26:686--695, 1994.

....to handle particular problems and particular input changes. Examples are incremental parsing [40] attribute evaluation [73, 85] data flow analysis [76] circuit evaluation [6] constraint solving [30] transitive closure [86] shortest path [70] minimum spanning tree [29, 23] connectivity [23, 71], and scheduling. Since these algorithms are manually derived to solve particular incremental problems, we say that they are ad hoc. The second category is called incremental execution frameworks. The goal is to study general methods for incremental problems. The idea is to allow different ....

M. Rauch. Improved data structures for fully dynamic biconnectivity. In Conference Proceedings of the 26th Annual ACM STOC. ACM, New York, 1994. 23

....log n) for (directed and undirected) bounded width grid graphs. 2.3. Lower bounds. As often in the analysis of algorithms, the arsenal of known lower bounds for these problems does not match the upper bounds. Let me give a detailed account of a reduction from [29] and later (but independently) [37]. The model is the cell probe model with logarithmic word size [47] Fredman and Saks give a lower bound of Omega Gamma 26 n= log log n) on the amortised complexity of the Dynamic Parity Prefix Problem: Given a vector x 1 ; xn of bits, maintain a data structure that is able to react to ....

Monika Rauch, Improved data structures for fully dynamic biconnectivity, 26th Ann. Symp. on Theory of Computing (STOC), ACM, 1994, pp. 686--695.

....and R 2 is 2 edge connectivity. We present a data structure to capture essential connectivity information of H 1 and H 2 simultaneously. This data structure can be efficiently updated after a new edge is added to G, and is useful for dynamically maintaining bi level connectivity information [19, 22, 31]. Using this data structure, we solve the bi level augmentation problem in O(n m) time. Our algorithm can be parallelized to run in O(log 2 n) time using n m processors on an EREW PRAM. We use the algorithm to solve several optimization problems for protecting sensitive information in cross ....

M. Rauch. Improved data structures for fully dynamic biconnectivity. In Proc. 26th Annual ACM Symp. on Theory of Computing, pages 686--695, 1994.

....with algorithms for updating these certificates. We thus obtain fully dynamic algorithms for biconnectivity in graphs that run in O( # n log n log# m n #) amortized time per operation, where m is the number of edges and n is the number of nodes in the graph. This improves upon the results in [12], in which algorithms were presented running in O( # m log n) amortized time, and solves the open problem to find certificates to speed up biconnectivity, as stated in [2] 1 Introduction The field of dynamic graph algorithms has become an important field in algorithmic research in recent years. ....

....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 ....

[Article contains additional citation context not shown here]

M. H. Rauch. "Improved Data Structures for Fully Dynamic Biconnectivity " Proc. 26 Annual Symp. on Theory of Computing, 1994, 686--695. A full version of the paper is available at http:\\www.research.digital.com\SRC\ personal\monika\papers.html.

.... r, where r denotes the length of the path. Lower bound. Our update operations are su#ciently versatile to admit a lower bound proof for the problem in the cell probe model with logarithmic word size. The proof is a reduction to the Parity Prefix Problem of [8] in the same fashion as [12] and [16]. Theorem 2. The Dynamic Transitive Closure Problem on spherical st graphs requires amortised time ##me# n log log n) in the cell probe model with logarithmic word size. 2.3 Related Work Restricted versions of the present problem have been studied by Tamassia and Preparata [20] who consider ....

Monika Rauch. Improved data structures for fully dynamic biconnectivity. In 26th STOC, pages 686--695. ACM, 1994.

....URL: http: www.info.uniroma2.it italiano . Part of this work was done while visiting the Max Planck Institut fur Informatik, Im Stadtwald, 66123 Saarbrucken, Germany. 1 Introduction In the last years research in dynamic graph algorithms has been a blossoming field (see e.g. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 24]) The main dynamic model that has been considered in the literature is the following. We are given a graph G = V; E) and we wish to maintain some property P in G during edge deletions and edge insertions. We refer to this as the dynamic edge model . If the graph represents a communication ....

M. Rauch. Improved data structures for fully dynamic biconnectivity. Proceedings 26th ACM Symposium on Theory of Computing, 1994, 686--695.

....as a function of the output change [17, 40] The main dynamic problems considered on directed graphs include shortest paths and transitive closure. For lack of space, we do not include in this chapter dynamic algorithms for planar graphs, which have received considerable attention in recent years [6, 7, 11, 12, 18, 21, 22, 28, 32, 34, 42, 46, 47, 48, 51], and focus our attention to general undirected graphs only. The remainder of the chapter is organized as follows. In Section 2 we give some preliminary definitions and a little terminology. Dynamic tree problems are considered in Section 3, while in Section 4 we describe partially dynamic ....

....forest [4, 10, 14, 45] This is an important problem on its own, but it has also impact on other problems as well. Indeed the data structures and techniques developed for dynamic minimum spanning forests have found applications also in other areas, such as dynamic edge and vertex connectivity [10, 15, 19, 26, 41, 42]. Thus, we will focus our attention to the fully dynamic maintenance of minimum spanning trees. 5.1 Clustering and Topology Trees Let G = V; E) be a graph, with a designated spanning tree S. Clustering is a method of partitioning the vertex set V , into connected subtrees in S, so that each ....

[Article contains additional citation context not shown here]

M. Rauch. Improved data structures for fully dynamic biconnectivity. In Proc. 26th Symp. Theory of Computing, 1994.

....best deterministic and the best randomized bounds. plane planar general, rand. general, det. connectivity O(log n) 3] O(log 2 n) 5] O(log 2 n) 10, 10] O( p n) 4] 2 edge connectivity O(log 2 n) 11] O(log 2 n) 5] O(log 3 n) 8, 10] O( p n) 4] 2 vertex connectivity O(log 2 n) [14] O( p n) 5] O( p n log 3=2 n) 9] 3 edge connectivity O( p n) 5] O(n 2=3 ) 4] 3 vertex connectivity O( p n) 5] O(n) 4] 4 edge connectivity O( p n) 5] O(nff(n) 4] planarity testing O(log 2 n) 12] O( p n) 5] Figure 1: The best upper bounds for fully dynamic problems. We ....

....bounds of Omega Gamma ff(m; n) per operation are known for connectivity, 2 edge connectivity, 2 vertex connectivity, 3 edge connectivity, 3 vertex connectivity, and planarity testing [17, 15] by reducing these problems to the union find problem. An earlier version of this work has appeared in [14]. In the next section we give the general ideas of the lower bound proofs. In Section 3 and Section 4 we present the proofs for fully dynamic planarity testing and k edge and k vertex connectivity. In Section 5 we remove the dependency on b in a more specific model of computation. 2 The General ....

M. H. Rauch. "Improved Data Structures for Fully Dynamic Biconnectivity", Proc. 26th Annual Symp. on Theory of Computing, 1994, 686--695.

....it is in Dyn FO. In [TY79] Tarjan and Yao propose a dynamic model whose complexity measure is the number of probes into a data structure and any other computation is free. A log n= log log n lower bound on a dynamic prefix multiplication problem was proved in [FS89] Other lower bounds [M93] R94] have been proved using these methods. Other work on dynamic complexity for databases includes the theory of maintaining materialized views upon updates ( J92] GMS93] Io85] and in integrity constraint simplification ( LST87] N82] The design of dynamic algorithms is an active field. See, ....

....Other work on dynamic complexity for databases includes the theory of maintaining materialized views upon updates ( J92] GMS93] Io85] and in integrity constraint simplification ( LST87] N82] The design of dynamic algorithms is an active field. See, for example, E 92] E 92b] R94] CT91] F85] F91] among many others. There is also a large amount of work in the programming language community on incremental computation, see for example [RR96, LT94] This paper is organized as follows. In Section 2, we begin with some background on Descriptive Complexity. In Section 3, ....

M. Rauch. " Improved Data Structures for Fully Dynamic Biconnectivity," ACM Symp. Theory Of Comput. (1994), 686--695.

....v) Return true if vertices u and v are in the same biconnected component and an articulation point separating u and v otherwise. We give an algorithm for general graphs with O(m 2=3 ) amortized update time and O(1) query time. In a later paper we reduce the update time further to O( p m) [16]. Independently Eppstein et al. 7] have developed an algorithm for this problem whose running time per update operation is O(n) and whose running time per query is O(1) They use a general technique to speed up dynamic graph algorithms. However, their technique cannot be applied to the algorithm ....

M. Rauch, "Improved Data Structures for Fully Dynamic Biconnectivity", Proc. 26nd Annual ACM Sympos. on Theory of Computing, 1994, 686--695.

....O(log 4 n) Our algorithm is a careful generalization of a recent O(log 2 n) deterministic fully dynamic connectivity algorithm [11] For biconnectivity, the previous results are a lot worse. The first non trivial result was a deterministic bound of O(m 2=3 ) from 1992 by Rauch [9] In 1994 [12], Rauch improved this bound to O(minf p m log n; ng) In 1995, Rauch) Henzinger and Poutr e further improved the deterministic bound to O( p n log n logdm=ne) In 1995 [6] Henzinger and King generalized their randomized algorithm from [7] to the biconnectivity problem to achieve an ....

Monika Rauch. Improved data structures for fully dynamic biconnectivity. In Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, may 1994.

....E mail: italiano unive.it. URL: http: www.dsi.unive.it italiano. tions (i.e. either insertions or deletions, but not both) is allowed. The area of dynamic graph algorithms has been a blossoming field of research in the last years, and it has produced a large body of algorithmic techniques [2, 5, 6, 7, 8, 9, 10, 11, 14]. Perhaps one of the most studied problems in this area is the fully dynamic maintenance of the minimum spanning tree of a graph [2, 5, 7, 8] This problem is important on its own, and it finds applications to other problems as well, including many dynamic vertex and edge connectivity problems. ....

M. Rauch, "Improved data structures for fully dynamic biconnectivity", Proc. 26th Symp. on Theory of Computing (1994), 686--695.

....Award. y Department of Computer Science, University of Victoria, Victoria, BC. Email: val csr.uvic.ca. This research was supported by an NSERC Grant. 1 Throughout the paper the logarithms are base 2. Previous Work. In recent years a lot of work has been done in fully dynamic algorithms (see [1, 3, 4, 6, 7, 8, 10, 11, 12, 15, 16, 18] for connectivity related work in undirected graphs) There is also a large body of work for restricted classes of graphs and for insertions only algorithms. Currently the best time bounds for fully dynamic algorithms in undirected n node graphs are: O( p n) per update for a minimum spanning ....

M. H. Rauch, "Improved Data Structures for Fully Dynamic Biconnectivity in Graphs". Proc. 26th Symp. on Theory of Computing, 1994, 686--695.

....answering a query. For the dynamic connectivity problem, for example, a query takes two nodes u and v as its arguments and returns True , if there is a path connecting u and v in the current graph. The field of dynamic graph algorithms has been a blossoming field of research in the last years [4, 9, 11, 13, 14, 15, 17, 19, 31, 33], motivated by theoretical and practical questions (see for instance [29] However, despite this blend of theoretical and practical interests, we are aware of no implementations and experimental studies in this field. In this paper, we aim at bridging this gap by studying the practical properties ....

M. Rauch, "Improved data structures for fully dynamic biconnectivity", Proc. 26th Symp. on Theory of Computing (1994), 686--695.

....cardinality. In the case of maximum matching a query outputs a current maximum matching. Alternatively, a query could also be: Is the edge e in the current graph in the current maximum matching Recently, a lot of work has been done on dynamic algorithms for various connectivity proper ties [10, 11, 12, 13, 16, 24, 25, 26]. The current best deterministic bound for maintaining connected or 2 edge connected components of a graph is O(x ) 10] The best randomized algorithm achieves O(1 3 resp. O(1 4 per update [17] It is an open problem if the connected or 2 edge con nected components of a graph can be ....

M. H. Rauch. Improved data structures for fully dynamic biconnectivity. In Proc. 26th Syrup. on Theory of Computing, pages 686 695, 1994.

....showed how to maintain information about the minimum spanning forest and the connected components of a plane graph in O(log n) time per operation. Hershberger et al. 9] gave an algorithm that maintains the 2 edge connected components of a plane graph in O(log 2 n) time per operation. Rauch [12] gave an algorithm that maintains information about 2 vertex connectivity in plane graphs in O(log 2 n) time. In all of these algorithms, insertions expect to be told in which face the edge is to be inserted. If it is not known beforehand that the edge insertion will preserve the given ....

....structure for maintaining information about the planar embedding of an n vertex graph during edge insertions and deletions. Our data structure supports queries and updates in O(log 2 n) worst case time each. Recently, Rauch proved a lower bound of Omega Gamma 24 n= log log n) for this problem [12], which holds in the cell probe model of computation [15] Our algorithm is a factor of log n log log n away from this lower bound, leaving open the question of whether our bounds can be improved. If the graph is allowed to change its planar embedding during edge insertions, this problem becomes ....

M. H. Rauch "Improved Data Structures for Fully Dynamic Biconnectivity." Proc. 26th Symp. on Theory of Computing, 1994.

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M. Rauch. Improved data structures for fully dynamic biconnectivity. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 686--695, 1994. Submitted to SIAM J. Computing.

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