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Improved Data Structures for Fully Dynamic Biconnectivity
- Proc. 26th ACM Symp. Theory of Computing
, 1997
"... We present fully dynamic algorithms for maintaining the biconnected components in general and plane graphs. A fully dynamic algorithm maintains a graph during a sequence of insertions and deletions of edges or isolated vertices. Let m be the number of edges and n be the number of vertices in a gr ..."
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We present fully dynamic algorithms for maintaining the biconnected components in general and plane graphs. A fully dynamic algorithm maintains a graph during a sequence of insertions and deletions of edges or isolated vertices. Let m be the number of edges and n be the number of vertices in a graph. The time per operation of the previously best deterministic algorithms were O(min{m 2/3 , n}) in general graphs and O( # n) in plane graphs for fully dynamic biconnectivity. We improve these running times to O( # m log n) in general graphs and O(log 2 n) in plane graphs. Our algorithm for general graphs can also find the biconnected components of all vertices in time O(n). 1 Introduction Many computing activities require the recomputation of a solution after a small modification of the input data. Thus algorithms are needed that update an old solution in response to a change in the problem instance. Dynamic graph algorithms are data structures that, given an input graph G, mai...
Equivalence and Related Problems
, 1997
"... Abstract In this paper we introduce a new class of dynamic graph algorithms called quasi-fully dynamic algorithms, which are much more general than the backtracking algorithms and are much simpler than the fully dynamic algorithms. These algorithms are especially suitable for applications in which a ..."
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Abstract In this paper we introduce a new class of dynamic graph algorithms called quasi-fully dynamic algorithms, which are much more general than the backtracking algorithms and are much simpler than the fully dynamic algorithms. These algorithms are especially suitable for applications in which a certain core connected portion of the graph remains fixed, and fully dynamic updates occur on the remaining edges in the graph. We present very simple quasi-fully dynamic algorithms with O(log n) worst case time, per operation, for 2-edge connectivity and cycle equivalence. The former is deterministic while the latter is Monte-Carlo type randomized. For 2-vertex connectivity, we give a randomized Las Vegas algorithm with O(log4 n) expected amortized time per operation. We introduce the concept of quasi-k-edge-connectivity, which is a slightly relaxed version of k-edge connectivity, and show that it can be maintained in O(log n) worst case time per operation. We also analyze the performance of a natural extension of our quasi-fully dynamic algorithms to fully dynamic algorithms. The quasi-fully dynamic algorithm we present for cycle equivalence (which has several applications in optimizing compilers) is of special interest since the algorithm is quite simple, and no special-purpose incremental or backtracking algorithm is known for this problem. 1 Introduction Dynamic graph algorithms have received a great deal of attention in the last few years (see e.g., [4]). These algorithms maintain a property of a given graph under a sequence of suitably restricted updates and queries. Throughout this paper we will be concerned with edge updates (insertions/deletions) only: insertion/deletion of isolated vertices can be implemented trivially in all the known dynamic graph algorithms. The existing dynamic algorithms can be classified into three types depending on the nature of (edge) updates allowed: ffl Partially Dynamic: Only insertions are allowed (Incremental) or only deletions are allowed (Decremental).
Quasi-Fully Dynamic Algorithms for Two-Connectivity, Cycle Equivalence and Related Problems
, 1997
"... In this paper weintroduce a new class of dynamic graph algorithms called quasi-fully dynamic algorithms, which are much more general than the backtracking algorithms and are much simpler than the fully dynamic algorithms. These algorithms are especially suitable for applications in which a certain c ..."
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In this paper weintroduce a new class of dynamic graph algorithms called quasi-fully dynamic algorithms, which are much more general than the backtracking algorithms and are much simpler than the fully dynamic algorithms. These algorithms are especially suitable for applications in which a certain core connected portion of the graph remains xed, and fully dynamic updates occur on the remaining edges in the graph. We present very simple quasi-fully dynamic algorithms with O(log n) worst case time, per operation, for 2-edge connectivity and cycle equivalence. The former is deterministic while the latter is Monte-Carlo type randomized. For 2-vertex connectivity, we give a randomized Las Vegas algorithm with O(log 4 n) expected amortized time per operation. We introduce the concept of quasi-k-edge-connectivity, which is a slightly relaxed version of k-edge connectivity, and show that it can be maintained in O(log n) worst case time per operation. We also analyze the performance of a natural extension of our quasi-fully dynamic algorithms to fully dynamic algorithms. The quasi-fully dynamic algorithm we present for cycle equivalence (which has several applications in optimizing compilers) is of special interest since the algorithm is quite simple, and no special-purpose incremental or backtracking algorithm is known for this problem. 1
Quasi-Fully Dynamic Algorithms for Two-Connectivity, Cycle Equivalence and Related Problems
, 1997
"... In this paper we introduce a new class of dynamic graph algorithms called quasi-fully dynamic algorithms, which are much more general than the backtracking algorithms and are much simpler than the fully dynamic algorithms. These algorithms are especially suitable for applications in which a certa ..."
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In this paper we introduce a new class of dynamic graph algorithms called quasi-fully dynamic algorithms, which are much more general than the backtracking algorithms and are much simpler than the fully dynamic algorithms. These algorithms are especially suitable for applications in which a certain core connected portion of the graph remains fixed, and fully dynamic updates occur on the remaining edges in the graph. We present very simple quasi-fully dynamic algorithms with O(log n) worst case time, per operation, for 2-edge connectivity and cycle equivalence. The former is deterministic while the latter is Monte-Carlo type randomized. For 2-vertex connectivity, we give a randomized Las Vegas algorithm with O(log 4 n) expected amortized time per operation. We introduce the concept of quasi-k-edge-connectivity, which is a slightly relaxed version of k-edge connectivity, and show that it can be maintained in O(logn) worst case time per operation. We also analyze the performa...
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, 2014
"... This thesis is made available online and is protected by original copyright. Please scroll down to view the document itself. Please refer to the repository record for this item for information to help you to cite it. Our policy information is available from the repository home page. ..."
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This thesis is made available online and is protected by original copyright. Please scroll down to view the document itself. Please refer to the repository record for this item for information to help you to cite it. Our policy information is available from the repository home page.
