Journal ArticleDOI
Maintaining bridge-connected and biconnected components on-line
TLDR
A modified version of the dynamic trees of Sleator and Tarjan is developed that is suitable for efficient recursive algorithms, and used to reduce the running time of the algorithms for both problems toO(mα(m,n), where α is a functional inverse of Ackermann's function.Abstract:
We consider the twin problems of maintaining the bridge-connected components and the biconnected components of a dynamic undirected graph. The allowed changes to the graph are vertex and edge insertions. We give an algorithm for each problem. With simple data structures, each algorithm runs inO(n logn +m) time, wheren is the number of vertices andm is the number of operations. We develop a modified version of the dynamic trees of Sleator and Tarjan that is suitable for efficient recursive algorithms, and use it to reduce the running time of the algorithms for both problems toO(mα(m,n)), where α is a functional inverse of Ackermann's function. This time bound is optimal. All of the algorithms useO(n) space.read more
Citations
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Journal ArticleDOI
Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity
TL;DR: Deterministic fully dynamic graph algorithms are presented for connectivity, minimum spanning tree, 2-edge connectivity, and biconnectivity.
Journal ArticleDOI
On-Line Planarity Testing
TL;DR: An efficient technique for on-line planarity testing of a graph is presented that uses O(n) space and supports tests and insertions of vertices and edges in O(\log n) time, where n is the current number of Vertices of G.
Journal ArticleDOI
Sparsification—a technique for speeding up dynamic graph algorithms
TL;DR: In this article, the authors provide data strutures that maintain a graph as edges are inserted and deleted, and keep track of the following properties with the following times: minimum spanning forests, graph connectivity, graph 2-edge connectivity, and bipartiteness in timeO(n 1/2) per change; 3-edge connections, in time O(n 2/3) per insertion; 4-edge connection, in O(na(n)) per insertion.
Proceedings ArticleDOI
Sparsification-a technique for speeding up dynamic graph algorithms
TL;DR: The authors provide data structures that maintain a graph as edges are inserted and deleted, and keep track of the following properties: minimum spanning forests, best swap, graph connectivity, and graph
Proceedings ArticleDOI
Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity
TL;DR: Deterministic fully dynamic graph algorithms are presented for connectivity, minimum spanning tree, 2-edge connectivity, and biconnectivity.
References
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Efficiency of a Good But Not Linear Set Union Algorithm
TL;DR: It is shown that, if t(m, n) is seen as the maximum time reqmred by a sequence of m > n FINDs and n -- 1 intermixed UNIONs, then kima(m), n is shown to be related to a functional inverse of Ackermann's functmn and as very slow-growing.
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Self-adjusting binary search trees
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Journal ArticleDOI
A data structure for dynamic trees
TL;DR: An O(mn log n)-time algorithm is obtained to find a maximum flow in a network of n vertices and m edges, beating by a factor of log n the fastest algorithm previously known for sparse graphs.