Results 1 
3 of
3
Randomized Fully Dynamic Graph Algorithms with Polylogarithmic Time per Operation
 JOURNAL OF THE ACM
, 1999
"... This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion. The algorithms are designed using a new d ..."
Abstract

Cited by 70 (0 self)
 Add to MetaCart
This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion. The algorithms are designed using a new dynamic technique which combines a novel graph decomposition with randomization. They are LasVegas type randomized algorithms which use simple data structures and have a small constant factor. Let n denote the number of nodes in the graph. For a sequence of \Omega\Gamma m 0 ) operations, where m 0 is the number of edges in the initial graph, the expected time for p updates is O(p log 3 n) 1 for connectivity and bipartiteness. The worstcase time for one query is O(log n= log log n). For the kedge witness problem ("Does the removal of k given edges disconnect the graph?") the expected time for p updates is O(p log 3 n) and expected time for q queries is O(qk log 3 n). Given a grap...
Randomized Dynamic Graph ALgorithms with Polylogarithmic Time per Operation
 PROC. 33RD ANNUAL SYMP. ON FOUNDATIONS OF COMPUTER SCIENCE
, 1995
"... ..."
Maintaining Dynamic Graph Properties Deterministically
"... In this paper we present deterministic fully dynamic algorithms for maintaining several properties on undirected graphs subject to edge insertions and deletions, in polylogarithmic time per operation. Combining techniques from [6, 10], we can maintain a minimum spanning forest of a graph with k di#e ..."
Abstract
 Add to MetaCart
In this paper we present deterministic fully dynamic algorithms for maintaining several properties on undirected graphs subject to edge insertions and deletions, in polylogarithmic time per operation. Combining techniques from [6, 10], we can maintain a minimum spanning forest of a graph with k di#erent edge weights in O(k log time per update; maintain an 1+#approximation of the minimum spanning forest in O(log n log(U(1 + #))/ log(1 + #)) amortized time per update, where edge weights are between 1 and U ; test if a graph is bipartite in O(1) worstcase time, supporting updates in O(log n) amortized time; test if the removal of k given edges disconnect the graph (kedge witness problem) in O(k log n) amortized time, supporting updates in O(log n) amortized time; maintain a maximal spanning forest decomposition of order k in O(k log per update. For all these problems, our algorithms match the previous best randomized bounds, and improve substantially over the best deterministic bounds.