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Maintaining Center and Median in Dynamic Trees
, 2000
"... We show how to maintain centers and medians for a collection of dynamic trees where edges may be inserted and deleted and node and edge weights may be changed. All updates are supported in O(log n) time, where n is the size of the tree(s) involved in the update. ..."
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Cited by 17 (4 self)
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We show how to maintain centers and medians for a collection of dynamic trees where edges may be inserted and deleted and node and edge weights may be changed. All updates are supported in O(log n) time, where n is the size of the tree(s) involved in the update.
Fast Incremental Planarity Testing
 19 th International Colloquium on Automata, Languages and Programming (ICALP), volume 623 of LNCS
, 1992
"... The incremental planarity testing problem is to perform the following operations on a biconnected planar graph G of at most n vertices: test if an edge can be added between two vertices while preserving planarity; add edges and vertices that preserve planarity. Let m be the total number of operation ..."
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Cited by 16 (0 self)
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The incremental planarity testing problem is to perform the following operations on a biconnected planar graph G of at most n vertices: test if an edge can be added between two vertices while preserving planarity; add edges and vertices that preserve planarity. Let m be the total number of operations. We present fast data structures for this problem that can be used in conjunction with the previous algorithm of Di Battista and Tamassia to achieve an O(ff(m; n)) worstcase amortized time per test operation. If the graph is biconnected, a sequence of n additions can be performed in total time O(mff(m;n)) worstcase plus O(n) expected time. Our tree data structure is flexible and can answer in O(1) time queries about parents, roots, and nearest common ancestors while performing tree modifications such as inserting nodes, cutting edges, and merging or splitting nodes. If the graph is not biconnected then insertions of edges and vertices require O(log n) amortized expected time per operat...
Certificates and Fast Algorithms for Biconnectivity in FullyDynamic Graphs
 Third Annual European Symposium on Algorithms (ESA`95
, 1997
"... In this paper, we present sparse certificates for biconnectivity together with algorithms for updating these certificates. We thus obtain fullydynamic 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 i ..."
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Cited by 8 (1 self)
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In this paper, we present sparse certificates for biconnectivity together with algorithms for updating these certificates. We thus obtain fullydynamic 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. Currently, several results exist for incremental and fullydynamic graph problems, like for maintaining spanning trees, the 2edge or the 2vertexconnected 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], algorith...
Undirected VertexConnectivity Structure and Smallest FourVertexConnectivity Augmentation
 Proc. 6th ISAAC
, 1995
"... In this paper, we study properties for the structure of an undirected graph that is not 4vertexconnected. We also study the evolution of this structure when an edge is added to optimally increase the vertexconnectivity of the underlying graph. Several properties reported here can be extended t ..."
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Cited by 4 (0 self)
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In this paper, we study properties for the structure of an undirected graph that is not 4vertexconnected. We also study the evolution of this structure when an edge is added to optimally increase the vertexconnectivity of the underlying graph. Several properties reported here can be extended to the case of a graph that is not kvertex connected, for an arbitrary k. Using properties obtained here, we solve the problem of finding a smallest set of edges whose addition 4vertexconnects an undirected graph. This is a fundamental problem in graph theory and has applications in network reliability and in statistical data security. We give an O(n \Delta log n + m)time algorithm for finding a set of edges with the smallest cardinality whose addition 4vertexconnects an undirected graph, where n and m are the number of vertices and edges in the input graph, respectively. This is the first polynomial time algorithm for this problem when the input graph is not 3vertexconnecte...
Combine and Conquer
 Department of Computer Science, Brown University, Providence, RI
, 1992
"... We present a general technique for dynamizing a class of problems whose underlying structure is a computation graph embedded in a tree. We associate values, called attributes, with the nodes, paths, and subtrees of our trees. Path attributes form a path attribute system, if they are maintained in ..."
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Cited by 2 (0 self)
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We present a general technique for dynamizing a class of problems whose underlying structure is a computation graph embedded in a tree. We associate values, called attributes, with the nodes, paths, and subtrees of our trees. Path attributes form a path attribute system, if they are maintained in constant time under path concatenation. Additionally, attributes form a tree attribute system if the tree attributes of the tail of a path \Pi are determined in constant time from the path attributes of \Pi. We also introduce a new data structure called a linear attribute grammar. An attribute grammar is a treebased expression where the values a node are calculated from the values at the parent, siblings, and/or the children of . A linear attribute grammar, is an attribute grammar where all dependencies are linear. Our contributions can be summarized as follows. We provide a framework for maintaining attribute systems on trees in a fully dynamic environment. We show that given a ...
Decremental Biconnectivity on Planar Graphs
"... : In this paper we present a (randomized) algorithm for maintaining the biconnected components of a dynamic planar graph of n vertices under deletions of edges. The biconnected components can be maintained under any sequence of edge deletions in a total of O(n log n) time, with high probability. Thi ..."
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: In this paper we present a (randomized) algorithm for maintaining the biconnected components of a dynamic planar graph of n vertices under deletions of edges. The biconnected components can be maintained under any sequence of edge deletions in a total of O(n log n) time, with high probability. This gives O(logn) amortized time per edge deletion, which improves previous (deterministic) results from [6] due to Giammarresi and Italiano, where O(n log 2 n) amortized time is needed. Our work describes a simplification of the data structures from [6] and uses dynamic perfect hashing to reduce the running time. As in [6], we only need O(n) space. Finally we describe some simply additional operations on the decremental data structure. By aid of them this the data structure is applicable for finding efficiently a \Deltaspanning tree in a biconnected planar graph with a maximum degree 2\Delta \Gamma 2 do to Czumaj and Strothmann [2]. Key Words: dynamic algorithms, graph algorithms, graph c...