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13
A Framework for Dynamic Graph Drawing
 CONGRESSUS NUMERANTIUM
, 1992
"... Drawing graphs is an important problem that combines flavors of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows ..."
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Cited by 515 (40 self)
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Drawing graphs is an important problem that combines flavors of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows: ffl We devise a model for dynamic graph algorithms, based on performing queries and updates on an implicit representation of the drawing, and we show its applications. ffl We present several efficient dynamic drawing algorithms for trees, seriesparallel digraphs, planar stdigraphs, and planar graphs. These algorithms adopt a variety of representations (e.g., straightline, polyline, visibility), and update the drawing in a smooth way.
Faster ShortestPath Algorithms for Planar Graphs
 STOC 94
, 1994
"... We give a lineartime algorithm for singlesource shortest paths in planar graphs with nonnegative edgelengths. Our algorithm also yields a lineartime algorithm for maximum flow in a planar graph with the source and sink on the same face. The previous best algorithms for these problems required\O ..."
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Cited by 158 (13 self)
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We give a lineartime algorithm for singlesource shortest paths in planar graphs with nonnegative edgelengths. Our algorithm also yields a lineartime algorithm for maximum flow in a planar graph with the source and sink on the same face. The previous best algorithms for these problems required\Omega\Gamma n p log n) time where n is the number of nodes in the input graph. For the case where negative edgelengths are allowed, we give an algorithm requiring O(n 4=3 log nL) time, where L is the absolute value of the most negative length. Previous algorithms for shortest paths with negative edgelengths required \Omega\Gamma n 3=2 ) time. Our shortestpath algorithm yields an O(n 4=3 log n)time algorithm for finding a perfect matching in a planar bipartite graph. A similar improvement is obtained for maximum flow in a directed planar graph.
Dynamic Graph Algorithms
, 1999
"... Introduction In many applications of graph algorithms, including communication networks, graphics, assembly planning, and VLSI design, graphs are subject to discrete changes, such as additions or deletions of edges or vertices. In the last decade there has been a growing interest in such dynamicall ..."
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Cited by 54 (1 self)
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Introduction In many applications of graph algorithms, including communication networks, graphics, assembly planning, and VLSI design, graphs are subject to discrete changes, such as additions or deletions of edges or vertices. In the last decade there has been a growing interest in such dynamically changing graphs, and a whole body of algorithms and data structures for dynamic graphs has been discovered. This chapter is intended as an overview of this field. In a typical dynamic graph problem one would like to answer queries on graphs that are undergoing a sequence of updates, for instance, insertions and deletions of edges and vertices. The goal of a dynamic graph algorithm is to update efficiently the solution of a problem after dynamic changes, rather than having to recompute it from scratch each time. Given their powerful versatility, it is not surprising that dynamic algorithms and dynamic data structures are often more difficult to design and analyze than their static c
A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs
, 1998
"... In this paper we give a fully dynamic approximation scheme for maintaining allpairs shortest paths in planar networks. Given an error parameter ε such that 0 < ε, our algorithm maintains approximate allpairs shortest paths in an undirected planar graph G with nonnegative edge lengths. The approxi ..."
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Cited by 17 (1 self)
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In this paper we give a fully dynamic approximation scheme for maintaining allpairs shortest paths in planar networks. Given an error parameter ε such that 0 < ε, our algorithm maintains approximate allpairs shortest paths in an undirected planar graph G with nonnegative edge lengths. The approximate paths are guaranteed to be accurate to within a 1 + ε factor. The time bounds for both query and update for our algorithm is O(ε−1n2/3 log2 n log D), where n is the number of nodes in G and D is the sum of its edge lengths. The time bound for the queries is worst case, while that for the additions is amortized. Our approximation algorithm is based upon a novel technique for approximately representing allpairs shortest paths among a selected subset of the nodes by a sparse substitute graph.
Dynamic and Static Algorithms for Optimal Placement of Resources in a Tree
 THEORETICAL COMPUTER SCIENCE
, 1996
"... We consider the problem of placing resources in trees. We give algorithms for the static and dynamic version of the problem. The static algorithms are faster than the algorithms found in literature, while the dynamic algorithms are the first for this problem and run in polylogarithmic time. ..."
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Cited by 15 (1 self)
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We consider the problem of placing resources in trees. We give algorithms for the static and dynamic version of the problem. The static algorithms are faster than the algorithms found in literature, while the dynamic algorithms are the first for this problem and run in polylogarithmic time.
Graph Planarization and Skewness
"... The problem of finding a maximum spanning planar subgraph of a nonplanar graph is NPComplete. Several heuristics for the problem have been devised but their worstcase performance is unknown, although a trivial lower bound of 1/3 the optimum number of edges is easily shown. We discuss a new heurist ..."
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Cited by 8 (0 self)
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The problem of finding a maximum spanning planar subgraph of a nonplanar graph is NPComplete. Several heuristics for the problem have been devised but their worstcase performance is unknown, although a trivial lower bound of 1/3 the optimum number of edges is easily shown. We discuss a new heuristic, based on spanning trees, for generating a subgraph with size at least 2/3 of the optimum for any input graph. The skewness of the ndimensional hypercube Qn is also derived. Finally, we explore the relationship between the skewness and crossing number of a graph.
Dynamically Switching Vertices in Planar Graphs
 ALGORITHMICA
, 2000
"... We consider graphs whose vertices may be in one of two different states: either on or off. We wish to maintain dynamically such graphs under an intermixed sequence of updates and queries. An update may reverse the status of a vertex, by switching it either on or off, and may insert a new edge or ..."
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Cited by 6 (0 self)
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We consider graphs whose vertices may be in one of two different states: either on or off. We wish to maintain dynamically such graphs under an intermixed sequence of updates and queries. An update may reverse the status of a vertex, by switching it either on or off, and may insert a new edge or delete an existing edge. A query tests whether any two given vertices are connected in the subgraph induced by the vertices that are on. We give efficient algorithms that maintain information about connectivity on planar graphs in O(log³ n) amortized time per query, insert, delete, switchon and switchoff operation over sequences of at least\Omega\Gamma n) operations, where n is the number of vertices of the graph.
Fully Dynamic Planarity Testing with Applications
"... The fully dynamic planarity testing problem consists of performing an arbitrary sequence of the following three kinds of operations on a planar graph G: (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without ..."
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Cited by 6 (0 self)
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The fully dynamic planarity testing problem consists of performing an arbitrary sequence of the following three kinds of operations on a planar graph G: (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without violating planarity. We show how to support each of the above operations in O(n2=3) time, where n is the number of vertices in the graph. The bound for tests and deletions is worstcase, while the bound for insertions is amortized. This is the first algorithm for this problem with sublinear running time, and it affirmatively answers a question posed in [11]. The same data structure has further applications in maintaining the biconnected and triconnected components of a dynamic planar graph. The time bounds are the same: O(n2=3) worstcase time per edge deletion, O(n2=3) amortized time per edge insertion, and O(n2=3) worstcase time to check whether two vertices are either biconnected or triconnected.
A Uniform Approach to SemiDynamic Problems on Digraphs
 Theoretical Computer Science
, 1998
"... In this paper we propose a uniform approach to deal with incremental problems on digraphs and with decremental problems on dags generalizing a technique used by La Poutr'e and van Leeuwen in [17] for updating the transitive closure and the transitive reduction of a dag. We define a propagation pr ..."
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Cited by 5 (1 self)
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In this paper we propose a uniform approach to deal with incremental problems on digraphs and with decremental problems on dags generalizing a technique used by La Poutr'e and van Leeuwen in [17] for updating the transitive closure and the transitive reduction of a dag. We define a propagation property on a binary relationship over the vertices of a digraph as a simple sufficient condition to apply this approach. The proposed technique is suitable for a very simple implementation which does not depend on the particular problem; in other words, the same procedures can be used to deal with different problems by simply setting appropriate boundary conditions.