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RANDOMIZED Õ(M(V)) ALGORITHMS FOR PROBLEMS IN Matching Theory
, 1997
"... A randomized (Las Vegas) algorithm is given for finding the Gallai–Edmonds decomposition of a graph. Let n denote the number of vertices, and let M(n) denote the number of arithmetic operations for multiplying two n × n matrices. The sequential running time (i.e., number of bit operations) is within ..."
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Cited by 12 (0 self)
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A randomized (Las Vegas) algorithm is given for finding the Gallai–Edmonds decomposition of a graph. Let n denote the number of vertices, and let M(n) denote the number of arithmetic operations for multiplying two n × n matrices. The sequential running time (i.e., number of bit operations) is within a polylogarithmic factor of M(n). The parallel complexity is O((log n) 2) parallel time using a number of processors within a polylogarithmic factor of M(n). The same complexity bounds suffice for solving several other problems: (i) finding a minimum vertex cover in a bipartite graph, (ii) finding a minimum X→Y vertex separator in a directed graph, where X and Y are specified sets of vertices, (iii) finding the allowed edges (i.e., edges that occur in some maximum matching) of a graph, and (iv) finding the canonical partition of the vertex set of an elementary graph. The sequential algorithms for problems (i), (ii), and (iv) are Las Vegas, and the algorithm for problem (iii) is Monte Carlo. The new complexity bounds are significantly better than the best previous ones, e.g., using the best value of M(n) currently known, the new sequential running time is O(n2.38) versus the previous best O(n2.5 /(log n)) or more.
Geometric Representations of Graphs
 IN PAUL ERDÖS, PROC. CONF
, 1999
"... The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We survey the surprisingly large variety of graph properties related to geometric representations, c ..."
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The study of geometrically defined graphs, and of the reverse question, the construction of geometric representations of graphs, leads to unexpected connections between geometry and graph theory. We survey the surprisingly large variety of graph properties related to geometric representations, construction methods for geometric representations, and their applications in proofs and algorithms.
Directional Routing via Generals stNumberings
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 2000
"... We present a mathematical model for network routing based on generating paths in a consistent direction. Our development is based on an algebraic and geometric framework for defining a directional coordinate system for real vector spaces. Our model, which generalizes graph stnumberings, is based on ..."
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Cited by 3 (1 self)
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We present a mathematical model for network routing based on generating paths in a consistent direction. Our development is based on an algebraic and geometric framework for defining a directional coordinate system for real vector spaces. Our model, which generalizes graph stnumberings, is based on mapping the nodes of a network to points in multidimensional space and ensures that the paths generated in di#erent directions from the same source are nodedisjoint. Such directional embeddings encode the global disjoint path structure with very simple local information. We prove that all 3connected graphs have 3directional embeddings in the plane so that each node outside a set of extreme nodes has a neighbor in each of the three directional regions defined in the plane. We conjecture that the result generalizes to kconnected graphs. We also showthat a directed acyclic graph (dag) that is kconnected to a set of sinks has a kdirectional embedding in (k  1)space with the sink set as the extreme nodes.
Higher Dimensional Representations of Graphs
, 1995
"... Graphs are often used to model complex systems and to visualize relationships, and this often involves drawing a graph in the plane. For this, a variety of algorithms and mathematical tools have been used with varying success. We demonstrate why it is often more natural and more meaningful to view h ..."
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Cited by 2 (0 self)
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Graphs are often used to model complex systems and to visualize relationships, and this often involves drawing a graph in the plane. For this, a variety of algorithms and mathematical tools have been used with varying success. We demonstrate why it is often more natural and more meaningful to view higher dimensional representations of graphs. We present some of the theory and problems associated with constructing such representations, and we briefly describe some visualization tools which are now available for experimental research in this area. Figure 1: Picture of a graph 1 Introduction For many people the right picture is the key to understanding. The various ways of visualizing a graph provide different insights, and often hidden relationships are revealed. Also, the representation of a graph enfluences how well it can be used. We focus on several mathematical problems associated with drawing graphs. The problems and methods of solution are as diverse as the objectives includin...
and
, 1996
"... The vertex connectivity � of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known deterministic algorithm for finding the vertex connectivity and a corresponding Ž 3 separator. The time for a digraph having n vertices and m e ..."
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The vertex connectivity � of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known deterministic algorithm for finding the vertex connectivity and a corresponding Ž 3 separator. The time for a digraph having n vertices and m edges is O min � � n, � nm; 4. for an undirected graph the term m can be replaced by � n. A randomized algorithm finds � with error probability 1�2 in time Onm. Ž. If the vertices have nonnegative weights the weighted vertex connectivity is found in time Ž Ž 2 O � nmlog n �m.. 1 where �1�m�n is the unweighted vertex connectivity or in Ž Ž 2 expected time Onmlog n �m.. with error probability 1�2. The main algorithm combines two previous vertex connectivity algorithms and a generalization of the
Computing Vertex Connectivity: New Bounds from Old Techniques
, 1999
"... Abstract The vertex connectivity ^ of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known deterministic algorithm for finding the vertex connectivity and a corresponding separator. The time for a digraph having n vertices an ..."
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Abstract The vertex connectivity ^ of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known deterministic algorithm for finding the vertex connectivity and a corresponding separator. The time for a digraph having n vertices and m edges is O(minf^3 + n; ^ngm); for an undirected graph the term m can be replaced by ^n. A randomized algorithm finds ^ with error probability 1=2 in time O(nm). If the vertices have nonnegative weights the weighted vertex connectivity is found in time O(^1nm log (n2=m)) where ^1 ^ m=n is the unweighted vertex connectivity, or in expected time O(nm log (n2=m)) with error probability 1/2. The main algorithm combines two previous vertex connectivity algorithms and a generalization of the preflowpush algorithm of Hao and Orlin [13] that computes edge connectivity.