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98
Optimal reduction of twoterminal directed acyclic graphs
 SIAM Journal on Computing
, 1992
"... Abstract. Algorithms for seriesparallel graphs can be extended to arbitrary twoterminal dags if node reductions are used along with series and parallel reductions. A node reduction contracts a vertex with unit indegree (outdegree) into its sole incoming (outgoing) neighbor. This paper gives an O ..."
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Cited by 14 (1 self)
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Abstract. Algorithms for seriesparallel graphs can be extended to arbitrary twoterminal dags if node reductions are used along with series and parallel reductions. A node reduction contracts a vertex with unit indegree (outdegree) into its sole incoming (outgoing) neighbor. This paper gives an O(n2"5) algorithm for minimizing node reductions, based on vertex cover in a transitive auxiliary graph. Applications include the analysis of PERT networks, dynamic programming approaches to network problems, and network reliability. For NPhard problems one can obtain algorithms that are exponential only in the minimum number of node reductions rather than the number of vertices. This gives improvements if the underlying graph is nearly seriesparallel.
A New Approach for Visualizing UML Class Diagrams
"... UML diagrams have become increasingly important in the engineering and reengineering processes for software systems. Of particular interest are UML class diagrams whose purpose is to display class hierarchies (generalizations), associations, aggregations, and compositions in one picture. The combina ..."
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Cited by 13 (0 self)
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UML diagrams have become increasingly important in the engineering and reengineering processes for software systems. Of particular interest are UML class diagrams whose purpose is to display class hierarchies (generalizations), associations, aggregations, and compositions in one picture. The combination of hierarchical and nonhierarchical relations poses a special challenge to a graph layout tool. Existing layout tools treat hierarchical and nonhierarchical relations either alike or as separate tasks in a twophase process as in, e.g., [Seemann 1997]. We suggest a new approach for visualizing UML class diagrams leading to a balanced mixture of the following aesthetic criteria: Crossing minimization, bend minimization, uniform direction within each class hierarchy, no nesting of one class hierarchy within another, orthogonal layout, merging of multiple inheritance edges, and good edge labelling. We have realized our approach within the graph drawing library GoVisual. Experiments show the superiority to stateoftheart and industrial standard layouts.
OutputSensitive Reporting of Disjoint Paths
, 1996
"... A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For ..."
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Cited by 11 (2 self)
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A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes kpath queries in outputsensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.
Directed st Numberings, Rubber Bands, and Testing Digraph kVertex Connectivity
"... Let G = (V, E) be a directed graph and n denote V. We show that G is kvertex connected iff for every subset X of V with IX I = k, there is an embedding of G in the (k I)dimensional space Rkl, ~ : V ~Rkl, such that no hyperplane contains k points of {~(v) \ v G V}, and for each v E V – X, f( ..."
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Cited by 10 (2 self)
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Let G = (V, E) be a directed graph and n denote V. We show that G is kvertex connected iff for every subset X of V with IX I = k, there is an embedding of G in the (k I)dimensional space Rkl, ~ : V ~Rkl, such that no hyperplane contains k points of {~(v) \ v G V}, and for each v E V – X, f(v) is in the convex hull of {~(w) I (v, W) G E}. This result generalizes to directed graphs the notion of convex embedding of undirected graphs introduced by Linial, LOV6SZ and Wigderson in ‘Rubber bands, convex embedding and graph connectivity, ” Combinatorics 8 (1988), 91102. Using this characterization, a directed graph can be tested for kvertex connectivity by a Monte Carlo algorithm in time O((M(n) + nkf(k)). (log n)) with error probability < l/n, and by a Las Vegas algorithm in expected time O((lf(n)+nM(k)).k), where M(n) denotes the number of arithmetic steps for multiplying two n x n matrices (Al(n) = 0(n2.3755)). Our Monte Carlo algorithm improves on the best previous deterministic and randomized time complexities for k> no. *9; e.g., for k = @, the factor of improvement is> n0.G2. Both algorithms have processor efficient parallel versions that run in O((log n)2) time on the EREW PRAM model of computation, using a number of processors equal to (logn) times the respective sequential time complexities. Our Monte Carlo parallel algorithm improves on the number of processors used by the best previous (Monte Carlo) parallel algorithm by a factor of at least (n2/(log n)3) while having the same running time. Generalizing the notion of st numberings, we give a combinatorial construction of a directed st nulmberiug for any 2vertex connected directed graph.
Pitfalls of using PQTrees in Automatic Graph Drawing
, 1997
"... A number of erroneous attempts involving PQtrees in the context of automatic graph drawing algorithms have been presented in the literature in recent years. In order to prevent future research from constructing algorithms with similar errors we point out some of the major mistakes. In particula ..."
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A number of erroneous attempts involving PQtrees in the context of automatic graph drawing algorithms have been presented in the literature in recent years. In order to prevent future research from constructing algorithms with similar errors we point out some of the major mistakes. In particular, we examine erroneous usage of the PQtree data structure in algorithms for computing maximal planar subgraphs and an algorithm for testing leveled planarity of leveled directed acyclic graphs with several sources and sinks.
Practical Toroidality Testing
 Proc. of the Eighth Annual ACMSIAM Symposium on Discrete Algorithms
, 1996
"... We describe an algorithm for embedding graphs on the torus (doughnut) which we implemented first in C, and then in C++. Although the algorithm is exponential in the worst case, it was very effective for indicating the small graphs which are torus obstructions. We have completed examination of the gr ..."
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Cited by 10 (2 self)
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We describe an algorithm for embedding graphs on the torus (doughnut) which we implemented first in C, and then in C++. Although the algorithm is exponential in the worst case, it was very effective for indicating the small graphs which are torus obstructions. We have completed examination of the graphs on up to 10 vertices and the 11 vertex ones up to 24 edges, and of these 3884 are topological obstructions, and 2249 are also minor order obstructions. A cursory search of 12 and 13 vertex graphs resulted in several more. We purport that this approach has proved practical as it has permitted us to compile what we believe to be the biggest collection of torus obstructions in the world to date. 1 Introduction A graph is said to be embedded on a surface if it is drawn there with no crossing edges. A graph is planar if it can be drawn on the sphere, and is toroidal if it can be drawn on the torus (a sphere with one handle). The genus of a planar graph is zero, and a nonplanar graph which ...
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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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.
Certifying Algorithms
, 2010
"... A certifying algorithm is an algorithm that produces, with each output, a certificate or witness (easytoverify proof) that the particular output has not been compromised by a bug. A user of a certifying algorithm inputs x, receives the output y and the certificate w, and then checks, either manual ..."
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A certifying algorithm is an algorithm that produces, with each output, a certificate or witness (easytoverify proof) that the particular output has not been compromised by a bug. A user of a certifying algorithm inputs x, receives the output y and the certificate w, and then checks, either manually or by use of a program, that w proves that y is a correct output for input x. In this way, he/she can be sure of the correctness of the output without having to trust the algorithm. We put forward the thesis that certifying algorithms are much superior to noncertifying algorithms, and that for complex algorithmic tasks, only certifying algorithms are satisfactory. Acceptance of this thesis would lead to a change of how algorithms are taught and how algorithms are researched. The widespread use of certifying algorithms would greatly enhance the reliability of algorithmic software. We survey the state of the art in certifying algorithms and add to it. In particular, we start a