A cocolouring of a graph G is a partition of the vertices such that each set of the partition induces either a clique or an independent set in G. The cochromatic number of G is the smallest cardinality of a cocolourin...
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A cocolouring of a graph G is a partition of the vertices such that each set of the partition induces either a clique or an independent set in G. The cochromatic number of G is the smallest cardinality of a cocolouring of G. We show that the cochromatic number problem remains NP-complete for line graphs of comparability graphs (and hence for line graphs and K1.3-free graphs), and we present polynomial time algorithms for computing the cochromatic numbers of chordal graphs and cographs.
Gallo et al. [4] recently examined the problem of computing on line a sequence of k maximum flows and minimum cuts in a network of n nodes, where certain edge capacities change between each flow. They showed that for ...
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Gallo et al. [4] recently examined the problem of computing on line a sequence of k maximum flows and minimum cuts in a network of n nodes, where certain edge capacities change between each flow. They showed that for an important class of networks, the k maximum flows and minimum cuts can be computed simply in O(n(3) + kn(2)) total time, provided that the capacity changes are made ''in order.'' Using dynamic trees their time bound is O(nm log(n(2)/m)+ km log(n(2)/m)). We show how to reduce the total time, using a simple algorithm, to O(n(3) + kn) for explicitly computing the k minimum cuts and implicitly representing the k flows. Using dynamic trees our bound is O(nm log(n(2)/m)+ kn log(n2/m)). We further reduce the total time to O(n(2) root m) if k is at most O(n). We also apply the ideas from [10] to show that the faster bounds hold even when the capacity changes are not ''in order,'' provided we only need the minimum cuts;if the flows are required then the times are respectively O(n(3) + km) and O(n(2) root m). We illustrate the utility of these results by applying them to the rectilinear layout problem.
In this paper we present a randomized parallel algorithm for finding a maximal independent set in a random graph with n vertices in which the edges are chosen with probability p such that 2/(n - 1) less than or equal ...
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In this paper we present a randomized parallel algorithm for finding a maximal independent set in a random graph with n vertices in which the edges are chosen with probability p such that 2/(n - 1) less than or equal to p < 1. The algorithm has O(log(2)n) expected time using only O(n) processors on the EREW PRAM model.
A classic PMC (Preparata, Metze, and Chien) multiprocessor system [F. ***, G. Metze, and R. T. Chien, IEEE Trans. Electr. Comput., EC-16 (1967), pp. 848-854] composed of n units is said to be t/(t + 1) diagnosable [A....
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A classic PMC (Preparata, Metze, and Chien) multiprocessor system [F. ***, G. Metze, and R. T. Chien, IEEE Trans. Electr. Comput., EC-16 (1967), pp. 848-854] composed of n units is said to be t/(t + 1) diagnosable [A. D. Friedman, A new measure of digital system diagnosis, in Dig. 1975 Int. Symp. Fault-Tolerant Comput., 1975, pp. 167-170] if, given a syndrome (complete collection of test results), the set of faulty units can be isolated to within a set of at most t + 1 units, assuming that at most t units in the system are faulty. This paper presents a methodology for determining when a unit v can belong to an allowable fault set of cardinality at most t. Based on this methodology, for a given syndrome in a t/(t + 1)-diagnosable system, the authors establish a necessary and sufficient condition for a vertex v to belong to an allowable fault set of cardinality at most t and certain properties of t/(t + 1)-diagnosable systems. This condition leads to an O(n(3.5))t/(t + 1)-diagnosis algorithm. This t/(t + 1)-diagnosis algorithm complements the t/(t + 1)-diagnosability algorithm of Sullivan [The complexity of system-level fault diagnosis and diagnosability, Ph.D. thesis, Yale University, New Haven, CT, 1986].
In graphs of bounded arboricity, the total complexity of all maximal complete bipartite subgraphs us O(n). We described a linear time algorithm to list such subgraphs. The arboricity bound is necessary: for any consta...
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In graphs of bounded arboricity, the total complexity of all maximal complete bipartite subgraphs us O(n). We described a linear time algorithm to list such subgraphs. The arboricity bound is necessary: for any constant k and any n there exists and n-vertex graph with P(n) edges and (n/log n)(k) maximal complete bipartite subgraphs K-k,K-l.
We present an O(\V\ + \E\) algorithm for finding the minimum cost between two vertices of a Manhattan graph (V, E) - a graph in which the vertices are points in Z(d) and the weight on each edge is the Manhattan distan...
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We present an O(\V\ + \E\) algorithm for finding the minimum cost between two vertices of a Manhattan graph (V, E) - a graph in which the vertices are points in Z(d) and the weight on each edge is the Manhattan distance between its two incident vertices.
The modification of the greedy algorithm for the weighted vertex cover problem (WVCP) suggested by Clarkson yields an algorithm with better worst-case performance. The results presented in this paper indicate, however...
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The modification of the greedy algorithm for the weighted vertex cover problem (WVCP) suggested by Clarkson yields an algorithm with better worst-case performance. The results presented in this paper indicate, however, that the generic greedy algorithm performs far better on the average than the modified algorithm. Even so, the generic greedy algorithm produces solutions which are on the average several percent above optimum, and which cannot be improved by additional runs due to the deterministic nature of the algorithm. The nondeterministic Simulated Annealing algorithm has therefore been specialized to the WVCP. The Simulated Annealing algorithm yields solutions of higher quality, even when only a moderate number of iterations are performed.
This paper examines the class of vertex-colored graphs that can be triangulated without the introduction of edges between vertices of the same color. This is related to a fundamental and long-standing problem for nume...
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This paper examines the class of vertex-colored graphs that can be triangulated without the introduction of edges between vertices of the same color. This is related to a fundamental and long-standing problem for numerical taxonomists, called the Perfect Phylogeny Problem. These problems are known to be polynomially equivalent and NP-complete. This paper presents a dynamic programming algorithm that can be used to determine whether a given vertex-colored graph can be so triangulated and that runs in O((n + m(k - 2))k+1) time, where the graph has n vertices, m edges, and k colors. The corresponding algorithm for the Perfect Phylogeny Problem runs in O(r(k+1)k(k+1) + sk2) time, where s species are defined by k r-state characters.
Let an undirected graph G be given, along with a specified depth-first spanning tree T. Almost-linear-time algorithms are given to solve the following two problems. First, for every vertex v, compute the number of des...
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Let an undirected graph G be given, along with a specified depth-first spanning tree T. Almost-linear-time algorithms are given to solve the following two problems. First, for every vertex v, compute the number of descendants w of v for which some descendant of w is adjacent (in G) to v. Second, for every vertex v, compute the number of ancestors of v that are adjacent (in G) to at least one descendant of v. These problems arise in Cholesky and QR factorizations of sparse matrices. The authors' algorithms can be used to determine the number of nonzero entries in each row and column of the triangular factor of a matrix from the zero/nonzero structure of the matrix. Such a prediction makes storage allocation for sparse matrix factorizations more efficient. The authors' algorithms run in time linear in the size of the input times a slowly growing inverse of Ackermann's function. The best previously known algorithms for these problems ran in time linear in the sum of the nonzero counts, which is usually much larger. Experimental results are given demonstrating the practical efficiency of the new algorithms.
This paper presents a VLSI array for labeling the connected components of a graph on N nodes in O(r) steps using a reconfigurable bus of width m bits, such that and 1≤r≤m. The network architecture consists of an arr...
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This paper presents a VLSI array for labeling the connected components of a graph on N nodes in O(r) steps using a reconfigurable bus of width m bits, such that and 1≤r≤m. The network architecture consists of an array of simple logic nodes which are connected by a reconfigurable bus. To solve a problem on N nodes, the array uses N processors and N(N−1)/2 switches. The proposed connectivity and transitive closure algorithms are based on a processor indexing scheme which employs constant-weight codes. It is shown that when r is a constant, then the algorithm takes O(1) time using a bus of width O(N 1/r ), and when r=[ log N/ loglog N], the algorithm takes O( log N/ loglog N) time using a bus of width O( log N) bits.
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