A vertex ranking of an undirected graph G is a labeling of the vertices of G with integers such that every path connecting two vertices with the same label i contains an intermediate vertex with label j > i. A vert...
详细信息
A vertex ranking of an undirected graph G is a labeling of the vertices of G with integers such that every path connecting two vertices with the same label i contains an intermediate vertex with label j > i. A vertex ranking of G is called optimal if it uses the minimum number of distinct labels among all possible vertex rankings. The problem of finding an optimal vertex ranking for general graphs is NP-hard, and NP-hard even for chordal graphs which form a superclass of block graphs. In this paper, we present the first polynomial algorithm which runs in O(n(2) log Delta) time for finding an optimal vertex ranking of a block graph G, where n and Delta denote the number of vertices and the maximum degree of G, respectively. (c) 2008 Elsevier Inc. All rights reserved.
We present I/O-efficient algorithms for computing optimal separator partitions of planar graphs. Our main result shows that, given a planar graph G with N vertices and an integer r > 0, a vertex separator of size O...
详细信息
We present I/O-efficient algorithms for computing optimal separator partitions of planar graphs. Our main result shows that, given a planar graph G with N vertices and an integer r > 0, a vertex separator of size O(N/root r) that partitions G into O(N/r) subgraphs of size at most r and boundary size O(v r) can be computed in O(sort(N)) I/Os. This bound holds provided that M >= 56r log(2) B. Together with an I/O-efficient planar embedding algorithm presented in [N. Zeh, I/O-Efficient algorithms for Shortest Path Related Problems, Ph. D. thesis, School of Computer Science, Carleton University, Ottawa, ON, Canada, 2002], this result is the basis for I/O-efficient solutions to many other fundamental problems on planar graphs, including breadth-first search and shortest paths [L. Arge, G. S. Brodal, and L. Toma, J. algorithms, 53 (2004), pp. 186 - 206;L. Arge, L. Toma, and N. Zeh, I/O-efficient algorithms for planar digraphs, in Proceedings of the 15th ACM Symposium on Parallelism in algorithms and Architectures, ACM, New York, 2003, pp. 85 - 93], depth-first search [L. Arge et al., J. graph algorithms Appl., 7 (2003), pp. 105 - 129;L. Arge and N. Zeh, I/O-efficient strong connectivity and depth-first search for directed planar graphs, in Proceedings of the 44th IEEE Symposium on Foundations of Computer Science, IEEE Press, Piscataway, NJ, 2003, pp. 261 - 270], strong connectivity [L. Arge and N. Zeh, I/O-efficient strong connectivity and depth-first search for directed planar graphs, in Proceedings of the 44th IEEE Symposium on Foundations of Computer Science, IEEE Press, Piscataway, NJ, 2003, pp. 261 - 270], and topological sorting [L. Arge and L. Toma, Simplified external memory algorithms for planar DAGs, in Proceedings of the 9th Scandinavian Workshop on Algorithm Theory, Lecture Notes in Comput. Sci. 3111, Springer-Verlag, Berlin, New York, 2004, pp. 493 - 503;L. Arge, L. Toma, and N. Zeh, I/O-efficient algorithms for planar digraphs, in Proceedings of the 15th ACM
Let nu be a finite set of n elements and F = {X-1, X-2, X-m} a family of m subsets of nu. Two sets X-i and X-i of F overlap if X-i boolean AND X-i not equal empty set, X-j\X-i not equal empty set, and X-i \ X-j not eq...
详细信息
Let nu be a finite set of n elements and F = {X-1, X-2, X-m} a family of m subsets of nu. Two sets X-i and X-i of F overlap if X-i boolean AND X-i not equal empty set, X-j\X-i not equal empty set, and X-i \ X-j not equal empty set. Two sets X, Y epsilon F are in the same overlap class if there is a series X = X-1, X-2, .... X-k = Y of sets of F in which each XiXj+(1) overlaps. In this note, we focus on efficiently identifying all overlap classes in O(n + Sigma(m)(i=1) vertical bar X-i vertical bar) time. We thus revisit the clever algorithm of Dahlhaus [E. Dahlhaus, Parallel algorithms for hierarchical clustering and applications to split decomposition and parity graph recognition, J. algorithms 36 (2) (2000) 205-240] of which we give a clear presentation and that we simplify to make it practical and implementable in its real worst case complexity. An useful variant of Dahlhaus's approach is also explained. (C) 2008 Elsevier B.V. All rights reserved.
The problem of finding a spanning tree with few leaves is motivated by the design of communication networks, where the cost of the devices depends on their routing functionality (ending, forwarding, or routing a conne...
详细信息
The problem of finding a spanning tree with few leaves is motivated by the design of communication networks, where the cost of the devices depends on their routing functionality (ending, forwarding, or routing a connection). Besides this application, the problem has its own theoretical importance as a generalization of the Hamiltonian path problem. Lu and Ravi showed that there is no constant factor approximation for minimizing the number of leaves of a spanning tree, unless P = NP. Thus instead of minimizing the number of leaves, we are going to deal with maximizing the number of non-leaves: we give a linear-time 2-approximation for arbitrary graphs, a 3/2-approximation for claw-free graphs, and a 6/5-approximation for cubic graphs. (c) 2007 Elsevier B.V. All fights reserved.
Given an arbitrary graph G = (V, E) and a proper interval graph H = (V, F) with E C F we say that H is a proper interval completion of G. The graph H is called a minimal proper interval completion of G if, for any san...
详细信息
Given an arbitrary graph G = (V, E) and a proper interval graph H = (V, F) with E C F we say that H is a proper interval completion of G. The graph H is called a minimal proper interval completion of G if, for any sandwich graph H, = (V, 171) with E subset of F' subset of F, H' is not a proper interval graph. In this paper we give a Omicron(n + in) time algorithm computing a minimal proper interval completion of an arbitrary graph. The output is a proper interval model of the completion. (c) 2007 Elsevier B.V. All rights reserved.
We consider all-optical networks that use wavelength-division multiplexing and employ wavelength conversion at specific nodes in order to maximize their capacity usage. We investigate the effect of allowing reroutings...
详细信息
We consider all-optical networks that use wavelength-division multiplexing and employ wavelength conversion at specific nodes in order to maximize their capacity usage. We investigate the effect of allowing reroutings on the number of necessary wavelength converters. We disprove a claim of Wilfong and Winkler [G. Wilfong, P. Winkler, Ring routing and wavelength translation, in: Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete algorithms, SODA '98, 1998, pp. 333-341] according to which reroutings do not have any effect on the number of necessary wavelength converters on bidirected networks. We show that there exist (bidirected) networks on n nodes that require Theta(n) converters without reroutings, but only O(1) converters if reroutings are allowed. We also address the cases of undirected networks and networks with shortest-path routings. In each case, we resolve the complexity of computing optimal placements of converters. (C) 2007 Elsevier B.V. All rights reserved.
Suppose that D is an acyclic orientation of the graph G. An arc of D is dependent if its reversal creates a directed cycle. Let d(D) denote the number of dependent arcs in D. Define d(min)(G) (d(max)(G)) to be the min...
详细信息
Suppose that D is an acyclic orientation of the graph G. An arc of D is dependent if its reversal creates a directed cycle. Let d(D) denote the number of dependent arcs in D. Define d(min)(G) (d(max)(G)) to be the minimum (maximum) number of d(D) over all acyclic orientations D of G. We call G fully orientable if G has an acyclic orientation with exactly k dependent arcs for every k satisfying d(min)(G) <= k <= d(max)(G). We prove that every 2-degenerate graph is fully orientable and give interpretations to their d(min). (c) 2007 Elsevier B.V. All rights reserved.
In this paper, we prove polynomial running time bounds for an Ant Colony Optimization (ACO) algorithm for the single-destination shortest path problem on directed acyclic graphs. More specifically, we show that the ex...
详细信息
In this paper, we prove polynomial running time bounds for an Ant Colony Optimization (ACO) algorithm for the single-destination shortest path problem on directed acyclic graphs. More specifically, we show that the expected number of iterations required for an ACO-based algorithm with n ants is O(1/rho n(2) log n) for graphs with n nodes and nt edges, where rho is an evaporation rate. This result can be modified to show that an ACO-based algorithm for One-Max with multiple ants converges in expected O(1/rho n(2) log n) iterations, where n is the number of variables. This result stands in sharp contrast with that of Neumann and Witt, where a single-ant algorithm is shown to require an exponential running time if rho = O(n(-1-epsilon)) epsilon > 0. (C) 2007 Elsevier B.V. All fights reserved.
We say that a graph has a matching cutset if its vertices can be coloured in red and blue in such a way that there exists at least one vertex coloured in red and at least one vertex coloured in blue, and every vertex ...
详细信息
We say that a graph has a matching cutset if its vertices can be coloured in red and blue in such a way that there exists at least one vertex coloured in red and at least one vertex coloured in blue, and every vertex has at most one neighbour Coloured in the opposite colour. In this paper we study the algorithmic complexity of a problem of recognizing graphs which possess a matching cutset. In particular we present a polynomial-time algorithm which solves this problem for graphs of diameter two. (C) 2008 Elsevier B.V. All rights reserved.
We propose a novel hierarchical clustering algorithm for data-sets in which only pairwise distances between the points are provided. The classical Hungarian method is an efficient algorithm for solving the problem of ...
详细信息
We propose a novel hierarchical clustering algorithm for data-sets in which only pairwise distances between the points are provided. The classical Hungarian method is an efficient algorithm for solving the problem of minimal-weight cycle cover. We utilize the Hungarian method as the basic building block of our clustering algorithm. The disjoint cycles, produced by the Hungarian method, are viewed as a partition of the data-set. The clustering algorithm is formed by hierarchical merging. The proposed algorithm can handle data that is arranged in non-convex sets. The number of the clusters is automatically found as part of the clustering process. We report an improved performance of our algorithm in a variety of examples and compare it to the spectral clustering algorithm. (c) 2008 Elsevier B.V. All rights reserved.
暂无评论