The longest path problem is the one that finds a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are kn...
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The longest path problem is the one that finds a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. Among those, for trees, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that the longest path problem can be solved efficiently for some tree-like graph classes by this approach. We next propose two new graph classes that have natural interval representations, and show that the longest path problem can be solved efficiently on these classes.
The connectivity problem is a fundamental problem in graph theory. The best known algorithm to solve the connectivity problem on general graphs with n vertices and m edges takes O( K(G) mn(1.5)) time, where K(G) is th...
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The connectivity problem is a fundamental problem in graph theory. The best known algorithm to solve the connectivity problem on general graphs with n vertices and m edges takes O( K(G) mn(1.5)) time, where K(G) is the vertex connectivity of G. In this paper, an efficient algorithm is designed to solve vertex connectivity problem, which takes O(n(2)) time and O(n) space for a trapezoid graph.
The Pattern Matching problem with Swaps consists in finding all occurrences of a pattern P in a text T, when disjoint local swaps in the pattern are allowed. In the Approximate Pattern Matching problem with Swaps one ...
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The Pattern Matching problem with Swaps consists in finding all occurrences of a pattern P in a text T, when disjoint local swaps in the pattern are allowed. In the Approximate Pattern Matching problem with Swaps one seeks, for every text location with a swapped match of P, the number of swaps necessary to obtain a match at the location. In this paper we devise two general algorithms for both Standard and Approximate Pattern Matching with Swaps, named CROSS-SAMPLING and BACKWARD-CROSS-SAMPLING, with a O(nm) and O(nm(2)) worst-case time complexity, respectively. Then we provide efficient implementations of them, based on bit-parallelism, which achieve O(n) and O(nm) worst-case time complexity, with patterns v-hose length is comparable to the word-size of the target machine. From an extensive comparison with. some of the most Live algorithms for the matching problem, it turns out that our algorithms a) very flexible and achieve very good results in practice.
Based on the geometric representation, an efficient algorithm is designed to find all articulation points of a permutation graph. The proposed algorithm takes only O(n log n) time and O(n) space, where n represents th...
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Based on the geometric representation, an efficient algorithm is designed to find all articulation points of a permutation graph. The proposed algorithm takes only O(n log n) time and O(n) space, where n represents the number of vertices. The proposed sequential algorithm can easily be implemented in parallel which takes O(log n) time and O(n) processors on an EREW PRAM. These are the first known algorithms for the problem on this class of graph.
An efficient parallel algorithm is presented to find a maximum weight independent set of a permutation graph which takes O(log n) time using O(n(2)/log n) processors on an EREW PRAM, provided the graph has at most O(n...
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An efficient parallel algorithm is presented to find a maximum weight independent set of a permutation graph which takes O(log n) time using O(n(2)/log n) processors on an EREW PRAM, provided the graph has at most O(n) maximal independent sets. The best known parallel algorithm takes O(log(2) n) time and O(n(3)/log n) processors on a CREW PRAM.
In this paper, an algorithm is designed to find a maximum weight independent set of a circular-arc graph with n vertices. The weights considered here are all non-negative real numbers and associated with each of the v...
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In this paper, an algorithm is designed to find a maximum weight independent set of a circular-arc graph with n vertices. The weights considered here are all non-negative real numbers and associated with each of the vertex of the graph. The proposed algorithm runs in time O(n(2)). Here we shown that the program slots of television channels during 24 hours can be modeled as a circular-arc graph. Each program represents a vertex and number of viewers of that program represents the weight of the corresponding vertex. Two vertices are connected by an edge iff the corresponding program slots have a common program time, i.e., if I-i and I-j are the program slots of two programs i and j then the corresponding vertices i and j are connected by an edge iff I-i boolean AND I-j not equal phi. We also shown that the non-overlapping program slots with maximum number of viewers can be selected by computing maximum weight independent set on the corresponding circular-arc graph.
In this paper, we consider a variant of the well-known Steiner tree problem. Given a complete graph G = (V, E) with a cost function c : E -> R+ and two subsets R and R' satisfying R' subset of R subset of V...
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In this paper, we consider a variant of the well-known Steiner tree problem. Given a complete graph G = (V, E) with a cost function c : E -> R+ and two subsets R and R' satisfying R' subset of R subset of V, a selected-internal Steiner tree is a Steiner tree which contains (or spans) all the vertices in R such that each vertex in R' cannot be a leaf. The selected-internal Steiner tree problem is to find a selected-internal Steiner tree with the minimum cost. In this paper, we present a 2 rho-approximation algorithm for the problem, where p is the best-known approximation ratio for the Steiner tree problem. (c) 2007 Elsevier B.V. All rights reserved.
A popular way to describe and build the DAWG or Directed Acyclic Word Graph of a string is by transformation of the corresponding subword tree. This transformation, which is not difficult to reverse, is easy to grasp ...
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A popular way to describe and build the DAWG or Directed Acyclic Word Graph of a string is by transformation of the corresponding subword tree. This transformation, which is not difficult to reverse, is easy to grasp and almost trivial to implement except for the assumed implication of a standard tree isomorphism algorithm. Here we point out a simple property of subword trees that makes checking tree isomorphism in this context a straightforward process, thereby simplifying the transformation significantly. Subword trees and DAWGs arise rather ubiquitously in applications of string processing, where they often play complementary roles. Efficient conversions are thus especially desirable. (C) 2001 Elsevier Science B.V. All rights reserved.
Let T = (V, E) be a tree with n nodes such that each node v is associated with a value-weight pair (val(v), w(v)), where the value val(v) is a real number and the weight w(v) is a positive integer. The density of a pa...
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Let T = (V, E) be a tree with n nodes such that each node v is associated with a value-weight pair (val(v), w(v)), where the value val(v) is a real number and the weight w(v) is a positive integer. The density of a path P = (v(1), v(2).....v(k)) is defined as Sigma(k)(i=1) val(i)/Sigma(k)(i=1)w(i). The weight of P, denoted by w(P), is Sigma(k)(i=1)w(i). Given a tree of n nodes, two integers w(min) and w(max), and a length lower bound L, we present a pseudo-polynomial O(w(max)nL)-time algorithm to find a maximum-density path P in T such that W-min <= w(P) <= w(max) and the length of P is at least L. (c) 2007 Elsevier B.V. All rights reserved.
Given a complete graph G = (V,E), with nonnegative edge costs, two subsets R subset of V and R'subset of R, a partition R = {R-1,R-2, ... ,R-k} of R, R-i boolean AND R-j = phi, i not equal j and Script capital R&#...
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Given a complete graph G = (V,E), with nonnegative edge costs, two subsets R subset of V and R'subset of R, a partition R = {R-1,R-2, ... ,R-k} of R, R-i boolean AND R-j = phi, i not equal j and Script capital R' = {R-1',R-2', ...,R-k'} of R', R-i'subset of R-i, a clustered Steiner tree is a tree T of G that spans all vertices in R such that T can be cut into k subtrees T-i by removing k - 1 edges and each subtree T-i spans all vertices in R-i, 1 <= i <= k. The cost of a clustered Steiner tree is defined to be the sum of the costs of all its edges. A clustered selected-internal Steiner tree of G is a clustered Steiner tree for R if all vertices in R-i' are internal vertices of T-i. The clustered selected-internal Steiner tree problem is concerned with the determination of a clustered selected-internal Steiner tree T for R and R' in G with minimum cost. In this paper, we present the first known approximation algorithm with performance ratio (rho + 4) for the clustered selected-internal Steiner tree problem, where rho is the best-known performance ratio for the Steiner tree problem.
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