K. Subramani

A faster algorithm for the single source shortest path problem with few distinct positive lengths

In this paper, we propose an efficient method for implementing Dijkstras algorithm for the Single Source Shortest Path Problem (SSSPP) in a graph whose edges have positive length, and where there are few distinct edge lengths. The SSSPP is one of the most widely studied problems in theoretical computer science and operations research. On a graph with n vertices, m edges and K distinct edge lengths, our algorithm runs in O(m) time if nK=<2m, and O(mlognKm) time, otherwise. We tested our algorithm against some of the fastest algorithms for SSSPP on graphs with arbitrary but positive lengths. Our experiments on graphs with few edge lengths confirmed our theoretical results, as the proposed algorithm consistently dominated the other SSSPP algorithms, which did not exploit the special structure of having few distinct edge lengths.

A Zero-Space algorithm for Negative Cost Cycle Detection in networks

This paper is concerned with the problem of checking whether a network with positive and negative costs on its arcs contains a negative cost cycle. The Negative Cost Cycle Detection (NCCD) problem is one of the more fundamental problems in network design and finds applications in a number of domains ranging from Network Optimization and Operations Research to Constraint Programming and System Verification. As per the literature, approaches to this problem have been either Relaxation-based or Contraction-based. We introduce a fundamentally new approach for negative cost cycle detection; our approach, which we term as the Stressing Algorithm, is based on exploiting the connections between the NCCD problem and the problem of checking whether a system of difference constraints is feasible. The Stressing Algorithm is an incremental, comparison-based procedure which is as efficient as the fastest known comparison-based algorithm for this problem. In particular, on a network with n vertices and m edges, the Stressing Algorithm takes O(m@?n) time to detect the presence of a negative cost cycle or to report that none exists. A very important feature of the Stressing Algorithm is that it uses zero extra space; this is in marked contrast to all known algorithms that require@W(n)extra space. It is well known that the NCCD problem is closely related to the Single Source Shortest Paths (SSSP) problem, i.e., the problem of determining the shortest path distances of all the vertices in a network, from a specified source; indeed most algorithms in the literature for the NCCD problem are modifications of approaches to the SSSP problem. At this juncture, it is not clear whether the Stressing Algorithm could be extended to solve the SSSP problem, even if O(n) extra space is available.

Randomized algorithms for finding the shortest negative cost cycle in networks

Abstract In this paper, we design and analyze a fast, randomized algorithm for the problem of finding a negative cost cycle having the smallest number of edges in a directed, weighted graph. This problem will henceforth be referred to as the Shortest Negative Cost Cycle problem (SNCC). SNCC is closely related to the problem of checking whether a directed, weighted graph contains a negative cost cycle (NCCD). NCCD is an extremely well-studied problem within the domains of operations research and theoretical computer science. SNCC is important in its own right and finds several applications in program verification (Satisfiability modulo theories), abstract interpretation and real-time scheduling. It is also an important subroutine in solving the generalized submodular flow problem, which has applications in trading networks. The randomized algorithm presented in this paper for SNCC determines a shortest negative cost cycle with probability at least ( 1 − e − 1 ) in O ( m ⋅ n ⋅ log n ) time, on a network with n vertices and m edges. This is, in general, a significant improvement over the best deterministic bound of O ( m ⋅ n ⋅ | C ∗ | ) over the same parameters, where C ∗ is a shortest negative cost cycle. This algorithm requires Ω ( n ⋅ log n ) random bits. We then propose a second randomized algorithm that runs in O ( m ⋅ n ⋅ log n ) expected time and requires O ( n ) random bits (expected).

Improved Algorithms for Detecting Negative Cost Cycles in Undirected Graphs

In this paper, we explore the design of algorithms for the problem of checking whether an undirected graph contains a negative cost cycle (UNCCD). It is known that this problem is significantly harder than the corresponding problem in directed graphs. Current approaches for solving this problem involve reducing it to either the b -matching problem or the T -join problem. The latter approach is more efficient in that it runs in O (n 3) time on a graph with n vertices and m edges, while the former runs in O (n 6) time. This paper shows that instances of the UNCCD problem, in which edge weights are restricted to be in the range { *** K ··K } can be solved in O (n 2.75·logn ) time. Our algorithm is basically a variation of the T -join approach, which exploits the existence of extremely efficient shortest path algorithms in graphs with integral positive weights. We also provide an implementation profile of the algorithms discussed.

On the negative cost girth problem in planar networks

In this paper, we discuss an efficient divide-and-conquer algorithm for the negative cost girth (NCG) problem in planar networks. Recall that the girth of an unweighted graph (directed or undirected) is the length of the shortest cycle in the graph. We extend the notion of girth to arbitrarily weighted networks as follows: The negative cost girth of a graph is the number of edges in a shortest negative cost simple cycle, i.e., a negative cost cycle having the fewest number of edges. The NCG problem in general networks has been well-studied, and there exist several algorithms for this problem. Clearly, the extant algorithms for the NCG problem can be used when the input network is restricted to be planar. However, the techniques used in this paper result in an algorithm with a superior running time. Our algorithm is based on the well-known Lipton-Tarjan planar separator theorem. On a directed, weighted, planar network G = { V , E , c } with n vertices, m = O ( n ) edges, and negative cost girth k, the algorithm runs in O ( n 1.5 ? k ) time. This is a significant improvement over the O ( m ? n ? k ) = O ( n 2 ? k ) running time that results, when the fastest known topology-oblivious algorithm for the NCG problem is restricted to planar networks. Additionally, our algorithm can be extended to find the NCG in selected generalizations of planar networks.

On Clustering Without Replication in Combinatorial Circuits

In this paper, we consider the problem of clustering combinatorial circuits for delay minimization, when logic replication is not allowed CN. The problem of delay minimization when logic replication is allowed CA has been well studied, and is known to be solvable in polynomial-time [8]. However, unbounded logic replication can be quite expensive. Thus, CN is an important problem. We show that selected variants of CN are NP-hard. We also obtain approximability and inapproximability results for these problems.

A new algorithm for the minimum spanning tree verification problem

This paper proposes a new algorithm for the minimum spanning tree verification (MSTV) problem in undirected graphs. The MSTV problem is distinct from the minimum spanning tree construction problem. The above problems have been studied extensively, and there exist several papers in the literature devoted to them. Our algorithm for the MSTV problem combines the insights of Borůvka’s algorithm for constructing a minimum spanning tree with a bucketing scheme. The principal idea underlying this combination is the efficient identification of edges that cannot be part of any minimum spanning tree. Although the proposed algorithm imposes no restrictions on the input graph, it was designed to exploit the case in which the number of distinct edge weights is small. On a graph with

Two-level heaps: a new priority queue structure with applications to the single source shortest path problem

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