Showing papers on "Longest path problem published in 1986"

An O ( ND ) difference algorithm and its variations

Abstract: The problems of finding a longest common subsequence of two sequencesA andB and a shortest edit script for transformingA intoB have long been known to be dual problems. In this paper, they are shown to be equivalent to finding a shortest/longest path in an edit graph. Using this perspective, a simpleO(ND) time and space algorithm is developed whereN is the sum of the lengths ofA andB andD is the size of the minimum edit script forA andB. The algorithm performs well when differences are small (sequences are similar) and is consequently fast in typical applications. The algorithm is shown to haveO(N+D 2) expected-time performance under a basic stochastic model. A refinement of the algorithm requires onlyO(N) space, and the use of suffix trees leads to anO(N logN+D 2) time variation.

On shortest paths in polyhedral spaces

Abstract: We consider the problem of computing the shortest path between two points in two- or three-dimensional space bounded by polyhedral surfaces. In the 2-D case the problem is easily solved in time $O(n^2 \log n)$. In the general 3-D case the problem is quite hard to solve, and is not even discrete; we present a doubly-exponential procedure for solving the discrete subproblem of determining the sequence of boundary edges through which the shortest path passes. Finally we consider a favorable special case of the 3-D shortest path problem, namely that of finding the shortest path between two points along the surface of a convex polyhedron, and solve it in time $O(n^3 \log n)$.

The complexity of reliability computations in planar and acyclic graphs

Abstract: We show that the problem of computing source-sink reliability is NP-hard, in fact # P-complete, even for undirected and acyclic directed source-sink planar graphs having vertex degree at most three. Thus the source-sink reliability problem is unlikely to have an efficient algorithm, even when the graph can be laid out on a rectilinear grid.

Shortest paths in networks with exponentially distributed arc lengths

Abstract: This paper develops methods for the exact computation of the distribution of the length of the shortest path from a given source node s to a given sink node t in a directed network in which the arc lengths are independent and exponentially distributed random variables. A continuous time Markov chain with a single absorbing state is constructed from the original network such that the time until absorption into this absorbing state starting from the initial state is equal to the length of the shortest path in the original network. It is shown that the state space of this Markov chain is the set of all minimal (s, t) cuts in the network and that its generator matrix is upper triangular. Algorithms are described for computing the distribution and moments of the length of the shortest path based on this Markov chain representation. Algorithms are also developed for computing the probability that a given (s, t) path is the shortest path in the network and for computing the conditional distribution of the length of a path given that it is the shortest (s, t) path in the network. All algorithms are numerically stable and are illustrated by several numerical examples.

Shortest Path Solves Translation Separability of Polygons

Abstract: A magnetohydrodynamic generator is shown in which a superconductive magnet is employed to create a magnetic field normal to the direction of flow of a high temperature, conductive gas stream. The magnet comprises opposed coils embraced by U-shaped iron frames which essentially neutralize the forces of attraction and repulsion created in these coils. The coils are mounted in Dewars, the inner casings and outer casings of which are interconnected by a spoke rod system which minimizes heat losses of liquid helium which is circulated through the inner casings to maintain the coils in a super cooled condition.

Extension of the parallel nested dissection algorithm to path algebra problems

Abstract: This paper extends the author's parallel nested dissection algorithm of [PR] originally devised for solving sparse linear systems. We present a class of new applications of the nested dissection method, this time to path algebra computations, (in both cases of single source and all pair paths), where the path algebra problem is defined by a symmetric matrix A whose associated graph G with n vertices is planar. We substantially improve the known algorithms for path algebra problems of that general class: {fx470-1}

Connectivity of finite anisotropic random graphs and directed graphs

Abstract: For graphs on a finite set of vertices with arbitrary probabilities of independently occurring edges, the reliability is defined as the probability that the graph is connected, and the redundancy as the expected number of spanning trees of the graph. Analogous measures of connectivity are defined for random finite directed graphs with arbitrary probabilities of independently occurring directed edges. Recursive formulas for computing the reliability are known. Determinantal formulas, based on matrix-tree theorems, for computing the redundancy are given here. Among random graphs with a given sum of edge probabilities, the more evenly the probabilities are distributed over potential edges, the larger the redundancy. This inequality, proved using the theory of majorization, in combination with examples shows unexpectedly that conficts between reliability and redundancy can arise in the design of communication networks modelled by such random graphs. The significance of these calculations for the command and control of nuclear forces is sketched.

Minimum Number Of Arcs In Conditional Monte Carlo Sampling Of Stochastic Networks

Abstract: Consider a stochastic activity network, such as PERT network. This paper deals with the problem of estimating the distribution function (df) of the duration of the longest path in the stochastic ne...

Generalised arrays and shortest path problems

Abstract: Real world problems are often formulated as flow optimization problems on large networks, with thousands of nodes but with a restricted number of arcs. The average number of arcs having a given node as origin or terminal node (node degree) may be limited even to a few units. Standard optimization methods, mainly matrix methods for complete or strongly connected graphs, may become unpractical; recent results have shown the superiority of advanced methods based on suitable data structures.In this study the classical shortest path problem, one of the basic techniques in network flow optimization, is approached, in APL environment, using: i) standard matrix methods, ii) methods that handle irregular structures element by element, and iii) advanced methods using generalized arrays. Preliminary results indicate which approach should be preferred, according to problem characteristics and size.

A note on the shortest path through a given link in a network

Abstract: A network, or a graph, can be described by a set of nodes or vertices and a set of links or edges. Let uv be two vertices and xy be a link in a given undirected network. We show that a shortest path between u and v through xy can be found by transforming the problem into a minimum cost network flow problem. It may be noted that when the network is directed, the problem is NP -complete.