Showing papers on "Longest path problem published in 1991"
TL;DR: A stochastic version of the classical shortest path problem whereby for each node of a graph, the authors must choose a probability distribution over the set of successor nodes so as to reach a certain destination node with minimum expected cost is considered.
Abstract: We consider a stochastic version of the classical shortest path problem whereby for each node of a graph, we must choose a probability distribution over the set of successor nodes so as to reach a certain destination node with minimum expected cost. The costs of transition between successive nodes can be positive as well as negative. We prove natural generalizations of the standard results for the deterministic shortest path problem, and we extend the corresponding theory for undiscounted finite state Markovian decision problems by removing the usual restriction that costs are either all nonnegative or all nonpositive.
578 citations
TL;DR: It is shown that the computational problem of devising a strategy that achieves a given worst-case ratio to the optimum path in a graph is a universal two-person game, and thus PSPACE-complete, whereas optimizing the expected ratio is #P-hard.
478 citations
11 Nov 1991
TL;DR: A framework which allows various previously proposed path sensitization criteria to be compared with each other in a unified way is presented and an exact path sensitized criterion and a looser path sensitizing criterion based on the framework are proposed.
Abstract: Since the delay of a circuit is determined by the delay of its longest sensitizable paths (such paths are called critical paths), the problem of estimating the delay of a circuit is called critical path problem One important aspect of the critical path problem is to decide whether a path is sensitizable A framework which allows various previously proposed path sensitization criteria to be compared with each other in a unified way is presented An exact path sensitization criterion and a looser path sensitization criterion based on the framework are also proposed >
229 citations
TL;DR: The problem of maintaining on-line a solution to the All Pairs Shortest Paths Problem in a directed graph G = (V,E) where edges may be dynamically inserted or have their cost decreased is considered and a new data structure is introduced which is able to answer queries concerning the length of the shortest path between any two vertices in constant time.
180 citations
TL;DR: It is demonstrated that, in general, there exist cases in which no finite path is optimal leading us to define an infinite path in such a way that the minimum weight problem always has a solution.
Abstract: We investigate the minimum weight path problem in networks whose link weights and link delays are both functions of time. We demonstrate that, in general, there exist cases in which no finite path is optimal leading us to define an infinite path (naturally, containing loops) in such a way that the minimum weight problem always has a solution. We also characterize the structure of an infinite optimal path. In many practical cases, finite optimal paths do exist. We formulate a criterion that guarantees the existence of a finite optimal path and develop an algorithm to find such a path. Some special cases, e.g., optimal loopless paths, are also discussed.
165 citations
TL;DR: A new technique is introduced for proving that the H -coloring problem is polynomial time decidable for some fixed graphs H .
81 citations
TL;DR: A simple O(m) algorithm is given to solve the maximum capacity path problem to find a path joining two fixed vertices of an edge weighted graph such that the minimum edge weight is maximized.
54 citations
TL;DR: The class of P4-extendible graphs is introduced, which properly contains the class of cographs, and admits a tree representation unique up to isomorphism, and can be obtained in polynomial time.
53 citations
TL;DR: A branch and bound algorithm is developed treating the general case of finding a clique of largest cardinality in an arbitrary graph and Computational experiments on random graphs show that this algorithm compares favorable to the fastest known method.
Abstract: The classical problem of finding a clique of largest cardinality in an arbitrary graph is NP-complete For that reason earlier work diverges into two directions The first concerns algorithms solving the problem for arbitrary graphs in reasonable (but exponential) time, the other restricts to special classes of graphs where polynomial methods can be found Here, the two directions are combined in a way A branch and bound algorithm is developed treating the general case Computational experiments on random graphs show that this algorithm compares favorable to the fastest known method Furthermore, it consumes only polynomial time for quite a few graph classes For some of them no polynomial solution method is given so far
51 citations
01 Jun 1991
TL;DR: Efficient sequential and parallel solutions to the problem of answer on-line queries requesting the shortest distance between any two vertices in G, based on the hammock decomposition of a planar digraph and the use of separators for computing quickly internal distances in the graph.
Abstract: We provide here efficient sequential and parallel solutions to the following problem: given a planar digraph G (with real edge weights but no negative cycles) for preprocessing, answer on-line queries requesting the shortest distance (or path) between any two vertices in G. Our algorithms for preprocessing need O(n log n + q2) space and O(n log n + q2) sequential time. (Here q is the cardinality of a set of faces of a planar embedding of G that cover all vertices.)A parallel implementation on a CREW PRAM needs also O(n log n + q2) space and runs in O(log2n) time using O(n + M(q)) processors (where M(q) is the number of processors required to multiply two q × q matrices in O(log q) time), provided that the q faces are given by the input.This enables us to achieve O(log n) time using a single processor for a “distance” query, or O(L + log n) time for a “path” query (where L is the length of the path). Note that this is a considerable improvement over previous results in the case where q = o(n). Our techniques are based on the hammock decomposition of a planar digraph and the use of separators for computing quickly internal distances in the graph. Several other results are achieved. For outerplanar graphs, our algorithms preprocess the graph in O(n logn) space and run either in O(n log n) sequential time, or in O(log2n) time using O(n) processors on a CREW PRAM. A “distance” query can be answered in O(log n) time using a single processor. A “path” query is answered in O(L + log n) time. An optimal solution is given in the case of trees. We achieve O(1) time per “distance” query andwe need O(n) sequential time, or O(log n) time and O(n/log n) processors (on an EREW PRAM) for preprocessing. A “path” query is answered in O(L) time.
48 citations
Book•
01 Jan 1991
TL;DR: In this article, the Hungarian method was used to solve the problem of optimal assignment of independent paths in a graph model, where the objective is to minimize the cost of the critical path.
Abstract: STRUCTURE OF THE GRAPH MODEL The abstract graph Geometrical realization of graphs Components Leaves Blocks The strongly connected components of directed graphs Problems OPTIMAL FLOWS Two basic problems Maximal set of independent paths The optimal assignment problem The Hungarian method Max flow-min cut Dynamic flow The mobilization problem The synthesis of flow problems Optical planning The role of the critical path Minimal cost transportation Minimal cost flows Problems GRAPHS AND MATRICES The adjacency matrix The incidence matrix The circuit matrix Interrelations between the matrices of graphs The spectrum of graphs, the complexity Linear electrical networks Further matrices associated with graphs Problems and solutions
17 Jun 1991
TL;DR: An asymptotic improvement of Fredman's algorithm by the factor of (logn/loglogn) 1Is that solves DMM in O(n3(loglog n/log n) ~/2) time, meaning that the APSP problem can be solved with the same time complexity.
Abstract: The all pairs shortest path (APSP) problem is to compute shortest paths between all pairs of vertices of a directed graph with non-negative edge costs. We present an algorithm that computes shortest distances between all pairs of vertices, since shortest paths can be computed easily as by-products as in most other algorithms. It is well-known that the time complexity of (n, n)-distance matrix multiplication (DMM) is asymptotically equal to that of the APSP problem for a graph with n vertices. See Aho, ttopcroft and Ullman [1] for example. Based on this fact, Fredma~ [5] invented an algorithm for DMM of O(n3(log log n/log n) 1/3) time, which is o(n3), meaning that the APSP problem can be solved with the same time complexity. In the average case, Moffat and Takaoka [6] solved this problem with O(n 9 log n) expected time. Our algorithm in this paper solves DMM in O(n3(loglog n/log n) ~/2) time, meaning that the APSP problem can be solved with the same time complexity. This is an asymptotic improvement of Fredman's algorithm by the factor of (logn/loglogn) 1Is. Another merit of our algorithm is that it is simple and easy to implement, whereas Fredman's algorithm is complicated and difficult to implement. Also a possible parallel implementation is mentioned. The base of logarithm is assumed to be two in this paper and fractions are rounded up if necessary.
09 Apr 1991
TL;DR: The problem of generating a path between any two points for a point object in a 2D plane filled with unknown obstacles of arbitrary shapes is discussed and the universal lower bound on the worst case path length of any algorithm is found.
Abstract: The problem of generating a path between any two points for a point object in a 2D plane filled with unknown obstacles of arbitrary shapes is discussed. This problem is termed P1. The issue of worst-case path lengths is analysed in a general setting, independent of any particular algorithm. It is shown that there are two distinct approaches available to solve P1, dividing the set of all possible algorithms that solve P1 into two disjoint classes. The minimum worst-case path length possible in each class is determined and the universal lower bound on the worst case path length of any algorithm is found. The results are shown to be useful in developing algorithms and more general problem models. >
TL;DR: A novel technique of stream contraction is used to accelerate by a factor of log n and with no increase of the processor bound the authors' previous parallel algorithm for path algebra computation in the case of the minimum cost path computation in an n -vertex graph.
TL;DR: The proposed extended Hough transform can be tuned to equivalent to well-known robust least-squares techniques, and allows efficient, approximate M-estimation.
Abstract: The problem of determining the optimal straight path between a planar set of points is considered Each point contributes to the cost of a path a value that depends on the distance between the path and the point The cost function, quantifying this dependence, can be arbitrary and may be different for different points An algorithm to solve this problem using an extension of the Hough transform is described The range of applications includes straight-line fitting to a set of points in the presence of outliers, navigation, and path planning The proposed extended Hough transform can be tuned to equivalent to well-known robust least-squares techniques, and allows efficient, approximate M-estimation >
TL;DR: It is observed that the class of functions computable in polynomial-time with one free evaluation of unbounded parallel queries to NP oracle sets, captures that complexity.
Abstract: We study the computational complexity of computing optimal solutions (the solutions themselves, not just their cost) for NP optimization problems where the costs of feasible solutions are bounded above by a polynomial in the length of their instances (we simply denote by NPOP such an NP optimization problem). It is of particular interest to find a computational structure (or equivalently, a complexity class) which. captures that complexity, if we consider the problems of computing optimal solutions for NPOP’s as a class of functions giving those optimal solutions. In this paper, we will observe that the class of functions computable in polynomial-time with one free evaluation of unbounded parallel queries to NP oracle sets, captures that complexity. We first show that for any NPOP Π, there exists a polynomial-time bounded randomized algorithm which, given an instance of Π, uses one free evaluation of parallel queries to an NP oracle set and outputs some optimal solution of the instance with very high probability. We then show that for several natural NPOP’s, any function giving those optimal solutions is at least as computationally hard as all functions in . To show the hardness results, we introduce a property of NPOP’s, called paddability, and we show a general result that if Π is a paddable NPOP and its associated decision problem is NP-hard, then all functions in are computable in polynomial-time with one free evaluation of an arbitrary function giving optimal solutions for instances of Π. The hardness results are applications of this general result. Among the NPOP’s, we include MAXIMUM CLIQUE, MINIMUM COLORING, LONGEST PATH, LONGEST CYCLE, 0–1 TRAVELING SALESPERSON, and 0–1 INTEGER PROGRAMMING.
TL;DR: This work considers some classes of network minimax problems that generalize combinatorial versions of shortest and longest path problems in networks and proposes effective polynomial-time algorithms for solving these problems.
Abstract: We consider some classes of network minimax problems that generalize combinatorial versions of shortest and longest path problems in networks. Effective polynomial-time algorithms are proposed for solving these problems.
11 Nov 1991
TL;DR: A sensitization criterion for the short path is proposed and an accurate model for calculation of signal stable range can be created, which will allow the output stable range of a gate to be the union of its inputs when the input leads hold a controlling value.
Abstract: The estimation of signal stable ranges in a combinational circuit is an important issue for determining clock time in a synchronous system. An optimal clocking period time highly depends on the accuracy of the shortest path length as well as the longest path length in a combinational circuit. In this paper, a sensitization criterion for the short path is first proposed. Based on this sensitization criterion, an accurate model for calculation of signal stable range can be created. This will allow the output stable range of a gate to be the union of its inputs when the input leads hold a controlling value, rather than to be always the intersection. Then, an LS-algorithm for calculation of signal stable ranges is presented in which both the sensitizable shortest path and the sensitizable longest path are considered. It avoids the exhaustive search by tracing the path sensitization and eliminates some conservative restriction to get more accurate results in a more efficient way, compared to the previous approaches. The speedup and the improved accuracy of the proposed LS-algorithm showed promising experimental results. >
01 Jun 1991
TL;DR: In this article, the problem of finding the shortest path between two points in a plane containing obstacles is considered, and a finite list of homotopy classes is obtained whose union contains a shortest path.
Abstract: : The problem of finding the shortest path between two points in a plane containing obstacles is considered. The set of such paths is uncountably infinite, making an exhaustive search impossible. This difficulty is overcome by reducing the size of the search space. The search is first restricted to a countably infinite set by focusing attention on the set of homotopy classes. By applying simple optimality principles, a finite list of such classes is obtained whose union contains the shortest path. This process of simplification is accomplished by modeling the topology of the region with a graph. Optimality principles come into play during a graph traversal which is used to produce the finite search list. In addition, a computational investigation of two methods by which homotopy classes can be named is discussed, and properties of the graph models are investigated. The thesis of CPT Andre M. Cuerington, U.S. Army, calculates the actual shortest path using the search list produced here.
TL;DR: An efficient reduction process for path problems on circular-arc graphs is introduced, which gives anO(n+m) algorithm for proper circular- Arc graphs, and an O(n-m) algorithms for general circular-Arc graphs.
Abstract: An efficient reduction process for path problems on circular-arc graphs is introduced. For the parity path problem, this reduction gives anO(n+m) algorithm for proper circular-arc graphs, and anO(n+m) algorithm for general circular-arc graphs. This reduction also gives anO(n+m) algorithm for the two path problem on circular-arc graphs.
08 Jan 1991
TL;DR: An interconnection network based on a star graph of arbitrary order n, which belongs to a family of Cayley graphs, in which vertices correspond to elements of a permutation group, and edges correspond to generators of the group, is analysed.
Abstract: An interconnection network based on a star graph of arbitrary order n, is analysed. Star graphs belong to a family of Cayley graphs, in which vertices correspond to elements of a permutation group, and edges correspond to generators of the group. Generators of the star graph of order n are n-1 transpositions, which may be represented with a star transposition tree. The algebraic properties of these graphs are investigated through analysis of the corresponding transposition tree. The algebraic expression for the shortest path between two nodes is found, and it is shown that every shortest path consists of a number of subpaths, which may be combined in an arbitrary order. From the analysis of shortest paths follows the minimum number of virtual channels, into which every physical channel must be split in order to avoid deadlock. Finally, a routing algorithm is developed which repetitively extracts subpaths, by sorting the source permutation to the destination permutation. >
TL;DR: The lexicographically first maximal (lfm) induced path problem is shown and the problem restricted dags with degree 3 is shown in NC2 while the problem for degree 4 is P-complete.
11 Jun 1991
TL;DR: In this article, an exact algorithm for module allocation on module allocation graphs that are comparability graphs is presented, which takes into account the first-order effects of the interconnection weights, and a description of the constraints on the library and the schedule for the module allocation graph to be comparability graph is presented.
Abstract: A description is presented of the constraints on the library and the schedule for the module allocation graphs to be comparability graphs. An exact algorithm for module allocation on module allocation graphs that are comparability graphs is presented. This algorithm takes into account the first-order effects of the interconnection weights. >
TL;DR: In this paper, the reaction and chemical distances between isomeric molecular graphs are studied and the principle of minimum structural change, often used in mechanistic organic chemistry, is introduced.
Abstract: The distance between molecular graphsG andG′ is equal to the length of a minimum path connecting these molecular graphs in the so-called graphs of distances. The graph of distances has vertices which are molecular graphs taken from the same family of isomeric graphs, and two molecular graphs are adjacent if there exists a prototype reaction graph which transforms one into the other. Distances may alternatively be determined by applying the concept of common supergraphs. In particular, the reaction and chemical distances between isomeric molecular graphs are studied. These distances allow us to simply incorporate the principle of minimum structural change, often used in mechanistic organic chemistry.
17 Jun 1991
TL;DR: The proposed data structures allow us to compute after an update the shortest-path tree rooted at an arbitrary query node in time O(n√log log n) and to perform an update in O((log n)3).
Abstract: We propose data structures for maintaining shortest path in planar graphs in which the weight of an edge is modified. Our data structures allow us to compute after an update the shortest-path tree rooted at an arbitrary query node in time O(n√log log n) and to perform an update in O((log n)3). Our data structure can be also applied to the problem of maintaining the maximum flow problem in an s − t planar network.
08 Apr 1991
TL;DR: To efficiently process recursive queries in a DBMS (database management system), a parallel, direct transitive closure algorithm is proposed by reorganizing the computation order of Warren's algorithm.
Abstract: To efficiently process recursive queries in a DBMS (database management system), a parallel, direct transitive closure algorithm is proposed. Efficiency is obtained by reorganizing the computation order of Warren's algorithm. The number of transfers among processors depends only on the number of processors and does not depend on the depth of the longest path. The evaluation shows an improvement due to the parallelism and the superiority of the proposed algorithm over recent propositions. The speed of the production of new tuples is very high and the volume of transfers between the sites is reduced. >
TL;DR: A fast algorithm is given for the all-pairs shortest paths problem for banded matrices having band-width b that solves the negative-cycle problem and calculates all path lengths within the band in O(nb2) time.
01 Jan 1991
TL;DR: A semiautonomous digit collection and call state detection subsystem is provided for use in a time division switching system such as a PBX.
Abstract: A semiautonomous digit collection and call state detection subsystem is provided for use in a time division switching system such as a PBX. Each active (call serving) system port circuit applies a hook state signal to a specially provided bus during each occurrence of the time slot to which the port circuit is assigned. The subsystem's logic analyzes these hook state signals to identify certain predetermined call states and to count dial pulses. Output messages representing identified call states and dialed digits are made available to the switching system under control of a system scanner.