Showing papers on "Longest path problem published in 1988"
TL;DR: An approximate solution to the weighted-graph-matching problem is discussed for both undirected and directed graphs and an analytic approach is used instead of a combinatorial or iterative approach to the optimum matching problem.
Abstract: An approximate solution to the weighted-graph-matching problem is discussed for both undirected and directed graphs. The weighted-graph-matching problem is that of finding the optimum matching between two weighted graphs, which are graphs with weights at each arc. The proposed method uses an analytic instead of a combinatorial or iterative approach to the optimum matching problem. Using the eigendecompositions of the adjacency matrices (in the case of the undirected-graph-matching problem) or Hermitian matrices derived from the adjacency matrices (in the case of the directed-graph-matching problem), a matching close to the optimum can be found efficiently when the graphs are sufficiently close to each other. Simulation results are given to evaluate the performance of the proposed method. >
777 citations
TL;DR: Eight algorithms which solve the shortest path tree problem on directed graphs are presented, together with the results of wide-ranging experimentation designed to compare their relative performances on different graph topologies.
Abstract: Theshortest path problem is considered from a computational point of view. Eight algorithms which solve theshortest path tree problem on directed graphs are presented, together with the results of wide-ranging experimentation designed to compare their relative performances on different graph topologies. The focus of this paper is on the implementation of the different data structures used in the algorithms. A "Pidgin Pascal" description of the algorithms is given, containing enough details to allow for almost direct implementation in any programming language. In addition, Fortran codes of the algorithms and of the graph generators used in the experimentation are provided on the diskette.
364 citations
TL;DR: Efficient deterministic techniques for breaking symmetry in parallel are used to construct fast linear processor algorithms for several problems, including the problem of $(Delta + 1)$-coloring constant-degree graphs and 5-colors planar graphs.
Abstract: This paper describes efficient deterministic techniques for breaking symmetry in parallel. These techniques work well on rooted trees and graphs of constant degree or genus. The primary technique allows us to 3-color a rooted tree in $O( \lg^* n )$ time on an EREW PRAM using a linear number of processors. These techniques are used to construct fast linear processor algorithms for several problems, including the problem of $( \Delta + 1)$-coloring constant-degree graphs and 5-coloring planar graphs. Lower bounds for 2-coloring directed lists and for finding maximal independent sets in arbitrary graphs are also proved.
266 citations
13 Jun 1988
TL;DR: The test results from the measurement of sample programs show that the knowledge of the critical path in a program's execution helps users identify performance problems and better understand the behavior of a program.
Abstract: The authors present the design, implementation, and testing of the critical path analysis technique using the IPS performance measurement tool for parallel and distributed programs. They create a precedence graph of a program's activities (program activity graph) with the data collected during the execution of a program. The critical path, the longest path in the program activity graph, represents the sequence of the program activities that take the longest time to execute. Various algorithms are developed to track the critical path from this graph. The events in this path are associated with the entities in the source program, and the statistical results are displayed on the basis of the hierarchical structure of the IPS. The test results from the measurement of sample programs show that the knowledge of the critical path in a program's execution helps users identify performance problems and better understand the behavior of a program. >
160 citations
TL;DR: This paper restricts ourselves to directed acyclic graphs (DAGs), and shows how to return an arbitrarily chosen path between couples of nodes (if it exists) during the deletion of edges in ark efficient manner.
111 citations
90 citations
IBM1
TL;DR: It is shown that for directed graphs, reachability can not be expressed by an existential monadic second-order sentence, but for directedGraphs with degree at most k, reachable is expressible by anexistential monadicSecond- order sentence.
Abstract: It is shown that for directed graphs, reachability can not be expressed by an existential monadic second-order sentence. The proof makes use of Ehrenfeucht-Fraisse games, along with probabilistic. However, it is shown that for directed graphs with degree at most k, reachability is expressible by an existential monadic second-order sentence. One reason for the interest in the main result is that while there is considerable empirical evidence (in terms of the efficiency of algorithms that have been discovered) that reachability in directed graphs is 'harder' than reachability in undirected graphs, this is the first proof in a precise technical sense that this is so. >
88 citations
TL;DR: In this paper, a reduction of the problem to an unrestricted shortest path problem in a special network is constructed in O(n logn) steps wheren is the number of obstacles.
Abstract: It is well known that job-shop scheduling problems with two jobs can be formulated as shortest path problems with obstacles in the plane. A reduction of this problem to an unrestricted shortest path problem in a special networkN is constructed inO (n logn) steps wheren is the number of obstacles. The shortest path inN can be found in timeO (n).
85 citations
TL;DR: An efficient and flexible algorithm is presented for finding a k shortest loopless path with distinct initial links from one node to each other node, showing it to offer a substantial improvement over applying known algorithms to the problem.
Abstract: An efficient and flexible algorithm is presented for finding a k shortest loopless path with distinct initial links from one node to each other node. Low-order polynomial bounds are established for the worst-case time complexity of the algorithm, showing it to offer a substantial improvement over applying known algorithms to the problem. The algorithm can incorporate various extensions, including the ability to handle an algebraic objective, which enhance its applicability to diverse network models. In addition, the k shortest path formulation and algorithm are proposed as a basis for network survivability measures where path length bounds exist. >
85 citations
TL;DR: It is observed that the optimum solution of the minimum cost-reliability ratio path problem is an efficient extreme solution of a bicriteria path problem and the algorithm is shown to be pseudo-polynomial.
55 citations
TL;DR: Balasubramanian et al. as discussed by the authors extended the characteristic polynomials of ordinary (nonweighted) graphs to edge-weighted graphs, heterographs, graphs with loops, directed graphs, and signed graphs.
Abstract: The computer code developed previously (K. Balasubramanian, J. Computational Chern., 5,387 (1984)) for the characteristic polynomials of ordinary (nonweighted) graphs is extended in this investigation to edge-weighted graphs, heterographs (vertex-weighted), graphs with loops, directed graphs, and signed graphs. This extension leads to a number of important applications of this code to several areas such as chemical kinetics, statistical mechanics, quantum chemistry of polymers, and unsaturated systems containing heteroatoms which include bond alternation. The characteristic polynomials of several edgeweighted graphs which may represent conjugated systems with bond alternations, heterographs (molecules with heteroatoms), directed graphs (chemical reaction network), and signed graphs and lattices are obtained for the first time. I. INTRODUCTION The evaluation of the characteristic polynomials of graphs has been the subject of numerous inve~tigationsl-~~ in recent years. The evaluation of these polynomials for graphs is generally regarded as a tedious problem owing to the combinatorial complexity involved in this problem. There are a number of chemical applications of these polynomials, many of which are discussed in the literature,'-34 particularly the more recent references. In summary, the characteristic polynomials have applications in chemical kinetics,37 quantum chemistry, dynamics of oscillatory reactions, and in determining the stabilities of reaction networks,35 lattice statistic^,^^ estimating the stabilities of conjugated system~,~~ formulation of TEMO theorem,39 enumeration of walks and self-returning walks,3 and electronic structure of organic polymers and periodic stu~tures.~~" In earlier investigations, we discussed an algorithm2 and computer code' to generate the characteristic polynomials of graphs containing a large number of vertices with very little computer time using an elegant recur
01 Jan 1988
TL;DR: In this article, a path-planning algorithm for a mobile robot between start and goal positions in a workspace is presented, which runs on the quadtree representation via a path graph.
Abstract: Determination of the shortest collision- free path for a mobile robot between start and goal positions in a workspace is central to design of an autonomous mobile robot. This paper presents a feasible path-planning algorithm which runs on the quadtree representation via a path graph. The quadtree representing the workspace is obtained from fast conversion of a real image taken through a camera on the ceiling. Thus, the algorithm can run even when obstacles in the workspace are shifted frequently. The quadtree also integrates both obstacle regions and other regions in the workspace with a hierarchical structure in positioning. By using the hierarchical structure, the mobile robot is reduced to a point and then the forbidden regions where the point robot cannot enter into are also understood in the quadtree. Hence, the algorithm can select the shortest collision-free path out of the quadtree, which is to be a line between given two positions. Moreover, the algorithm searches in the quadtree by way of a path graph when it selects the shortest collision-free line out of the quadtree. Namely, the path graph is initially defined by an arc with start and goal position's nodes, and the proposed algorithm gradually builds the path graph on the quadtree by the following processes synchronously: 1) Selection of the shortest path out of the path graph; and 2) Expansion of a part of the path graph along the shortest path so as to avoid the forbidden regions in the quadtree. By this synchronism, the algorithm can keep the size of the path graph as small as possible in connection with given two positions, and consequently it runs fast enough to be used practically even in any cluttered workspace. Finally, several experimental results show that the proposed algorithm is superior to some conventional algorithms in respect of calculation time.
TL;DR: Finite automata theory is made use of in order to solve some shortest path problems under constraint in valuated graphs.
01 Jan 1988
TL;DR: In this article, it was shown that there is a Hamilton path and a cycle of length at least p-1 in any regular multipartite tournament of order p, where p is the number of players.
Abstract: In this paper, the following results will be shown: 1 There is a Hamilton path and a cycle of length at least p —1 in any regular multipartite tournament of order p; (i) There is a longest path U O ,…, u t in any oriented graph such that d − (u O ) + d + (u t ) ≤ t.
01 Jun 1988
TL;DR: A new algorithm is given which solves the longest path problem in the constraint graph and resolves all positive cycles either by constraint relaxation or by jog generation and the amount of additional effort required for the over-constraint resolution is very low.
Abstract: The occurrence and resolution of over-constraints in one-dimensional layout compaction will be discussed. A new algorithm is given which solves the longest path problem in the constraint graph and resolves all positive cycles either by constraint relaxation or by jog generation. The presented algorithm eliminates the positive cycles one at a time and reuses the intermediate results, which had been obtained up to the point when the cycle was detected. Thus, the amount of additional effort required for the over-constraint resolution is very low. Some typical over-constrained situations and their solutions are shown. The algorithm has been implemented in a mask compaction program that is currently in use for process migration and design rule error correction at SIEMENS. CPU-times gained during the process migration of a microcontroller layout are given.
TL;DR: Two subproblems that arise when routing channels with interchangeable terminals are shown to be NP-hard are determining whether there is a net-to-terminal assignment that results in an acyclic vertical and constraint graph and obtaining net- To-Terminal assignments for which the length of the longest path in the vertical constraint graph is minimum.
Abstract: Two subproblems that arise when routing channels with interchangeable terminals are shown to be NP-hard. These problems are: (1) determining whether there is a net-to-terminal assignment that results in an acyclic vertical and constraint graph and (2) for instances with acyclic vertical constraint graphs, obtaining net-to-terminal assignments for which the length of the longest path in the vertical constraint graph is minimum. >
15 Jun 1988
TL;DR: In this article, two new proofs of the theorem are given, one of which has the merit that it uses only the property that an isomorphism from one tree to another maps a path of length d to a path with length d. The other proof is by induction, and it rapidly shows that it is sufficient to examine the unicyclic graphs with the condition that every second vertex on the cycle has valency two.
Abstract: A theorem characterizing graphs with only one isomorphism class of spanning trees was given in [2], [3], [5], [6], and [14]. Here, two new proofs of the theorem are given, one of which has the merit that it uses only the property that an isomorphism from one tree to another maps a path of length d to a path of length d. The other proof is by induction, and it is interesting because it rapidly shows that it is sufficient to examine the unicyclic graphs with the property that every second vertex on the cycle has valency two. Two new analogues of the theorem are stated: one with labelled vertices (given in the induction proof) and one for oriented graphs. A counterexample shows that the characterization does not hold for infinite graphs, but a conjecture is stated.
04 Jul 1988
TL;DR: This paper discusses some results on maintaining paths in dynamic graphs, considers paths problems on dynamic labeled graphs and shows how to maintain path expressions in the acyclic case when insertions of new arcs are allowed.
Abstract: In several applications it is necessary to deal with data structures that may dinamically change during a sequence of operations. In these cases the classical worst case analysis of the cost of a single operation may not adequately describe the behaviour of the structure but it is rather more meaningful to analyze the cost of the whole sequence of operations. In this paper we first discuss some results on maintaining paths in dynamic graphs. Besides, we consider paths problems on dynamic labeled graphs and we show how to maintain path expressions in the acyclic case when insertions of new arcs are allowed.
21 Dec 1988
TL;DR: The two path problem (TPP) is to determine whether there exist two vertex disjoint paths connecting s with t and u with v and to find such paths if they exist.
Abstract: Let G= (V,E) be a finite undirected graph with four distinguished vertices s, t, u, v. The two path problem (TPP) is to determine whether there exist two vertex disjoint paths connecting s with t and u with v and to find such paths if they exist.
TL;DR: The concept of Fibonacci graphs introduced and developed by this author is critically reviewed in this paper, which leads to modulation of the polynomial of a graph with the path of a path, leading to an easypencil-and-paper method of calculating characteristic, matching, counting, sextet, rook, color and king polynomials.
Abstract: The concept of Fibonacci graphs introduced and developed by this author is critically reviewed. The concept has been shown to provide an easypencil-and-paper method of calculating characteristic, matching, counting, sextet, rook, color and king polynomials of graphs of quite large size with limited connectivities. For example, the coefficients of the matching polynomial of 18-annuleno—18-annulene can be obtainedby hand using the definition of Fibonacci graphs. They are (in absolute magnitudes): 1, 35, 557, 5337, 34 361, 157 081, 525 296, 1304 426, 2 416 571, 3 327 037, 3 362 528, 2 440 842, 1 229 614, 407 814, 81936, 8652, 361, 3. The theory of Fibonacci graphs is reviewed in an easy and detailed language. The theory leads to modulation of the polynomial of a graph with the polynomial of a path.
TL;DR: A succinct linear notation system to encode the structure of a graph requires a prior canonical numbering of the graph nodes based on the lengths of a longest path and path branches, and uses locants to designate branch positions and cyclicity.
Abstract: A succinct linear notation system to encode the structure of a graph is exemplified. The notation requires a prior canonical numbering of the graph nodes based on the lengths of a longest path and path branches, and uses locants to designate branch positions and cyclicity. An algorithm and computer program to obtain the longest paths and a spanning tree containing a longest path is described. An index which measures the similarity of a pair of graphs is defined based on a comparison of their linear codes.
14 Nov 1988
TL;DR: A method is proposed that divides the collision free space into convex polygons that can be considered as path segments that leads to a connective graph indicating the collisionfree paths between the free edges.
Abstract: A method is proposed that divides the collision free space into convex polygons that can be considered as path segments. A path segment is generated for each pair of free edges on the same polygon. The path segments sharing a common free edge are then linked. This leads to a connective graph indicating the collision free paths between the free edges. When new data about the world are obtained, it is only necessary to update the convex polygons and their path segments on which new information is available and to modify the links connecting these path segments to others. >
TL;DR: In this article, a hierarchical approach for the shortest collision-free path construction problem is proposed, which generalizes some earlier algorithms and allows for incremental improvement of the planned path, and introduces the concept of k-th neighborhood, k -th visibility graph, and k −th shortest path.
01 Jan 1988
TL;DR: This paper considers the shortest-path problem in networks in which the delay of the edges changes with time according to continuous functions and investigates and analyzes protocols that exchange local time-delay functions only within limited intervals yet allow every node to calculate its representation in the shortest path in time.
Abstract: This paper addresses algorithms for networks whose behavior changes with time according to known functions. Because the computation depends on the same functions it attempts to compute, its execution must obey strict timing constraints. When distributed versions of such algorithms are considered, a key difficulty is how to transfer local timing functions among the participating nodes. To that end it is necessary to characterize the parameterization of the functions and accommodate this parameterization in the computation. In particular, we consider the shortest-path problem in networks in which the delay of the edges changes with time according to continuous functions. We present distributed protocols for finding the shortest and minimum delay path under various waiting constraints. We investigate and analyze protocols that exchange local time-delay functions only within limited intervals yet allow every node to calculate its representation in the shortest path in time for it to be used.
08 Jul 1988
TL;DR: This paper presents the Guibas-Kung-Thompson systolic array for computing the reflexive and transitive closure of a binary relation and introduces a more general class of all-pairs shortest paths problems in complete semi-rings which can not be solved using the previous array.
Abstract: This paper deals with systolic algorithms for some path-finding problems. First we present the Guibas-Kung-Thompson systolic array for computing the reflexive and transitive closure of a binary relation. Then we introduce a more general class of all-pairs shortest paths problems in complete semi-rings which can not be solved using the previous array. We introduce the well-known Gauss-Jordan algorithm to solve this general class of problems, and we show how to map it onto a systolic array whose performances overcome those of all the systolic arrays previously introduced in the literature.
TL;DR: In this article, the authors discuss criticality with respect to properties and property types in general, and construct new critical graphs from old by replacing edges by other graphs, in particular the property type thickness.
Abstract: We discuss criticality with respect to properties and property types in general, and we construct new critical graphs from old by replacing edges by other graphs. In particular, the property type thickness is investigated.
01 Jan 1988
TL;DR: An approach to robot path planning is presented which can be applied to any holonomically constrained mechanical system and a bound on the level of the search graph required for convergence is presented.
Abstract: An approach to robot path planning is presented which can be applied to any holonomically constrained mechanical system. The resulting path n composed of an n-dimensional web of stepwise movements along free edges of hypercubes. The main computational function required for the algorithm is the calculation of free intervals for the configuration variables, starting from some initial position. The algorithm is shown to find a path if one exists, and a bound on the level of the search graph required for convergence is presented. The maximum number of computations required for convergence of the algorithm is shown to depend on the amount of hspace surrounding the obstacle-free path.
01 Jan 1988
TL;DR: The Critical Path is the longest path through the construction activities to complete the project and activities on the critical path have zero float, which means that they cannot slip without delaying the completion of the project.
Abstract: Today when speaking of construction scheduling we are normally referring to the critical path method. The Critical Path is the longest path through the construction activities to complete the project. Because of the complexity of most projects the computations are performed on computers using programs especially designed for this task. The critical path method provides information on the earliest and latest start and finish dates for each activity. For dates on the critical path the early and late dates are the same. We say that activities on the critical path have zero float, which means that they cannot slip without delaying the completion of the project.