Showing papers on "Longest path problem published in 1993"

Shortest paths among obstacles in the plane

Abstract: We give a subquadratic (O(n5/3+e) time and space) algorithm for computing Euclidean shortest paths in the plane in the presence of polygonal obstacles; previous time bounds were at least quadratic in n, in the worst-case. The method avoids use of visibility graphs, relying instead on the continuous Dijkstra paradigm. The output is a shortest path map (of size O(n)) with respect to a given source point, which allows shortest path length queries to be answered in time O(log n). The algorithm extends to the case of multiple source points, yielding a geodesic Voronoi diagram within the same time bound.

Optimal greedy algorithms for indifference graphs

Abstract: A fundamental problem in social sciences and management is understanding and predicting decisions made by individuals, various groups, or the society as a whole. In this context, one important concept is the notion of indifference. We characterize the class of indifference graphs, that is, graphs which arise in the process of quantifying indifference relations. In particular, we show that these graphs are characterized by the existence of a special ordering of their vertices. As it turns out, this ordering leads naturally to optimal greedy algorithms for a number of computational problems, including coloring, finding a shortest path between two vertices, computing a maximum matching, the center, and a Hamiltonian path.

Computation of floating mode delay in combinational circuits: theory and algorithms

Abstract: Addresses the problem of accurately computing the delay of a combinational logic circuit in the floating mode of operation. (In this mode the state of the circuit is considered to be unknown when a vector is applied at the inputs.) It is well known that using the length of the topologically longest path as an estimate of circuit delay may be pessimistic since this path may be false, i.e., it cannot propagate an event. Thus, the true delay corresponds to the length of the longest true path. This forces one to examine the conditions under which a path is true. The authors introduce the notion of static cosensitization of paths which leads to necessary and sufficient conditions for determining the truth or falsity of a single path, or a set of paths. The authors apply these results to develop a delay computation algorithm that has the unique feature that it is able to determine the truth or falsity of entire sets of paths simultaneously. This algorithm uses conventional stuck-at-fault testing techniques to arrive at a delay computation method that is both correct and computationally practical, even for particularly difficult circuits. >

Minimum padding to satisfy short path constraints

Abstract: Combinational circuits are often embedded in synchronous designs with memory elements at the input and output ports. A performance metric for a circuit is the cycle time of the clock signal. Correct circuit operation requires that all paths have a delay that lies between an upper bound and a lower bound. Traditional approaches in delay optimization for combinational circuits have dealt with methods to decrease the delay of the longest path. We address the issue of satisfying the lower bound constraints. Such a problem also arises in wave pipelining of circuits. We propose to handle short path constraints as a post processing step after traditional delay optimization techniques. There are two issues presented in this paper. We first discuss necessary and sufficient conditions for successful delay insertion without increasing delays of any long paths. In the second part, we present a naive approach to padding delays (greedy heuristic) and an algorithm based on linear programming. We describe an application of the theory to wave pipelining of circuits. Results are presented on a set of benchmark circuits, using two delay models.

On cuts and matchings in planar graphs

Abstract: We study the max cut problem in graphs not contractible toK5, and optimum perfect matchings in planar graphs. We prove that both problems can be formulated as polynomial size linear programs.

A routing and broadcasting scheme on faulty star graphs

Abstract: The authors present a routing algorithm that uses the depth first search approach combined with a backtracking technique to route messages on the star graph in the presence of faulty links. The algorithm is distributed and requires no global knowledge of faults. The only knowledge required at a node is the state of its incident links. The routed message carries information about the followed path and the visited nodes. The algorithm routes messages along the optimal, i.e., the shortest path if no faults are encountered or if the faults are such that an optimal path still exists. In the absence of an optimal path, the algorithm always finds a path between two nodes within a bounded number of hops if the two nodes are connected. Otherwise, it returns the message to the originating node. The authors provide a performance analysis for the case where an optimal path does not exist. They prove that for a maximum of n-2 faults on a graph with N=n! nodes, at most 2i+2 steps are added to the path, where i is O( square root n). Finally, they use the routing algorithm to present an efficient broadcast algorithm on the star graph in the presence of faults. >

Detour Distance in Graphs

Abstract: For vertices u and v in a connected graph G, the detour distance d* (u, v) between u and v is the length of a longest path P for which the subgraph induced by the vertices of P is P itself. A graph G is called a detour graph if d* (u, v) equals the standard distance between u and v in G for every pair u, v of vertices of G. Several results concerning detour distance and detour graphs are presented.

Finding a minimum weight K-link path in graphs with Monge property and applications

Abstract: Let G be a weighted, complete, directed acyclic graph (DAG), whose edge weights obey the Monge condition. We give an efficient algorithm for finding the minimum weight K-link path between a given pair of vertices for any given K. The time complexity of our algorithm is O(n~=) for the concave case and O (ncr (n) log3 n) for the convex case. Our algorithm uses some properties of DAGs with Monge property together with a refined parametric search technique. We apply our algorithm (for the concave case) to get efficient solutions for the following problems, improving on previous results: (1) Finding the largest K-gon contained in a given polygon. (2) Finding the smallest K-gon that is the intersection of K halfplanes out of of given set of halfplanes defining an n-gon. (3) Computing maximum K-cliques of an interval graph. (4) Computing length limited Huffman codes. (5) Computing optimal discrete quantization.

A linear-processor polylog-time algorithm for shortest paths in planar graphs

Abstract: We give an algorithm requiring polylog time and a linear number of processors to solve single-source shortest paths in directed planar graphs, bounded-genus graphs, and 2-dimensional overlap graphs More generally, the algorithm works for any graph provided with a decomposition tree constructed using size-O(/spl radic/n polylog n) separators >

Finding a longest path in a complete multipartite digraph

Abstract: A digraph obtained by replacing each edge of a complete m-partite graph with an arc or a pair of mutually opposite arcs with the same end vertices is called a complete m-partite digraph. An $O ( n^3 )$ algorithm for finding a longest path in a complete m-partite $( m \geq 2 )$ digraph with n vertices is described in this paper. The algorithm requires time $O( n^{2.5} )$ in case of testing only the existence of a Hamiltonian path and finding it if one exists. It is simpler than the algorithm of Manoussakis and Tuza [SIAM J. Discrete Math., 3 (1990), pp. 537–543], which works only for $m = 2$. The algorithm implies a simple characterization of complete m-partite digraphs having Hamiltonian paths that was obtained for the first time in Gutin [Kibernetica (Kiev), 4 (1985), pp. 124–125] for $m = 2$ and in Gutin [Kibernetica (Kiev), 1(1988), pp. 107–108] for $ m \geq 2 $.

Geometric Bicriteria Optimal Path Problems

Abstract: A bicriteria optimal path simultaneously satisfies two bounds on two measures of path quality. The complexity of finding such a path depends on the particular choices of path quality. This thesis studies bicriteria path problems in a geometric setting using several pairs of path quality, including: path length measured according to different norms $(L\sb{p}$ and $L\sb{q});$ Euclidean length within two or more classes of regions; total turn and Euclidean length; total turn and number of links; and Euclidean length and number of links. For several cases, finding the bicriteria optimal path is shown to be NP-hard. These NP-hard cases include minimizing path length in two different norms, minimizing travel through two regions, and minimizing length and total turn. In the last case, an $O(En\sp2N\sp2$) pseudo-polynomial time algorithm to find an approximate answer is presented. In contrast, when the two measures of path quality are total turn and number of links, an $O(E\sp3n$log$\sp2n)$ exact algorithm is given. A main result of this thesis examines minimizing the Euclidean length and number of links of a path. When the geometric setting of this problem is a polygon without holes, this thesis presents an $O(n\sp3k\sp3$log$(Nk/\epsilon\sp{1/k}))$ algorithm to find a k-link path with Euclidean length at most 1 + $\epsilon$ times the length of the shortest k-link path. A faster algorithm for a relaxed case, when the output path is allowed to have 2k links, is presented for a polygon with or without holes. Finally, some approximation algorithms are outlined for finding a minimum link path among polyhedral obstacles.

On Approximating the Longest Path in a Graph (Preliminary Version)

Abstract: We consider the problem of approximating the longest path in undirected graphs and present both positive and negative results. A simple greedy algorithm is shown to find long paths in dense graphs. We also present an algorithm for finding paths of a logarithmic length in weakly Hamiltonian graphs, and this result is the best possible. For sparse random graphs, we show that a relatively long path can be obtained. To explain the difficulty of obtaining better approximations, we provide some strong hardness results. For any e 0, then NP has a quasi-polynomial deterministic time simulation.

Finding the 3D shortest path with visibility graph and minimum potential energy

Abstract: Finding a three dimensional shortest path is of importance in the development of automatic path planning for mobile robots and robot manipulators, and for practical implementation, the algorithms need to be efficient. Presented is a method for shortest path planning in three-dimensional space in the presence of convex polyhedra. It is based on the visibility graph approach, extended from two to three-dimensional space. A collineation is introduced for the identification of visible edges in the three-dimensional visibility graph. The principle of minimum potential energy is adopted for finding a set of sub-shortest paths via different edge sequences, and from them the global shortest path is selected. The three dimensional visibility graph is constructed in O(n/sup 3/v/sup k/) time, where n is the number of vertices of the polyhedra, k is the number of obstacles and v is the largest number of vertices on any one obstacle. The process to determine the shortest path runs recursively in polynomial time. Results of a computer simulation are given, showing the versatility and efficiency of the approach.

Shortest Paths in Reachability Graphs

Abstract: We prove the following property for safe conflict-free Petri nets and live and safe extended free-choice Petri nets:

Efficient Approximate Shortest-Path Queries Among Isothetic Rectangular Obstacles

Abstract: In this paper we consider the problem of approximate rectilinear shortest-path query between two arbitrary points in the presence of n isothetic and disjoint rectangular obstacles. We present an algorithm which reports a path whose length is at most three times the optimal path length between two arbitrary corner points and at most seven times the optimal path length between two arbitrary points. Our algorithm takes O(nlog3n) preprocessing time, O(n log2n) space and O(log2n) query time for the distance problem. The actual path can be reported in O(log2n+k where k is the number of segments in the reported path. Thus we exhibit a tradeoff between a previous result in [6] in which an exact solution of this query problem is given at the expense of O(n√n) preprocessing and O(√n+k) query time.

Shortest path cost distribution in random graphs with positive integer edge costs

Abstract: The probability distribution of the shortest path cost from a source node to an arbitrary destination node is considered for a random network model consisting of a complete digraph with positive integer random edge costs. Edge costs are chosen according to a common probability distribution for each direction. For this model, the joint distribution of the number of nodes which have a given sequence of shortest path costs from an arbitrary source node is determined explicitly. An expression is then obtained for the distribution of the shortest path cost between two arbitrary nodes using this joint distribution. The main result is the derivation of tight bounds and a sharp limit result for the distribution of the shortest path cost as the number of nodes tends to infinity. Numerical examples are presented to illustrate these results. >

Long paths and cycles in tough graphs

Abstract: For a graphG, letp(G) andc(G) denote the length of a longest path and cycle, respectively. Let ?(t,n) be the minimum ofp(G), whereG ranges over allt-tough connected graphs onn vertices. Similarly, let ?(t,n) be the minimum ofc(G), whereG ranges over allt-tough 2-connected graphs onn vertices. It is shown that for fixedt>0 there exist constantsA, B such that ?(t,n)?A·log(n) and ?(t,n)·log(?(t,n))?B·log(n). Examples are presented showing that fort≤1 there exist constantsA?, B? such that ?(t,n)≤A?·log(n) and ?(t,n)≤B?· log(n). It is conjectured that ?(t,n) ?B?·log(n) for some constantB?. This conjecture is shown to be valid within the class of 3-connected graphs and, as conjectured in Bondy [1] forl=3, within the class of 2-connectedK 1.l-free graphs, wherel is fixed.