Showing papers on "Longest path problem published in 1989"

On the General False Path Problem in Timing Analysis

Abstract: The false path problem is often referred to as the problem of detecting the longest sensitizable path (A path which is not a false path is a sensitizable path). The term “false path” is not clearly defined. In this paper, we first give a clear and precise definition of a false path. Then the general false path problem is formulated. The general false path problem is to detect whether a given path (not necessarily the longest one) is a false path. We present an efficient algorithm for solving the general false path problem. We also propose another algorithm which generates all the possible sensitizable paths with the delays greater than a given threshold T. The efficiency and effectiveness of the proposed algorithm are demonstrated by the experimental results.Index Terms: Timing Verification, Logic Simulation, VLSI circuit, Timing Analysis, False path, Graph Theory.

Shortest Paths Without a Map

Abstract: We study several versions of the shortest-path problem when the map is not known in advanced, but is specified dynamically. We are seeking dynamic decision rules that optimize the worst-case ratio of the distance covered to the length of the (statically) optimal path. We describe optimal decision rules for two cases: Layered graphs of bounded width, and two-dimensional scenes with unit square obstacles. The optimal rules turn out to be intuitive, common-sense heuristics. For slightly more general graphs and scenes, we show that no bounded ratio is possible. We also show 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.

Rectilinear shortest paths in the presence of rectangular barriers

Abstract: In this paper we address the following shortest-path problem. Given a point in the plane andn disjoint isothetic rectangles (barriers), we want to construct a shortestL1 path (not crossing any of the barriers) from the source point to any given query point. A restricted version of this problem (where the source and destination points are knowna priori) had been solved earlier inO(n2) time. Our approach consists of preprocessing the source point and the barriers to obtain a planar subdivision where a query point can be located and a shortest path connecting it to the source point quickly transvered. By showing that any such path is monotone in at least one ofx ory directions, we are able to apply a plane sweep technique to divide the plane intoO(n) rectangular regions. This leads to an algorithm whose complexity isO(n logn) preprocessing time,O(n) space, andO(logn+k) query time, wherek is the number of turns on the reported path. If only the length of the path is sought,O(logn) query time suffices. Furthermore, we show an ?(n logn) time lower bound for the case where the source and destination points are known in advance, which implies the optimality of our algorithm in this case.

Solving k -shortest and constrained shortest path problems efficiently

Abstract: In this paper, we examine the problems of finding thek-shortest paths, thek-shortest paths without repeated nodes, and the shortest path with a single side constraint between an origin and destination pair. Distances are assumed to be non-negative, but networks may be either directed or undirected. Two versions of algorithms for all these problems are compared. The computational results show that, in each case, the advantage of the adaptive version (as measured by total number of permanent labels) grows with the problem size.

A bidirectional shortest-path algorithm with good average-case behavior

Abstract: The two-terminal shortest-path problem asks for the shortest directed path from a specified nodes to a specified noded in a complete directed graphG onn nodes, where each edge has a nonnegative length. We show that if the length of each edge is chosen independently from the exponential distribution, and adjacency lists at each node are sorted by length, then a priority-queue implementation of Dijkstra's unidirectional search algorithm has the expected running time Θ(n logn). We present a bidirectional search algorithm that has expected running time Θ(√n logn). These results are generalized to apply to a wide class of edge-length distributions, and to sparse graphs. If adjacency lists are not sorted, bidirectional search has the expected running time Θ(a√n) on graphs of average degreea, as compared with Θ(an) for unidirectional search.

Speeding up dynamic programming algorithms for finding optimal lattice paths

Abstract: Many problems can be reduced to finding an optimal lattice path. For example, minimum path integrals can be computed by discretizing to a lattice graph and then using the optimal lattice path to approximate the minimum continuous path. Finding an optimal alignment between two sequences can also be reduced to finding an optimal lattice path. Dynamic programming algorithms are generally, well-suited to such problems, but can be slow and require too much storage if the lattice is too large, for example, if the lattice dimension is too high. Faster algorithms requiring less computer storage can often be constructed by restricting calculations to a “computational volume” known to contain the optimal path. Upper and lower bounds on path distances from the problem domain can often help to produce small computational volumes.

Shortest paths for autonomous vehicles

Abstract: Paths that stay on a given network of line segments except to turn onto one segment from another by following a circular arc are studied. The problem of finding a shortest-length collision-free path of this form for an autonomous vehicle with a bound on its steering angle is considered. A polynomial-time algorithm to modify a given feasible path into a shortest-length path traversing the same sequence of lanes is given. However, if the sequence of lanes to be traversed is not fixed, then the general problem of finding a shortest-length path of the restricted form is shown to be NP-complete. >

Bar-representable visibility graphs and a related network flow problem

Abstract: A bar layout is a set of vertically oriented non-intersecting line segments in the plane called bars. The visibility graph associated with a layout is defined as the graph whose vertices correspond to the bars and whose edges represent the horizontal visibilities between pairs of bars. This dissertation is concerned with the characterization of bar-representable graphs: those graphs which are the visibility graphs of some bar layout. A polynomial time algorithm for determining if a given graph is bar-representable, and the subsequent construction of an associated layout are provided. Weighted and directed versions of the problem are also formulated and solved; in particular, polynomial time algorithms for the layout of such graphs are developed. The Planar Full Flow problem is to determine a plane embedding and an (acyclic) orientation of an undirected planar network that admits a feasible flow, that uses all arcs (except those incident upon the source or sink) to full capacity and maintains planarity. The connection of this flow problem to bar-representable graphs is exploited to solve the weighted case of the latter. As evidence that the problem is inherently difficult, two natural variants of the Full Flow problem are shown to be strongly NP-Complete.

On the main channel length‐magnitude formula for random networks: A solution to Moon's conjecture

Abstract: This is a technical note which solves Moon's conjecture for the exact asymptotic formula for the expected length of the longest path in Shreve's random network for large magnitude (source number).

Finding All Shortest Path Edge Sequences on a Convex Polyhedron

Abstract: In this paper, the problems of computing the Euclidean shortest path between two points on the surface of a convex polyhedron and finding all shortest path edge sequences are considered. We propose an O(n6logn) algorithm to find All Shortest Path Edge Sequences, construct n Edge Sequence Trees, and draw out n(n−1)/2 Visibility Relation Diagrams for a given convex polyhedron. According to these data structures, not only can we enumerate all shortest path edge sequences and draw out all maximal ones, but we can also find the shortest path between any two points lying on edges in O(k+logn) time where k is the number of edges crossed by the shortest path.

A Level Structure Approach on the Bandwidth Problem for Special Graphs

Abstract: The labelingfthat attains this minimum value is called an optimal labeling. As is well known, to determine the bandwidth of a graph is an NP-complete problem. However, some results for special graphs can be obtained. A common way is to find a lower bound and then to construct a labeling f that attains the bound. In this regard, the level structure approach could be significant. The idea of level structure (i.e., a partition of the vertex set satisfying some conditions) arose from several approximate algorithms, such as the CM, RCM, GPS algorithms. A purpose of these algorithms is rather to find a level structure whose width is as small as possible and whose depth is as large as possible. In the theoretic area, the level width is connected with the cardinality of boundary due to Harper [8], and the level depth is connected with the diameter of G. Regarding the former aspect, the Harper’s theorem [8] and its variants seem to be efficient for some graph products, say P,,, x Pn [6], P, x C,, [7], C, x C.[lO], K , x K,[14],then-cubeQ,[S],andthe(m,n)-multipathP,,~[11, 131. Regarding the latter aspect, the Chvhtal’s theorem [5] and its generalizations, as discussed in this paper, seem to be efficient for some trees. The next section provides a generalization of the Chvital’s theorem. The third section proposes a general algorithm that is suitable to a class of trees mentioned in the following sections. The fourth section includes the bandwidth problem of caterpillars and two-layer stars. Syslo and Zak [ 151 have solved the problem of caterpillars by using the concept of critical subgraphs. We discuss it here as a special case of our general algorithm. The result of two-layer stars is new. The fifth section concentrates on the complete binary trees and k-ary trees whose bandwidths have been indicated in surveys by Chung [3,4]. We discuss the labeling method. The last section discusses the

The weighted Voronoi diagram and its applications in least-risk motion planning

Abstract: The authors present a path-planning algorithm to find the least-cost path along the WVD (weighted Voronoi diagram) based on Dijkstra's shortest path algorithm, where the cost of an edge is, in this case, the risk of transversing the edge. By imposing a discrete graph over the area of interest, the authors obtain a reduction from a continuous problem to a combinatorial problem. If the start or goal position does not lie on this graph, it is retracted by either the path of shortest distance or the path of steepest descent. Unfortunately, the WVD is not connected in some cases, and they have to enforce connectivity. With a practical aim in mind, the authors give a simple and easily computed path, connecting the disconnected components by the shortest distance path or by a circular path. With suitable preprocessing, they keep the run-time cost of path planning to O(n/sup 2/ log n), including the cost of retraction of both the start and goal positions and the cost of Dijkstra's shortest path (or least-cost) algorithm. The authors demonstrate a potential application of the weighted Voronoi diagram as a heuristic in least-risk motion planning. >

Real time penetration of multiple moving threats

Abstract: A local hill-climbing heuristic based on the weighted Voronoi diagram (WVD) is illustrated in an example of military application in simulated air-to-air scenario. For the risk measure, defined as the maximum of the ratio of lethality over distance, the WVD from computational geometry gives an intuitive suggestion of the best place to be. To find the least-cost path along the WVD, the author presents a motion-planning algorithm based on Dijkstra's shortest path algorithm where the cost of an edge is the risk of traversing the edge. By imposing a discrete graph over the area of interest, the author reduces a continuous problem to a combinatorial problem. If the start or goal position does not lie on this graph, it is retracted by either the path of shortest distance or the path of steepest descent. A simple and easily computed path connects the disconnected components by the shortest distance or by a circular path. On the basis of this heuristic, the run-time cost of motion planning is kept to O(n/sup 2/logn). >

and its Applications in Least-Risk Motion Planning

Abstract: Let S be a set of n points (p;} in a plane with a positive weight (or cost) w; associated with each point p; in S. If we consider S to be a set of threatening forces and the cost to be the risk measure, we define the least-risk motion planning problem, a problem which is continuous and very difficult. For the risk measure defined as the maximum of the ratio of weight over distance, the weighted Voronoi diagram (WVD) from computational geonietry gives an intuitive suggestion of the best place to be. We will give a path planning algorithm to find the least-cost path along the WVD based on Dijkstra’s shortest path algorithm where the cost of an edge is, in this case, the risk of traversing the edge. By imposing a discrete graph over the area of interest, we obtain a reduction from a continuous problem to a combinatorial problem. If the start or goal position does not lie on this graph, it is retracted by either the path of shortest distance or the path of steepest descent. Unfortunately, the WVD is not connected in some cases and we have to enforce connectivity. With a practical aim in mind, we give a simple and easily computed path, connecting the disconnected components by the shortest distance path or by a circular path. With suitable preprocessing, we keep the run-time cost of path planning to O(n* log n), including the cost of retraction of both the start and goal positions and the cost of Dijkstra’s shortest path (or least-cost) algorithm. This paper demonstrates a potential application of the weighted Voronoi diagram as a heuristic in the least risk motion planning.

Cycle breaking and longest path in channel routing problem

Abstract: Channel routing is one of the automatic routing techniques in LSI design. In the channel routing by trunk-branch scheme, if a cycle exists in the relational graph between subnets, the trunk of a net must be divided to break the cycle. Reference [11] presented a method, which divides the trunk of a net at a terminal on the graph representing the routing requirement. the effectiveness and pseudo-effectiveness of the method are defined. Through the discussion of the necessary and sufficient condition for the realizability of the division, a condition is derived for the routing requirement to be realized. This paper discusses the degree of freedom remaining in the trunk division (i.e., the connection of free arcs in the modification of the requirement graph). the effect of the degree of freedom on the final result, especially on the longest path in the requirement graph, is discussed. As a result, it is shown that if the trunk division satisfies a certain condition, it does not increase much the longest path length.

Finding Out Critical Points For Real-Time Path Planning

Abstract: Path planning for a mobile robot is a classic topic, but the path planning under real-time environment is a different issue. The system sources including sampling time, processing time, processes communicating time, and memory space are very limited for this type of application. This paper presents a method which abstracts the world representation from the sensory data and makes the decision as to which point will be a potentially critical point to span the world map by using incomplete knowledge about physical world and heuristic rule. Without any previous knowledge or map of the workspace, the robot will determine the world map by roving through the workspace. The computational complexity for building and searching such a map is not more than O( n 2 ) The find-path problem is well-known in robotics. Given an object with an initial location and orientation, a goal location and orientation, and a set of obstacles located in space, the problem is to find a continuous path for the object from the initial position to the goal position which avoids collisions with obstacles along the way. There are a lot of methods to find a collision-free path in given environment. Techniques for solving this problem can be classified into three approaches: 1) the configuration space approach [1],[2],[3] which represents the polygonal obstacles by vertices in a graph. The idea is to determine those parts of the free space which a reference point of the moving object can occupy without colliding with any obstacles. A path is then found for the reference point through this truly free space. Dealing with rotations turns out to be a major difficulty with the approach, requiring complex geometric algorithms which are computationally expensive. 2) the direct representation of the free space using basic shape primitives such as convex polygons [4] and overlapping generalized cones [5]. 3) the combination of technique 1 and 2 [6] by which the space is divided into the primary convex region, overlap region and obstacle region, then obstacle boundaries with attribute values are represented by the vertices of the hypergraph. The primary convex region and overlap region are represented by hyperedges, the centroids of overlap form the critical points. The difficulty is generating segment graph and estimating of minimum path width. The all techniques mentioned above need previous knowledge about the world to make path planning and the computational cost is not low. They are not available in an unknow and uncertain environment. Due to limited system resources such as CPU time, memory size and knowledge about the special application in an intelligent system (such as mobile robot), it is necessary to use algorithms that provide the good decision which is feasible with the available resources in real time rather than the best answer that could be achieved in unlimited time with unlimited resources. A real-time path planner should meet following requirements: - Quickly abstract the representation of the world from the sensory data without any previous knowledge about the robot environment. - Easily update the world model to spell out the global-path map and to reflect changes in the robot environment. - Must make a decision of where the robot must go and which direction the range sensor should point to in real time with limited resources. The method presented here assumes that the data from range sensors has been processed by signal process unite. The path planner will guide the scan of range sensor, find critical points, make decision where the robot should go and which point is poten- tial critical point, generate the path map and monitor the robot moves to the given point. The program runs recursively until the goal is reached or the whole workspace is roved through.

Collision-Free Path Planning for a Reconfigurable Robot

Abstract: In most of the existing research on robot path planning, a robot is approximated by a fixed shape, i.e., a circle or a polygon. In this paper, we study the problem of a collision-free path planning for a reconfigurable robot whose shape can be changed during motion. The task of the robot is to carry a polygonal object from a starting point to a destination point in a littered environment. This path planning problem is solved by using two major algorithms: the collision-free feasible configuration finding algorithm and the collision-free path finding algorithm. The collision-free feasible configuration finding algorithm finds all collision-free feasible configurations for the robot when the position and orientation of the carried object are given. The collision-free path finding algorithm generates some candidate paths first and then uses a graph search method to find a collision-free path from all the collision-free feasible configurations along the candidate paths.

Efficient parallel algorithms for path problems in directed graphs

Abstract: In this paper we describe a technique for finding efficient parallel algorithms for problems on directed graphs that involve checking the existence of certain kinds of paths in the graph. This technique provides efficient algorithms for finding dominators in flow graphs, performing interval and loop analysis on reducible flow graphs, and finding the feedback vertices of a digraph. Each of these algorithms takesO(log2n) time using the same number of processors needed for fast matrix multiplication. All of these bounds are for an EREW PRAM.

Proposed algorithms for bayesian networks based on fuzzy set theory

Abstract: Scope and method of study. Bayesian networks are directed acyclic graphs in which the nodes represent propositions, an arcs signify direct dependencies between the linked propositions, and the strengths of these dependencies are quantified by conditional probabilities. Pearl (1986) developed the algorithms for Bayesian networks which impart the impact on nodes to the entire network via a local propagation in a time proportional to the longest path in the network. This study proposed the use of linguistic probability instead of the probability. Thus, this study attempts to develop the approaches to the problems occurring due to the adoption of linguistic probabilities. A simulation model is employed in determining the type of operations performed on linguistic probabilities and an approach to a linguistic approximation is proposed. Findings and conclusions. The linguistic probability can be treated as either a fuzzy number or the fuzzy set. An experiment based on a simulation model indicates that the linguistic probability is treated as a fuzzy number. This result supports the position taken by a number of researchers including Bonissone and Decker (1986). A proposed approach to a linguistic approximation shows that among four features representing each linguistic probability, the skewness is not a good feature, but other features such as first-moment, power, and entropy are good features.