A Dijkstra-like method computing all extreme supported non-dominated solutions of the biobjective shortest path problem

We address the problem of determining all extreme supported solutions of the biobjective shortest path problem. A novel Dijkstra-like method generalizing Dijkstra׳s algorithm to this biobjective case is proposed. The algorithm runs in O(N(m+n log n)) time to solve one-to-one and one-to-all biobjective shortest path problems determining all extreme supported non-dominated points in the outcome space and one supported efficient path associated with each one of them. Here n is the number of nodes, m is the number of arcs and N is the number of extreme supported points in outcome space for the one-to-all biobjective shortest path problem. The memory space required by the algorithm is O(n+m) for the one-to-one problem and O(N+m) for the one-to-all problem. A computational experiment comparing the performance of the proposed methods and state-of-the-art methods is included.     

Flying over a polyhedral terrain

We consider the problem of computing shortest paths in three-dimensions in the presence of a single-obstacle polyhedral terrain, and present a new algorithm that for any p?1, computes a (c+ε)-approximation to the Lp-shortest path above a polyhedral terrain in time and O(nlogn) space, where n is the number of vertices of the terrain, and c=²(p−1)/p. This leads to a FPTAS for the problem in L₁ metric, a -factor approximation algorithm in Euclidean space, and a 2-approximation algorithm in the general Lp metric.     

Optimal piecewise linear motion of an object among obstacles

We present an algorithm for determining the shortest restricted path motion of a polygonal object amidst polygonal obstacles. The class of motions which are allowed can be described as follows: a designated vertex,P, of the polygonal object traverses a piecewise linear path, whose breakpoints are restricted to the vertices of the obstacles. The distance measure being minimized is the length of the path traversed byP. Our algorithm runs in timeO(n ⁴kogn). We also discuss a variation of this algorithm which minimizes any positive linear combination of length traversed byP and angular rotation of the ladder aboutP. This variation requiresO(n ₅) time.Research partially supported by a grant from the Hughes Artificial Intelligence Center, part of Hughes Aircraft.     

Mobile robot path planning algorithm by equivalent conduction heat flow topology optimization

This paper addresses the path planning problem for a point robot moving in a planar environment filled with obstacles. Our approach is based on the principles of thermal conduction and structural topology optimization and rests on the observation that, by identifying the starting and ending configurations of a point robot as the heat source and sink of a conducting plate, respectively, the path planning problem can be formulated as a topology optimization problem that minimizes thermal compliance. Obstacles are modeled as regions of zero thermal conductivity; in fact, regions can be assigned varying levels of non-uniform conductivity depending on the application. We describe the details of our path planning algorithm, including the use of artificial mass constraints (particularly limits on the plate mass) to ensure convergence, and the choice of penalty exponents. The feasibility and practicality of our approach is validated through numerical experiments performed with several benchmarks, with intriguing possibilities for extension to more complex environments and real terrains. The benchmark problems of this paper mainly consist of obstacle-free path planning problems in two-dimensional space with maze-typed, symmetric, and spiral-type obstacles. We also address planning problems involving user-specified checkpoints, and also finding shortest paths on real three-dimensional terrain. To the authors’ knowledge, the path planning going through a stopover is considered for the first time.     

L 1 shortest paths among polygonal obstacles in the plane

We present an algorithm for computingL ₁ shortest paths among polygonal obstacles in the plane. Our algorithm employs the “continuous Dijkstra” technique of propagating a “wavefront” and runs in timeO(E logn) and spaceO(E), wheren is the number of vertices of the obstacles andE is the number of “events.” By using bounds on the density of certain sparse binary matrices, we show thatE =O(n logn), implying that our algorithm is nearly optimal. We conjecture thatE =O(n), which would imply our algorithm to be optimal. Previous bounds for our problem were quadratic in time and space. Our algorithm generalizes to the case of fixed orientation metrics, yielding anO(n?^?1/2 log² n) time andO(n?^?1/2) space approximation algorithm for finding Euclidean shortest paths among obstacles. The algorithm further generalizes to the case of many sources, allowing us to compute anL ₁ Voronoi diagram for source points that lie among a collection of polygonal obstacles.     

On-line motion planning: Case of a planar rod

In this paper we develop an algorithm for planning the motion of a planar rod (a line segment) amidst obstacles bounded by simple, closed polygons. The exact shape, number and location of the obstacles are assumed unknown to the planning algorithm, which can only obtain information about the obstacles by detecting points of contact with the obstacles. The ability to detect contact with obstacles is formalized by move primitives that we callguarded moves. We call ours theon-line motion planning problem as opposed to the usualoff-line version. This is a significant departure from the usual setting for motion planning problems. What we demonstrate is that the retraction method can be applied, although new issues arise that have no counterparts in the usual setting. We are able to obtain an algorithm with path complexityO(n ²) guarded moves, wheren is the number of obstacle corners. This matches the known lower bound. The computational complexityO(n ²logn) of our algorithm matches the best known algorithm for the off-line version.This work is supported by NSF grants #DCR-84-01633, #DCR-84-01898 and PSC-CUNY 669287.     

A new algorithm for shortest paths among obstacles in the plane

We introduce a new algorithm for computing Euclidean shortest paths in the plane in the presence of polygonal obstacles. In particular, for a given start points, we build a planar subdivision (ashortest path map) that supports efficient queries for shortest paths froms to any destination pointt. The worst-case time complexity of our algorithm isO(kn log² n), wheren is the number of vertices describing the polygonal obstacles, andk is a parameter we call the illumination depth of the obstacle space. Our algorithm usesO(n) space, avoiding the possibly quadratic space complexity of methods that rely on visibility graphs. The quantityk is frequently significantly smaller thann, especially in some of the cases in which the visibility graph has quadratic size. In particular,k is bounded above by the number of different obstacles that touch any shortest path froms.Partially supported by NSF Grants IRI-8710858 and ECSE-8857642 and by a grant from Hughes Research Laboratories, Malibu, CA.     

Reducing the Complexity of Robot's Scene for Faster Collision Detection

In many real-life industrial applications such as welding and painting, the hand tip of a robot manipulator must follow a desired Cartesian curve while its body avoids collisions with obstacles in its environment. Collision detection is an absolutely essential task for any robotic manipulators in order to operate safely and effectively in cluttered environments. A significant factor that influences the complexity of the collision detection problem is the obstacles' density, i.e., the total number of obstacles in the robot's environment.In this paper, a heuristic algorithm for approximating the collision detection problem into a simpler one is presented. The algorithm reduces the number of obstacles that must be examined during the robot's motion by applying efficient techniques from computational geometry. The algorithm runs in time O(max(n ² log m, n m)) , and uses O(n ² + m n) space; with n being the number of obstacles in the robot's workspace, and m the total number of obstacles' vertices. Both costs are worst-case bounds.     

Planning constrained motion

We consider the motion planning problem for a point constrained to move along a path with radius of curvature at least one. The point moves in a two-dimensional universe with polygonal obstacles. We show the decidability of the reachability question: “Given a source placement (position and direction pair) and a target placement, is there a curvature-constrained path from source to target avoiding obstacles?” The decision procedure has time and space complexity 2 ^{o(poly(n, m))} wheren is the number of corners andm is the number of bits required to specify the position of corners.     

L1 shortest paths among polygonal obstacles in the plane

We present an algorithm for computingL ₁ shortest paths among polygonal obstacles in the plane. Our algorithm employs the continuous Dijkstra technique of propagating a wavefront and runs in timeO(E logn) and spaceO(E), wheren is the number of vertices of the obstacles andE is the number of events. By using bounds on the density of certain sparse binary matrices, we show thatE =O(n logn), implying that our algorithm is nearly optimal. We conjecture thatE =O(n), which would imply our algorithm to be optimal. Previous bounds for our problem were quadratic in time and space.Our algorithm generalizes to the case of fixed orientation metrics, yielding anO(n^–1/2 log² n) time andO(n^–1/2) space approximation algorithm for finding Euclidean shortest paths among obstacles. The algorithm further generalizes to the case of many sources, allowing us to compute anL ₁ Voronoi diagram for source points that lie among a collection of polygonal obstacles.Partially supported by a grant from Hughes Research Laboratories, Malibu, California and by NSF Grant ECSE-8857642. Much of this work was done while the author was a Ph.D. student at Stanford University, under the support of a Howard Hughes Doctoral Fellowship, and an employee of Hughes Research Laboratories.     

An optimal time–space algorithm for dense stereo matching

The increasing demand for higher resolution images and higher frame rate videos will always pose a challenge to computational power when real-time performance is required to solve the stereo-matching problem in 3D reconstruction applications. Therefore, the use of asymptotic analysis is necessary to measure the time and space performance of stereo-matching algorithms regardless of the size of the input and of the computational power available. In this paper, we survey several classic stereo-matching algorithms with regard to time–space complexity. We also report running time experiments for several algorithms that are consistent with our complexity analysis. We present a new dense stereo-matching algorithm based on a greedy heuristic path computation in disparity space. A procedure which improves disparity maps in depth discontinuity regions is introduced. This procedure works as a post-processing step for any technique that solves the dense stereo-matching problem. We prove that our algorithm and post-processing procedure have optimal O(n) time–space complexity, where n is the size of a stereo image. Our algorithm performs only a constant number of computations per pixel since it avoids a brute force search over the disparity range. Hence, our algorithm is faster than “real-time” techniques while producing comparable results when evaluated with ground-truth benchmarks. The correctness of our algorithm is demonstrated with experiments in real and synthetic data.     

Minimal Adaptive Routing on the Mesh with Bounded Queue Size

An adaptive routing algorithm is one in which the path a packet takes from its source to its destination may depend on other packets it encounters. Such algorithms potentially avoid network bottlenecks by routing packets around “hot spots.” Minimal adaptive routing algorithms have the additional advantage that the path each packet takes is a shortest one. For a large class of minimal adaptive routing algorithms, we present an Ω(n²/k²) bound on the worst case time to route a static permutation of packets on ann×nmesh or torus with nodes that can hold up tok≥ 1 packets each. This is the first nontrivial lower bound on adaptive routing algorithms. The argument extends to more general routing problems, such as theh–hrouting problem. It also extends to a large class of dimension order routing algorithms, yielding an Ω(n²/k) time bound. To complement these lower bounds, we present two upper bounds. One is anO(n²/k+n) time dimension order routing algorithm that matches the lower bound. The other is the first instance of a minimal adaptive routing algorithm that achievesO(n) time with constant sized queues per node. We point out why the latter algorithm is outside the model of our lower bounds.     

New Visual Invariants for Terrain Navigation Without 3D Reconstruction

For autonomous vehicles to achieve terrain navigation, obstacles must be discriminated from terrain before any path planning and obstacle avoidance activity is undertaken. In this paper, a novel approach to obstacle detection has been developed. The method finds obstacles in the 2D image space, as opposed to 3D reconstructed space, using optical flow. Our method assumes that both nonobstacle terrain regions, as well as regions with obstacles, will be visible in the imagery. Therefore, our goal is to discriminate between terrain regions with obstacles and terrain regions without obstacles. Our method uses new visual linear invariants based on optical flow. Employing the linear invariance property, obstacles can be directly detected by using reference flow lines obtained from measured optical flow. The main features of this approach are: (1) 2D visual information (i.e., optical flow) is directly used to detect obstacles; no range, 3D motion, or 3D scene geometry is recovered; (2) knowledge about the camera-to-ground coordinate transformation is not required; (3) knowledge about vehicle (or camera) motion is not required; (4) the method is valid for the vehicle (or camera) undergoing general six-degree-of-freedom motion; (5) the error sources involved are reduced to a minimum, because the only information required is one component of optical flow. Numerous experiments using both synthetic and real image data are presented. Our methods are demonstrated in both ground and air vehicle scenarios.     

Constructing the visibility graph for n-line segments in O(n2) time

Given a set S of line segments in the plane, its visibility graph G_S is the undirected graph which has the endpoints of the line segments in S as nodes and in which two nodes (points) are adjacent whenever they ‘see’ each other (the line segments in S are regarded as nontransparent obstacles). It is shown that G_S can be constructed in O(n²) time and space for a set S of n nonintersecting line segments. As an immediate implication, the shortest path between two points in the plane avoiding a set of n nonintersecting line segments can be computed in O(n²) time and space     

Offline variants of the “lion and man” problem: —Some problems and techniques for measuring crowdedness and for safe path planning—

Consider the following safe path planning problem: Given a set of trajectories (paths) of k point robots with maximum unit speed in a bounded region over a (long) time interval [0,T], find another trajectory (if it exists) subject to the same maximum unit speed limit, that avoids (that is, stays at a safe distance of) each of the other k trajectories over the entire time interval. We call this variant the continuous model of the safe path planning problem. The discrete model of this problem is: Given a set of trajectories (paths) of k point robots in a graph over a (long) time interval 0,1,2,…,T, find a trajectory (path) for another robot, that avoids each of the other k at any time instant in the given time interval.We introduce the notions of the avoidance number of a region, and that of a graph, respectively, as the maximum number of trajectories which can be avoided in the region (respectively, graph). We give the first estimates on the avoidance number of the n×n grid G_n, and also devise an efficient algorithm for the corresponding safe path planning problem in arbitrary graphs. We then show that our estimates on the avoidance number of G_n can be extended for the avoidance number of a bounded (fat) region. In the final part of our paper, we consider other related offline questions, such as the maximum number of men problem and the spy problem.     

Minimum-link paths among obstacles in the plane

Given a set of nonintersecting polygonal obstacles in the plane, thelink distance between two pointss andt is the minimum number of edges required to form a polygonal path connectings tot that avoids all obstacles. We present an algorithm that computes the link distance (and a corresponding minimum-link path) between two points in timeO(Eα(n) log² n) (and spaceO(E)), wheren is the total number of edges of the obstacles,E is the size of the visibility graph, and α(n) denotes the extremely slowly growing inverse of Ackermann's function. We show how to extend our method to allow computation of a tree (rooted ats) of minimum-link paths froms to all obstacle vertices. This leads to a method of solving the query version of our problem (for query pointst).     

Minimum-link paths among obstacles in the plane

Given a set of nonintersecting polygonal obstacles in the plane, thelink distance between two pointss andt is the minimum number of edges required to form a polygonal path connectings tot that avoids all obstacles. We present an algorithm that computes the link distance (and a corresponding minimum-link path) between two points in timeO(E(n) log² n) (and spaceO(E)), wheren is the total number of edges of the obstacles,E is the size of the visibility graph, and (n) denotes the extremely slowly growing inverse of Ackermann's function. We show how to extend our method to allow computation of a tree (rooted ats) of minimum-link paths froms to all obstacle vertices. This leads to a method of solving the query version of our problem (for query pointst).Joseph Mitchell was partially supported by NSF Grants IRI-8710858 and ECSE-8857642, and by a grant from Hughes Research Laboratories. This work was begun while Günter Rote and Gerhard Woeginger were at the Freie Universität Berlin, Fachbereich Mathematik, Institut für Informatik, and it was partially supported by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM). Gerhard Woeginger acknowledges the support by the Fonds zur Förderung der Wissenschaftlichen Forschung, Projekt S32/01.     

Exploring Unknown Environments with Obstacles

Abstract. We study exploration problems where a robot has to construct a complete map of an unknown environment using a path that is as short as possible. In the first problem setting we consider, a robot has to explore n rectangles. We show that no deterministic or randomized online algorithm can be better than Ω(\sqrt n ) -competitive, solving an open problem by Deng et al. [7]. We also generalize this bound to the problem of exploring three-dimensional rectilinear polyhedra without obstacles. In the second problem setting we study, a robot has to explore a grid graph with obstacles in a piecemeal fashion. The piecemeal constraint was defined by Betke et al. [4] and implies that the robot has to return to a start node every so often. Betke et al. gave an efficient algorithm for exploring grids with rectangular obstacles. We present an efficient strategy for piecemeal exploration of grids with arbitrary rectilinear obstacles.     

Light-assisted A⁎ path planning

This paper introduces a novel variant of the A^? path planning algorithm, which we call Light-assisted A^? (or LA^? for short). The LA^? algorithm expands less nodes than A^? during the search process, especially in scenarios where there are complex-shaped obstacles in the path between the start and goal nodes. This is achieved using the concept of (virtual) light which identifies and demotes dead-end paths blocked by obstacles, thus ensuring that the search stays focused on promising paths. Three path planning problems are used to test the performance of LA^?. These include path finding in a grid cluttered by randomly placed obstacles, robot navigation in a map containing multiple solid walls, and finally mazes. The results of these experiments show that LA^? can achieve orders of magnitude improvement in performance over A^?. In addition, LA^? results in near-optimal solutions that are very close to the optimal path obtained by the conventional A^? algorithm.     

Optimal piecewise linear motion of an object among obstacles

We present an algorithm for determining the shortest restricted path motion of a polygonal object amidst polygonal obstacles. The class of motions which are allowed can be described as follows: a designated vertex,P, of the polygonal object traverses a piecewise linear path, whose breakpoints are restricted to the vertices of the obstacles. The distance measure being minimized is the length of the path traversed byP. Our algorithm runs in timeO(n ⁴kogn). We also discuss a variation of this algorithm which minimizes any positive linear combination of length traversed byP and angular rotation of the ladder aboutP. This variation requiresO(n ₅) time.