Results 1  10
of
19
Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of t ..."
Abstract

Cited by 147 (12 self)
 Add to MetaCart
Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
An Optimal Algorithm for Euclidean Shortest Paths in the Plane
 SIAM J. Comput
, 1997
"... We propose an optimaltime algorithm for a classical problem in plane computational geometry: computing a shortest path between two points in the presence of polygonal obstacles. Our algorithm runs in worstcase time O(n log n) and requires O(n log n) space, where n is the total number of vertice ..."
Abstract

Cited by 86 (1 self)
 Add to MetaCart
We propose an optimaltime algorithm for a classical problem in plane computational geometry: computing a shortest path between two points in the presence of polygonal obstacles. Our algorithm runs in worstcase time O(n log n) and requires O(n log n) space, where n is the total number of vertices in the obstacle polygons. The algorithm is based on an efficient implementation of wavefront propagation among polygonal obstacles, and it actually computes a planar map encoding shortest paths from a fixed source point to all other points of the plane; the map can be used to answer singlesource shortest path queries in O(logn) time. The time complexity of our algorithm is a significant improvement over all previously published results on the shortest path problem. Finally, we also discuss extensions to more general shortest path problems, involving nonpoint and multiple sources. 1 Introduction 1.1 The Background and Our Result The Euclidean shortest path problem is one of the o...
An Efficient Algorithm for Euclidean Shortest Paths Among Polygonal Obstacles in the Plane
, 1988
"... We give an algorithm to compute a (Euclidean) shortest path in a polygon with h holes and a total of n vertices. The algorithm uses O(n) space and requires O(n + h² log n) time. ..."
Abstract

Cited by 34 (1 self)
 Add to MetaCart
We give an algorithm to compute a (Euclidean) shortest path in a polygon with h holes and a total of n vertices. The algorithm uses O(n) space and requires O(n + h² log n) time.
Optimal Navigation and Object Finding Without Geometric Maps or . . .
 IN PROC. IEEE INTERNATIONAL CONFERENCE ON ROBOTICS AND AUTOMATION
, 2003
"... In this paper we present a dynamic data structure, useful for robot navigation in an unknown, simplyconnected planar environment. The guiding philosophy in this work is to avoid traditional problems such as complete map building and localization by constructing a minimal representation based entirel ..."
Abstract

Cited by 29 (12 self)
 Add to MetaCart
In this paper we present a dynamic data structure, useful for robot navigation in an unknown, simplyconnected planar environment. The guiding philosophy in this work is to avoid traditional problems such as complete map building and localization by constructing a minimal representation based entirely on critical events in online sensor measurements made by the robot. Furthermore, this representation provides a sensorfeedback motion strategy that guides the robot along an optimal trajectory between any two environment locations, and allows the search of static targets, even though there is no geometric map of the environment. We present algorithms for building the data structure in an unknown environment, and for using it to perform optimal navigation. We implemented these algorithms on a real mobile robot. Results are presented in which the robot builds the data structure online, and is able to use it without needing a global reference frame. Simulation results are also shown to demonstrate how the robot is able to find interesting objects in the environment.
Autonomous Vehicle Navigation Utilizing Electrostatic Potentional Fields and Fuzzy Logic
 IEEE Trans. Robotic. Autom
, 2001
"... for online handling of live objects, Part I: Analytical model, ” in ..."
Abstract

Cited by 15 (9 self)
 Add to MetaCart
for online handling of live objects, Part I: Analytical model, ” in
Theory and Experiments in Autonomous SensorBased Motion Planning with Applications for Flight Planetary Microrovers
, 1999
"... With the success of Mars Pathfinder's Sojourner rover, a new era of planetary exploration has opened, with demand for highly capable mobile robots. These robots must be able to traverse long distances over rough, unknown terrain autonomously, under severe resource constraints. Much prior work in mob ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
With the success of Mars Pathfinder's Sojourner rover, a new era of planetary exploration has opened, with demand for highly capable mobile robots. These robots must be able to traverse long distances over rough, unknown terrain autonomously, under severe resource constraints. Much prior work in mobile robot path planning has been based on assumptions that are not truly applicable to navigation through planetary terrains. Based on the author's firsthand experience with the Mars Pathfinder mission, this work reviews issues which are critical for successful autonomous navigation of planetary rovers. No current methodology addresses all of these constraints. We next develop the sensorbased "Wedgebug" motionplanning algorithm. This algorithm is complete, correct, requires minimal memory for storage of its world model, and uses only onboard sensors, which are guided by the algorithm to e#ciently sense only the data needed for motion planning, while avoiding unnecessary robot motion. The p...
Heuristically Driven Front Propagation for Fast Geodesic Extraction
, 2007
"... This paper presents a new method to quickly extract geodesic paths on images and 3D meshes. We use a heuristic to drive the front propagation procedure of the classical Fast Marching. This results in a modification of the Fast Marching algorithm that is similar to the A ∗ algorithm used in artificia ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
This paper presents a new method to quickly extract geodesic paths on images and 3D meshes. We use a heuristic to drive the front propagation procedure of the classical Fast Marching. This results in a modification of the Fast Marching algorithm that is similar to the A ∗ algorithm used in artificial intelligence. In order to find very quickly geodesic paths between any given pair of points, two methods are proposed to devise an heuristic that restrict the front propagation. The multiresolition heuristic computes the heuristic using a propagation on a coarse map. For applications where precomputation is acceptable, the landmarkbased heuristic precomputes distance maps to a sparse set of landmark points. We introduce various distortion metrics in order to quantify the errors introduced by the heuristically driven propagations. We show that both heuristic approaches bring a large speed up for large scale applications that require the extraction of geodesics on images and 3D meshes.
Computing Shortest Transversals
, 1991
"... We present an O(n log 2 n) time and O(n) space algorithm for computing the shortest line segment that intersects a set of n given line segments or lines in the plane. If the line segments do not intersect the algorithm may be trimmed to run in O(n log n) time. Furthermore, in combination with lin ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
We present an O(n log 2 n) time and O(n) space algorithm for computing the shortest line segment that intersects a set of n given line segments or lines in the plane. If the line segments do not intersect the algorithm may be trimmed to run in O(n log n) time. Furthermore, in combination with linear programming the algorithm will also find the shortest line segment that intersects a set of n isothetic rectangles in the plane in O(n log k) time, where k is the combinatorial complexity of the space of transversals and k 4n. These results find application in: (1) linefitting between a set of n data ranges where it is desired to obtain the shortest lineoffit, (2) finding the shortest line segment from which a convex nvertex polygon is weakly externally visible, and (3) determining the shortest lineofsight between two edges of a simple nvertex polygon, for which O(n) time algorithms are also given. All the algorithms are based on the solution to a new fundamental geometric optim...
SamplingBased Motion Planning
, 2006
"... There are two main philosophies for addressing the motion planning problem, in Formulation 4.1 from Section 4.3.1. This chapter presents one of the philosophies, samplingbased motion planning, which is outlined in Figure 5.1. The main idea is to avoid the explicit construction of Cobs, as described ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
There are two main philosophies for addressing the motion planning problem, in Formulation 4.1 from Section 4.3.1. This chapter presents one of the philosophies, samplingbased motion planning, which is outlined in Figure 5.1. The main idea is to avoid the explicit construction of Cobs, as described in Section 4.3, and instead conduct a search that probes the Cspace with a sampling scheme. This probing is enabled by a collision detection module, which the motion planning algorithm considers as a “black box. ” This enables the development of planning algorithms that are independent of the particular geometric models. The collision detection module handles concerns such as whether the models are semialgebraic sets, 3D triangles, nonconvex polyhedra, and so on. This general philosophy has been very successful in recent years for solving problems from robotics, manufacturing, and biological applications that involve thousands and even millions of geometric primitives. Such problems would be practically impossible to solve using techniques that explicitly represent Cobs. Notions of completeness It is useful to define several notions of completeness
Some ApertureAngle Optimization Problems
"... Let P and Q be two disjoint convex polygons in the plane with m and n vertices, respectively. Given a point ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
Let P and Q be two disjoint convex polygons in the plane with m and n vertices, respectively. Given a point