Introduction A natural and well-studied 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

This paper describes a new algorithm for constructing the set of free bitangents of a collection of n disjoint convex obstacles of constant complexity. The algorithm runs in time O(n log n + k), where k is the output size, and uses O(n) space. While earlier algorithms achieve the same optimal running time, this is the first optimal algorithm that uses only linear space. The visibility graph or the visibility complex can be computed in the same time and space. The only complicated data structure used by the algorithm is a splittable queue, which can be implemented easily using red--black trees. The algorithm is conceptually very simple, and should therefore be easy to implement and quite fast in practice. The algorithm relies on greedy pseudotriangulations, which are subgraphs of the visibility graph with many nice combinatorial properties. These properties, and thus the correctness of the algorithm, are partially derived from properties of a certain partial order on the faces of th...

We show that the k free bitangents of a collection of n pairwise disjoint convex plane sets can be computed in time O(k+n log n) and O(n) working space. The algorithm uses only one advanced data structure, namely a splittable queue. We introduce (weakly) greedy pseudo--triangulations, whose combinatorial properties are crucial for our method. 1 Introduction Consider a collection O of pairwise disjoint convex objects in the plane. We are interested in problems in which these objects arise as obstacles, either in connection with visibility problems where they can block the view from an other geometric object, or in motion planning, where these objects may prevent a moving object from moving along a straight line path. The visibility graph is a central object in such contexts. For polygonal obstacles the vertices of these polygons are the nodes of the visibility graph, and two nodes are connected by an arc if the corresponding vertices can see each other. [9] describes the first non-triv...

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.

Given a set of obstacles and two distinguished points in the plane the problem of finding a collision free path subject to a certain optimization function is a fundamental problem that arises in many fields, such as motion planning in robotics, wire routing in VLSI and logistics in operations research. In this survey we emphasize its applications to VLSI design and limit ourselves to the rectilinear domain in which the goal path to be computed and the underlying obstacles are all rectilinearly oriented, i.e., the segments are either horizontal or vertical. We consider different routing environments, and various optimization criteria pertaining to VLSI design, and provide a survey of results that have been developed in the past, present current results and give open problems for future research. 1 Introduction Given a set of obstacles and two distinguished points in the plane, the problem of finding a collision free path subject to a certain optimization function is a fundamental probl...

We present an approximation algorithm that, given a simple, possibly nonconvex polyhedron P with n vertices in R 3 , and two points s and t on its surface @P , constructs a path on @P between s and t whose length is at most 7(1 + ")ae, where ae is the length of the shortest path between s and t on @P , and " ? 0 is an arbitararily small positive constant. The algorithm runs in O(n 5=3 log 5=3 n) time. We also present a slightly faster algorithm that runs in O(n 8=5 log 8=5 n) time and returns a path whose length is at most 15(1 + ")ae. Work on this paper has been supported by National Science Foundation Grant CCR-93--01259, by an Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, by matching funds from Xerox Corporation, and by a grant from the U.S.--Israeli Binational Science Foundation. y Department of Computer Science, Box 90129, Duke University, krv@cs.duke.edu z Department of Computer Science, Box 90129, Duke University, pa...

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Motion planning is studied in a time-varying environment. Each obstacle is a convex polygon that moves in a fixed direction at a constant speed. The robot is a convex polygon that is subject to a speed bound. A method is presented to determine whether or not there is a translational collision-free motion for a polygonal robot from an initial position to a final position, and to plan such a motion, if it exists. Our method makes use of the concepts of configuration spaces and accessibility. An algorithm is given for motion planning in such an environment and its time complexity is analyzed.

We consider the problem of planning sensor strategies that enable a sensor to be automatically con g-ured for robot tasks. In this paper we present robust and e cient algorithms for computing the regions from which a sensor has unobstructed or partially obstructed views of a target in a goal. We apply these algorithms to the Error Detection and Recovery problem of recognizing whether a goal or failure region has been achieved. Based on these methods and strategies for visually-cued camera control, we have built a robot surveillance system in which one mobile robot navigates to a viewing position from which it has an unobstructed view of a goal region, and then uses visual recognition to detect when a speci c target has entered the room. 1

We give the first single-layer clock tree construction with exact zero skew according to the Elmore delay model. The previous Linear-Planar-DME method [11] guarantees a planar solution under the linear delay model. In this paper, we use a Linear-Planar-DME variant connection topology to construct a low-cost zero skew tree (ZST) according to the Elmore delay model. While a linear-delay ZST is trivially converted to an Elmore-delay ZST by "detouring" wires, the key idea is to defer this detouring as much as possible to reduce tree cost. Costs of our planar ZST solutions are comparable to those of the best previous non-planar ZST solutions, and substantially improve over previous planar clock routing methods.