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

High-performance rendering engines in computer graphics are often pipelined, and their speed is bounded by the rate at which triangulation data can be sent into the machine. To reduce the data rate, it is desirable to order the triangles so that consecutive triangles share a face, meaning that only one additional vertex need be transmitted to describe each triangle. Such an ordering exists if and only if the dual graph of the triangulation contains a Hamiltonian path. In this paper, we consider several problems concerning triangulations with Hamiltonian duals. Specifically, we ffl Show that any set of n points in the plane has a Hamiltonian triangulation, and give two optimal \Theta(n log n) algorithms for constructing such a triangulation. We have implemented and tested both algorithms. ffl Consider the special case of sequential triangulations, where the Hamiltonian cycle is implied, and prove that certain non-degenerate point sets in the plane do not admit a sequential triangulati...

Let S be a set of n non-intersecting line segments in the plane. The visibility graph Gs of S is the graph that has the endpoints of the segments in S as nodes and in which two nodes are adjacent whenever they can "see"each other (i.e., the open line segment join- ing them is disjoint from all segments or is contained in a segment). Two new methods are presented to construct Gs. Both methods are very simple to implement. The first method is based on a new solution to the following problem: given a set of points, for each point sort the other points around it by angle. It runs in time O(n2). The second method uses the fact that visibility graphs often are sparse and runs in time O(m log n) where m is the number of edges in Gs. Both methods use only O(n) storage.

One strategy for the enumeration of a class of objects is local transformation, in which new objects of the class are produced by means of a small modification of a previously-visited object in the same class. When local transformation is possible, the operation can be used to generate objects of the class via random walks, and as the basis for such optimization heuristics as simulated annealing. For general simple polygons on fixed point sets, it is still not known whether the class of polygons on the set is connected via a constant-size local transformation. In this paper, we exhibit a simple local transformation for which the following polygon classes are connected: monotone, x-monotone, star-shaped, (weakly) edge-visible and (weakly) externally visible. The latter class is particularly interesting as it is the most general polygon class known to be connected under local transformation.

We develop a new finger search tree with worst case constant update time in the Pointer Machine (PM) model of computation. This was a major problem in the field of Data Structures and was tantalizingly open for over twenty years, while many attempts by researchers were made to solve it. The result comes as a consequence of the innovative mechanism that guides the rebalancing operations, combined with incremental multiple splitting and fusion techniques over nodes.

In research literature and many scientific disciplines, solution to the common problem in path planning for an autonomous robot has been extensively developed. Almost all explored techniques assume the robot has complete and detailed overview of the environment he is moving in. In addition to, many methods work over the graph representation of this environment which can be very difficult to construct or obtain in the real applications. This paper introduces a hybrid technique combining graph and grid representations of an examined space and capable of planning paths in known, partially known, unknown and dynamic environment at the price of the pseudo optimality of results. 1

. Given a set S of n non-intersecting line segments in the plane, we present an algorithm that computes the 2n visibility polygons of the endpoints of S, in output sensitive time. The algorithm relies on the ordered endpoint visibility graph information to traverse the endpoints of S in a spiral-like manner using a combination of Jarvis' March and depth-first search. One extension of this result is an efficient algorithm for computing the full visibility graph of S, in which vertices correspond to segments and a pair of vertices are joined by an edge if the corresponding line segments are somewhere visible. 1. Introduction. Problems involving the visibility of objects in a given domain arise in several areas of computer science, such as, VLSI design, graphics and motion planning. Visibility problems involving line segments in the plane are fundamental and many problems involving more general objects can be reduced (or approximated) by this case. Frequently, it is the underlying structu...

This thesis investigates geographic routing in mobile ad hoc networks. In geo-graphic routing, each node knows the position of one-hop neighbors, and packets are forwarded to a neighbor that is closer to the destination. This simple forward-ing strategy fails when there are obstacles such as voids or physical obstructions that prevent radio communications between nodes. Then, a recovery procedure is used to get around the obstacle. Earlier recovery techniques require that nodes know the exact position of other nodes. However, inexpensive position devices do not provide sufficient accuracy for these algorithms and it is difficult to obtain the information on moving nodes. Our approach to geographic routing begins with an observation that, if nodes know the shape and position of obstacles, they can make better routing decisions so that packets can avoid the obstacles in advance. A challenge is that obstacles in networks have complex shapes and change as edge nodes move. Our engineering solution is to partition the network into a grid and represent the obstacles on the grid map. The grid approximation represents obstacles by grid edges that are

Abstract. Quasi-Monte-Carlo methods are well-known for solving di erent problems of numerical analysis such asintegration, optimization, etc. The error estimates for global optimization depend on the dispersion of the point sequence with respect to balls. In general, the dispersion of a point set with respect to various classes of range spaces, like balls, squares, triangles, axis-parallel and arbitrary rectangles, spherical caps and slices, is the area of the largest empty range, and it is a measure for the distribution of the points. The main purpose of our paper is to give a survey about this topic, including some folklore results. Furthermore, we prove several properties of the dispersion, generalizing investigations of Niederreiter and others concerning balls. For several well-known uniformly distributed point sets we estimate the dispersion with respect to triangles, and we also compare them computationally. For the dispersion with respect to spherical slices we mention an application to the polygonal approximation of curves in space. 1.

High-performance rendering engines in computer graphics are often pipelined, and their speed is bounded by the rate at which triangulation data can be sent into the machine. To reduce the data rate, it is desirable to order the triangles so that consecutive triangles share a face, meaning that only one additional vertex need be transmitted to describe each triangle. Such an ordering exists if and only if the dual graph of the triangulation contains a Hamiltonian path. In this paper, we consider several problems concerning triangulations with Hamiltonian duals. Speci cally, we Show that any set of n points in general position in the plane has a Hamiltonian triangulation, and give an optimal (n log n) algorithm for constructing such a triangulation. Consider the special case of sequential triangulations, where the Hamiltonian cycle is implied, and prove that certain non-degenerate point sets in the plane do not admit a sequential triangulation. Further, we give e cient algorithms for testing whether a given triangulation of a point set or polygon is sequential. Show how to test whether a given polygon P has a Hamiltonian triangulation in time linear in the size of its visibility graph, and show that the problem is NP-complete for polygons with holes. Show how to add Steiner points to a given triangulation in order to create Hamiltonian triangulations which avoid narrow angles, thereby yielding guaranteed-quality Hamiltonian mesh generation. Give an encoding sequence for any triangulation whose length is at most 9=4 that of optimal.