An (n; s) Davenport-Schinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly non-contiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between Davenport-Schinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A near-linear bound on the maximum length of Davenport-Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.

This paper explores the motion planning problem for multiple mov- ing objects. The approach taken consists of assigning priorities to the objects, then planning motions one object at a time. For each moving object, the planner constructs a configuration space-time that represents the time-varying constraints im- posed on the moving object by the other moving and stationary objects. The planner represents this space-time approximately, using two-dimensional slices. The space-time is then searched for a collision-free path. The paper demonstrates this approach in two domains. One domain consists of translating planar objects; the other domain consists of two-link planar articulated arms.

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

Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floating-point arithmetic. Such implementations have many well-known problems, here informally called "robustness issues". To reconcile theory and practice, authors have suggested that theoretical algorithms ought to be redesigned to become robust under fixed-precision arithmetic. We suggest that in many cases, implementors should make robustness a non-issue by computing exactly. The advantages of exact computation are too many to ignore. Many of the presumed difficulties of exact computation are partly surmountable and partly inherent with the robustness goal. This paper formulates the theoretical framework for exact computation based on algebraic numbers. We then examine the practical support needed to make the exact approach a viable alternative. It turns out that the exact computation paradigm encomp...

This paper gives approximation algorithms for solving the following motion planning problem: Given a set of polyhedral obstacles and points s and t, find a shortest path from s to t that avoids the obstacles. The paths found by the algorithms are piecewise linear, and the length of a path is the sum of the lengths of the line segments making up the path. Approximation algorithms will be given for versions of this problem in the plane and in three-dimensional space. The algorithms return an ɛ-short path, that is, a path with length within (1 + ɛ) of shortest. Let n be the total number of faces of the polyhedral obstacles, and ɛ a given value satisfying 0 < ɛ ≤ π. The algorithm for the planar case requires O(n log n)/ɛ time to build a data structure of size O(n/ɛ). Given points s and t, an ɛ-short path from s to t can be found with the use of the data structure in time O(n/ɛ + n log n). The data structure is associated with a new variety of Voronoi diagram. Given obstacles S ⊂ E 3 and points s, t ∈ E 3, an ɛ-short path between s and t can be found in O(n 2 λ(n) log(n/ɛ)/ɛ 4 + n 2 log nρ log(n log ρ)) time, where ρ is the ratio of the length of the longest obstacle edge to the distance between s and t. The function λ(n) = α(n) O(α(n)O(1)), where the α(n) is a form of inverse of Ackermann’s function. For log(1/ɛ) and log ρ that are O(log n), this bound is O(n 2 (log 2 n)λ(n)/ɛ 4). 1

In this paper, we show that the universal covering space of a surface can be used to unify previous results on computing paths in a simple polygon. We optimize a given path among obstacles in the plane under the Euclidean and link metrics and under polygonal convex distance functions. Besides revealing connections between the minimum paths under these three distance functions, the framework provided by the universal cover leads to simplified linear-time algorithms for shortest path trees, for minimum-link paths in simple polygons, and for paths restricted to c given orientations. 1 Introduction If a wire, a pipe, or a robot must traverse a path among obstacles in the plane, then one might ask what is the best route to take. For the wire, perhaps the shortest distance is best; for the pipe, perhaps the fewest straight-line segments. For the robot, either might be best depending on the relative costs of turning and moving. In this paper, we find shortest paths and shortest closed curve...

Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which itfolds. (3) Can every planar polygonal chain be straightened?

Consider a simple polyhedron P, possibly non-convex, composed of n triangular regions (faces), each assigned a positive weight indicating the cost of travel in that region. We present and experimentally study several algorithms to compute an approximate weighted geodesic shortest path, ß 0 (s; t), between two points s and t on the surface of P. Our algorithms are simple, practical, less prone to numerical problems, adaptable to a wide spectrum of weight functions, and use only elementary data structures. An additional feature of our algorithms is that execution time and space utilization can be traded off for accuracy; likewise, a sequence of approximate shortest paths for a given pair of points can be computed with increasing accuracy (and execution time) if desired. Dynamic changes to the polyhedron (removal, insertions of vertices or faces) are easily handled. The key step in these algorithms is the construction of a graph by introducing Steiner points on the edges of the given p...

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.

Given a convex polytope P with n faces in IR 3 , points s; t 2 @P , and a parameter 0 ! " 1, we present an algorithm that constructs a path on @P from s to t whose length is at most (1+ ")d P (s; t), where dP (s; t) is the length of the shortest path between s and t on @P . The algorithm runs in O(n log 1=" + 1=" 3 ) time, and is relatively simple to implement. The running time is O(n+1=" 3 ) if we only want the approximate shortest path distance and not the path itself. We also present an extension of the algorithm that computes approximate shortest path distances from a given source point on @P to all vertices of P . Work by the first and the fourth authors 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, and by matching funds from Xerox Corporation. Work by the first three authors has been supported by a grant from the U.S.--Israeli Binational Science ...