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.

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

We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixed-precision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next we survey some recent applications of this paradigm. Finally, we outline some basic theory and techniques in this paradigm. 1 This paper will appear as a chapter in the 2nd edition of Computing in Euclidean Geometry, edited by D.-Z. Du and F.K. Hwang, published by World Scientific Press, 1994. 1 1 Two Numerical Computing Paradigms Computation has always been intimately associated with numbers: computability theory was early on formulated as a theory of computable numbers, the first computers have been number crunchers and the original mass-produced computers were pocket calculators. Although one's first exposure to computers today is likely to be some non-numerical application, numeri...

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...

shortest path, kinodynamics, polyhedral obstacles Abstract: We consider the following problem: given a robot system, nd a minimal-time trajectory that goes from a start state to a goal state while avoiding obstacles by a speed-dependent safety-margin and respecting dynamics bounds. In [1] we developed a provably good approximation algorithm for the minimum-time trajectory problem for a robot system with decoupled dynamics bounds (e.g., a point robot in R 3). This algorithm di ers from previous work in three ways. It is possible (1) to bound the goodness of the approximation by an error term �(2) to polynomially bound the computational complexity of our algorithm � and (3) to express the complexity as a polynomial function of the error term. Hence, given the geometric obstacles, dynamics bounds, and the error term, the algorithm returns a solution that is-close to optimal and requires only a polynomial (in ( 1)) amount of time. We extend the results of [1] in two ways. First, we modifyittohalve the exponent inthe polynomial bounds from 6d to 3d, so that that the new algorithm is O c d N 1 3d, where N is the geometric complexity of the obstacles and c is a robot-dependent constant. Second, the new algorithm nds a trajectory that matches the optimal in time with an factor sacri ced in the obstacle-avoidance safety margin. Similar results hold for polyhedral Cartesian manipulators in polyhedral environments. The new results indicate that an implementation of the algorithm could be reasonable, and a preliminary implementation has been done for the planar case.

: Kinodynamic planning attempts to solve a robot motion problem subject to simultaneous kinematic and dynamics constraints. In the general problem, given a robot system, we must find a minimal-time trajectory that goes from a start position and velocity to a goal position and velocity while avoiding obstacles by a safety margin and respecting constraints on velocity and acceleration. We consider the simplified case of a point mass under Newtonian mechanics, together with velocity and acceleration bounds. The point must be flown from a start to a goal, amidst polyhedral obstacles in 2D or 3D. While exact solutions to this problem are not known, we provide the first provably good approximation algorithm, and show that it runs in polynomial time. 1 An early version of this work appeared as [CDRX] 2 Computer Science Department, Cornell University, Ithaca, NY 14853 3 Computer Science Division, University of California, Berkeley, CA 94720 4 Computer Science Department, Duke Universit...

Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section...

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

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 ...

) Cristian S. Mata Joseph S. B. Mitchell y Abstract We present a practical new algorithm for the problem of computing low-cost paths in a weighted planar subdivision or on a weighted polyhedral surface. The algorithm is based on constructing a relatively sparse graph, a "pathnet", that links selected pairs of subdivision vertices with locally optimal paths. The pathnet can be searched for paths that are provably close to optimal and approach optimal, as one varies the parameter that controls the sparsity of the pathnet. We analyze our algorithm both analytically and experimentally. We report on the results of a set of experiments comparing the new algorithm with other standard methods. 1 Introduction For a given weight function, F : ! 2 ! !, the weighted length of an s-t path ß in the plane is the path integral, R ß F (x; y)doe, of the weight function along the path ß, linking the start s to the goal t. The weighted region metric associated with F defines the distance dF (...