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Approximate Euclidean shortest path in 3-space

Papadimitrious approximation approach to the Euclidean shortest path (ESP) problem in 3-space is revisited. As this problem is NP-hard, his approach represents an important step towards practical algorithms. Unfortunately, there are non-trivial gaps in the original description. Besides giving a complete treatment, we also give an alternative to his subdivision method which has some nice properties. Among the tools needed are root-separation bounds and non-trivial applications of Brents complexity bounds on evaluation of elementary functions using floating point numbers.

Precision-Sensitive Euclidean Shortest Path in 3-Space

This paper introduces the concept of precision-sensitive algorithms, analogous to the well-known output-sensitive algorithms. We exploit this idea in studying the complexity of the 3-dimensional Euclidean shortest path problem. Specifically, we analyze an incremental approximation approach and show that this approach yields an asymptotic improvement of running time. By using an optimization technique to improve paths on fixed edge sequences, we modify this algorithm to guarantee a relative error of O(2-r) in a time polynomial in r and

Computing the largest empty anchored cylinder, and related problems

1/\delta

Precision-sensitive Euclidean shortest path in 3-space (extended abstract)

, where

Lower Bounds for Geometrical and Physical Problems

\delta

Smallest enclosing cylinders

denotes the relative difference in path length between the shortest and the second shortest path. Our result is the best possible in some sense: if we have a strongly precision-sensitive algorithm, then we can show that unambiguous SAT (USAT) is in polynomial time, which is widely conjectured to be unlikely. Finally, we discuss the practicability of this approach. Experimental results are provided.

On the Topological Structure of Configuration Spaces

Let S be a set of n points in ℝd, and let each point p of S have a positive weight w(p). We consider the problem of computing a ray R emanating from the origin (resp. a line l through the origin) such that minp ∈ s w(p) · d(p,R) (resp. minp ∈ S w(p) · d(p,l)) is maximal. If all weights are one, this corresponds to computing a silo emanating from the origin (resp. a cylinder whose axis contains the origin) that does not contain any point of S and whose radius is maximal. For d = 2, we show how to solve these problems in O(n log n) time, which is optimal in the algebraic computation tree model. For d = 3, we give algorithms that are based on the parametric search technique and run in O(n log4 n) time. The previous best known algorithms for these three-dimensional problems had almost quadratic running time. In the final part of the paper, we consider some related problems.

On the topological structure of configuration spaces

Precision-Sensitive Euclidean Shortest Path in 3-Space *

Computing a Largest Empty Anchored Cylinder, and Related Problems

Motion planning involving arbitrarily many degrees of freedom is known to be PSPACE-hard. In this paper, we examine the complexity of generalized motion-planning problems for planar mechanisms consisting of independently movable objects. Our constructions constitute a general framework for reducing problems in information processing to motion planning, leading to easy proofs of known PSPACE-hardness results and to exponential lower bounds for geometrical problems related to motion planning. Particulalrly, we show that the problem of deciding whether a given mechanism

Subquadratic algorithms for the weighted maximin facility location problem.

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