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The opposite direction, that for any given P, e, h and q, the language L satisfying (2) is Turing-computable, follows from the fact that we can enumerate all points of P that are reachable from a given point 2 e (the algorithm has to follow the structure of P inductively, simplex for simplex, by marking the already detected reachable areas). 5. Future work.(More)
This paper addresses the complexity of computing the smallest-radius infinite cylinder that encloses an input set of n points in 3-space. We show that the problem can be solved in time O(n 4 log O(1) n) in an algebraic complexity model. We also achieve a time of O(n 4 L⋅μ(L)) in a bit complexity model where L is the maximum bit size of input numbers and(More)
Let S be a set of n points in IR d , and let each p o i n t p of S have a p o s i t i v e 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 min p2S w(p) d(pp R) (resp. min p2S w(p) d(pp l)) is maximal. If all weights are one, this corresponds to computing a silo emanating from(More)
Papadimitriou's 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(More)
This paper introduces the concept of precision-sensitive algorithms, in analogy 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 in-cremental approximation approach based on ideas in [CSY], and show that this approach yields an(More)
Let S be a set of n points in the plane, and let each point p of S have a positive weight w(p). We consider the problem of positioning a point x inside a compact region R ⊆ R 2 such that min{ w(p) −1 · d(x, p) ; p ∈ S } is maximized. Based on the parametric search paradigm, we give the first subquadratic algorithms for this problem, with running time O(n(More)