This paper gives an algorithm for polytope covering: let L and U be sets of points in R d, comprising n points altogether. A cover for L from U is a set C⊂U with L a subset of the convex hull of C. Suppose c is the size of a smallest such cover, if it exists. The randomized algorithm given here finds a cover of size no more than c(5dln c), for c large enough. The algorithm requires O(c 2 n 1+δ) expected time. More exactly, the time bound is
, where γγ1/[d/2]. The previous best bounds were cO(log n) cover size in O(n d) time.[MS92b] A variant algorithm is applied to the problem of approximating the boundary of a polytope with the boundary of a simpler polytope. For an appropriate measure, an approximation with error ε requires c=O(d/ε)d−1 vertices, and the algorithm gives an approximation with c(5 d 3 ln(1/ε)) vertices. The algorithms apply ideas previously used for small-dimensional linear programming.
- Convex Hull
- Range Query
- Query Time
- Successful Iteration
- Expected Time
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Clarkson, K.L. (1993). Algorithms for polytope covering and approximation. In: Dehne, F., Sack, JR., Santoro, N., Whitesides, S. (eds) Algorithms and Data Structures. WADS 1993. Lecture Notes in Computer Science, vol 709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57155-8_252
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