Efficient algorithms for geometric optimization
ACM Computing Surveys (CSUR)
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The problem of detecting efficiently whether a query simplex is collision-free among polyhedral obstacles is considered. If $n$ is the number of vertices, edges, and faces of the polyhedral obstacles, and $m$ is the amount of storage allocated for the data structure $(n^{1 + \epsilon} \leq m \leq n^{4 + \epsilon})$, it is possible to solve collision-free placements queries for simplices in time $O(n^{1 + \epsilon}/m^{1/4})$ for any $\epsilon 0$, where the constants depend on $\epsilon$. In order to solve this problem the authors develop data structures to detect on-line intersections of query half planes with sets of lines and segments. Some nearest-neighbor problems for objects in 3-space are also considered. Given a set of $n$ lines in 3-space, the shortest vertical segment between any pair of lines is found in randomized expected time $O(n^{8/5 + \epsilon})$ for every $\epsilon 0$. The longest connecting vertical segment is found in time $O(n^{4/3 + \epsilon})$. The shortest connecting segment is found in time $O(n^{5/3 + \epsilon})$.