Probing a set of hyperplanes by lines and related problems

Aoki, Yasukazu; Imai, Hiroshi; Imai, Keisuke; Rappaport, David

doi:10.1007/3-540-57155-8_237

Cited by 1 publication

(1 citation statement)

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“…Motivated by the problem of having a robot discover the shape of an object by touching it [14], there is considerable amount of related work in the computational geometry literature on discovering polygonal and polyhedral shapes from probing (e.g., see [5,6,9,11,19,21,26,33,42,41]). Rather than review all of this work in detail, we refer the interested reader to the survey and book chapter by Skiena [40,43], and simply mention that, with the notable exception of work by Dobkin et al [19], this prior work is primarily directed at discovering obstacles in a two-dimensional environment using various kinds of contact probes.…”

Section: Related Workmentioning

confidence: 99%

Cloning Voronoi Diagrams via Retroactive Data Structures

Dickerson

Eppstein

Goodrich

2010

Algorithms – ESA 2010

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We address the problem of replicating a Voronoi diagram V (S) of a planar point set S by making proximity queries, which are of three possible (in decreasing order of information content):1. the exact location of the nearest site(s) in S 2. the distance to and label(s) of the nearest site(s) in S 3. a unique label for every nearest site in S. We provide algorithms showing how queries of Type 1 and Type 2 allow an exact cloning of V (S) with O(n) queries and O(n log 2 n) processing time. We also prove that queries of Type 3 can never exactly clone V (S), but we show that with O(n log 1 ε ) queries we can construct an ε-approximate cloning of V (S). In addition to showing the limits of nearest-neighbor database security, our methods also provide one of the first natural algorithmic applications of retroactive data structures, as several of our methods critically rely on the use of such structures, one of which is new to this paper.

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