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Guarding curvilinear art galleries with edge or mobile guards

Published: 02 June 2008 Publication History

Abstract

In this paper we consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves. We consider two types of guards: edge guards (these are edges of the polygon) and mobile guards (these are either edges or straight-line diagonals of the polygon). Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs. We reduce the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a constrained triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in the 2-dominating set. We show that, given a triangulation graph Tp of a polygon P with n ≥ 3 vertices: (1) ⌊n+1/3⌋ diagonal guards are always sufficient and sometimes necessary, and (2) ⌊2n+1/5⌋ edge guards are always sufficient and ⌊2n/5⌋ edge guards are sometimes necessary, in order to 2-dominate Tp. We also show that a diagonal (resp., edge) 2-dominating set of size ⌊2n+1/5⌋ can be computed in O(n2) time and O(n) space. Based on these results we prove that, in order to monitor a piecewise-convex polygon P witn n ≥ 2 vertices: (1) ⌊n+1/3⌋ mobile guards or ⌊2n+1/5⌋ edge guards are always suffcient, and (2) ⌊n/3⌋ mobile edge guards are sometimes necessary. A mobile (resp., edge) guard set for P of size ⌊n+1/3⌋ (resp., ⌊2n+1/5⌋ or ⌊3n/7⌋ can be computed in O(n log n + T(n)) time and O(n) space, where T(n) denotes the time for computing a diagonal (resp., edge) 2-dominating set of size ⌊n+1/3⌋ (resp., ⌊2n+1/5⌋ or ⌊3n/7⌋ for a trinagulation graph with n vertices.

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cover image ACM Conferences
SPM '08: Proceedings of the 2008 ACM symposium on Solid and physical modeling
June 2008
423 pages
ISBN:9781605581064
DOI:10.1145/1364901
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 02 June 2008

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Author Tags

  1. 2-dominance
  2. art gallery
  3. curvilinear polygons
  4. diagonal guards
  5. edge guards
  6. mobile guards
  7. piece-wise-convex polygons
  8. triangulation graphs

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