Tight bounds for the rectangular art gallery problem

J. Czyzowicz; E. Rivera-Campo; N. Santoro; J. Urrutia; J. Zaks

doi:10.1007/3-540-55121-2_10

Tight bounds for the rectangular art gallery problem

J. Czyzowicz, E. Rivera-Campo, N. Santoro, J. Urrutia, J. Zaks

Department of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

Consider a rectangular art gallery, subdivided into n rectangular rooms; any two adjacent rooms have a door connecting them. We show that ⌈n/2⌉ guards are always sufficient to protect all rooms in a rectangular art gallery; furthermore, their positioning can be determined in O(n) time. We show that the optimal positioning of the guards can be determined in linear time. We extend the result by proving that in an arbitrary orthogonal art gallery (not necessarily convex, possibly having holes) with n rectangular rooms and k walls, ⌈(n+k)/2⌉ guards are always sufficient and occasionally necessary to guard all the rooms in our gallery. A linear time algorithm to find the positioning of the guards is obtained.

Original language	English
Title of host publication	Graph-Theoretic Concepts in Computer Science - 17th International Workshop, WG 1991, Proceedings
Editors	Gunther Schmidt, Rudolf Berghammer
Publisher	Springer Verlag
Pages	105-112
Number of pages	8
ISBN (Print)	9783540551218
DOIs	https://doi.org/10.1007/3-540-55121-2_10
State	Published - 1992
Event	17th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 1991 - Fischbachau, Germany Duration: 17 Jun 1991 → 19 Jun 1991

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	570 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	17th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 1991
Country/Territory	Germany
City	Fischbachau
Period	17/06/91 → 19/06/91

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1992.

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/3-540-55121-2_10

Cite this

Czyzowicz, J., Rivera-Campo, E., Santoro, N., Urrutia, J., & Zaks, J. (1992). Tight bounds for the rectangular art gallery problem. In G. Schmidt, & R. Berghammer (Eds.), Graph-Theoretic Concepts in Computer Science - 17th International Workshop, WG 1991, Proceedings (pp. 105-112). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 570 LNCS). Springer Verlag. https://doi.org/10.1007/3-540-55121-2_10

Czyzowicz, J. ; Rivera-Campo, E. ; Santoro, N. et al. / Tight bounds for the rectangular art gallery problem. Graph-Theoretic Concepts in Computer Science - 17th International Workshop, WG 1991, Proceedings. editor / Gunther Schmidt ; Rudolf Berghammer. Springer Verlag, 1992. pp. 105-112 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "Consider a rectangular art gallery, subdivided into n rectangular rooms; any two adjacent rooms have a door connecting them. We show that ⌈n/2⌉ guards are always sufficient to protect all rooms in a rectangular art gallery; furthermore, their positioning can be determined in O(n) time. We show that the optimal positioning of the guards can be determined in linear time. We extend the result by proving that in an arbitrary orthogonal art gallery (not necessarily convex, possibly having holes) with n rectangular rooms and k walls, ⌈(n+k)/2⌉ guards are always sufficient and occasionally necessary to guard all the rooms in our gallery. A linear time algorithm to find the positioning of the guards is obtained.",

author = "J. Czyzowicz and E. Rivera-Campo and N. Santoro and J. Urrutia and J. Zaks",

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Czyzowicz, J, Rivera-Campo, E, Santoro, N, Urrutia, J & Zaks, J 1992, Tight bounds for the rectangular art gallery problem. in G Schmidt & R Berghammer (eds), Graph-Theoretic Concepts in Computer Science - 17th International Workshop, WG 1991, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 570 LNCS, Springer Verlag, pp. 105-112, 17th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 1991, Fischbachau, Germany, 17/06/91. https://doi.org/10.1007/3-540-55121-2_10

Tight bounds for the rectangular art gallery problem. / Czyzowicz, J.; Rivera-Campo, E.; Santoro, N. et al.
Graph-Theoretic Concepts in Computer Science - 17th International Workshop, WG 1991, Proceedings. ed. / Gunther Schmidt; Rudolf Berghammer. Springer Verlag, 1992. p. 105-112 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 570 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - Consider a rectangular art gallery, subdivided into n rectangular rooms; any two adjacent rooms have a door connecting them. We show that ⌈n/2⌉ guards are always sufficient to protect all rooms in a rectangular art gallery; furthermore, their positioning can be determined in O(n) time. We show that the optimal positioning of the guards can be determined in linear time. We extend the result by proving that in an arbitrary orthogonal art gallery (not necessarily convex, possibly having holes) with n rectangular rooms and k walls, ⌈(n+k)/2⌉ guards are always sufficient and occasionally necessary to guard all the rooms in our gallery. A linear time algorithm to find the positioning of the guards is obtained.

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Czyzowicz J, Rivera-Campo E, Santoro N, Urrutia J, Zaks J. Tight bounds for the rectangular art gallery problem. In Schmidt G, Berghammer R, editors, Graph-Theoretic Concepts in Computer Science - 17th International Workshop, WG 1991, Proceedings. Springer Verlag. 1992. p. 105-112. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/3-540-55121-2_10

Tight bounds for the rectangular art gallery problem

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