Recent Results in Art Galleries

Proceedings of the IEEE

Volumn 80, Issue 9, 1992, Pages 1384-1399

  Shermer, Thomas C ^a  

^a SIMON FRASER UNIVERSITY   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER GRAPHICS; GEOMETRY; GRAPH THEORY;

ART GALLERY THEOREMS; COMPUTATIONAL GEOMETRY; VISIBILITY GRAPHS;

IMAGE PROCESSING;

EID: 0026923143     PISSN: 00189219     EISSN: 15582256     Source Type: Journal    
DOI: 10.1109/5.163407     Document Type: Article

References (125)

1. J. Abello, and Omer Egecioglu, “Visibility graphs of staircase polygons with uniform step length,” manuscript, 1989.
2. A. Aggarwal, “The art gallery theorem: Its variations, applications, and algorithmic aspects,” Ph.D. thesis, Johns Hopkins University, Baltimore, 1984.
3. A. Aggarwal, S. Ghosh, and R. Shyamasundar, “Computational complexity of restricted polygon decompositions,” in Computational Morphology, G. Toussaint, Ed. New York: North-Holland, 1988, pp. 1–11.
4. M. Albertson and C. O'Keefe, “Covering regions with squares,” SIAM J. Algebraic Discrete Methods, vol. 2, pp. 240–243, 1981.
5. D. Avis and H. ElGindy, “A combinatorial approach to polygon similarity,” IEEE Trans. Inform. Theory, vol. 29, pp. 148–150, 1983.
6. D. Avis and G. Toussaint, “An optimal algorithm for determining the visibility of a polygon from an edge,” IEEE Trans. Comput., vol. 30, pp. 910–914, 1981.
7. D. Avis and G. Toussaint, “An efficient algorithm for decomposing a polygon into star-shaped pieces,” Pattern Recognition, vol. 13, pp. 295–298, 1981.
8. R. Bar-Yehuda and E. Ben-Chanoch, “An O(n log* n) time algorithm for covering simple polygons with squares,” inProc. 2nd Canadian Conf. Computational Geometry (Ottawa), 1990, pp. 186–190.
9. P. Belleville, “Computing two-covers of simple polygons,” M.Sc. thesis, McGill University, 1991.
10. M. Breen, “Clear visibility and the dimension of kernels of starshaped sets,” Proc. Amer. Math. Soc., vol. 85, pp. 414–418, 1982.
11. M. Breen, Characterizing compact unions of two starshaped sets in Rd,” J. Geometry, vol. 35, pp. 14–18, 1989.
12. W. Bronsvoort, “Boundary evaluation and direct display of CSG models,” Computer-Aided Design, vol. 20, pp. 416–419, 1988.
13. R. Burkard, G. Rote, E. Yao, and Z. Yu, “Shortest polygonal paths in space,” Computing, vol. 45, pp. 51–68, 1990.
14. S. Chaiken, D. Kleitman, M. Saks, and J. Shearer, “Covering regions by rectangles,” SIAM J. Algebraic Discrete Methods, vol. 2, pp. 394–410, 1981.
15. B. Chazelle, “Triangulating a simple polygon in linear time,” in Proc. 31th Symp. Foundations of Computer Science, 1990, pp. 220–230.
16. B. Chazelle and D. Dobkin, “Optimal convex decompositions,” in Computational Geometry, G. Toussaint, Ed. New York: North-Holland, 1985.
17. B. Chazelle and J. Incerpi, “Triangulation and shape-complexity,” ACM Trans. Graphics, vol. 3, pp. 135–152, 1984.
18. W.-P. Chin and S. Ntafos, “Shortest watchman routes in simple polygons,” Discrete and Computational Geometry, to be published.
19. W.-P. Chin and S. Ntafos, “Optimal zoo-keeper routes,” in Proc. Southeastern CGTC Conf, 1987, pp. 257–266.
20. W.-P. Chin and S. Ntafos, “Optimum watchman routes,” Information Processing Lett., vol. 28, pp. 39–44, 1988.
21. V. Chvatal, “A combinatorial theorem in plane geometry,” J. Combinatorial Theory Series B, vol. 18, pp. 39–41, 1975.
22. H. Conn and J. O'Rourke, “Some restricted rectangle covering problems,” in Proc. 1987 Allerton Conf
23. C. Coullard, B. Gamble, W. Lenhart, W. Pulleyblank, and G. Toussaint, “On illuminating a set of disks,” manuscript, 1989.
24. C. Coullard and A. Lubiw, “Distance visibility graphs,” in Proc. 7th Annual ACM Symp. Computational Geometry (North Conway), 1991, pp. 289–296.
25. D. Crass, I. Suzuki, and M. Yamashita, “The polygon search problem,” Tech. Rep. 90-11-01, Dept. Electrical Engineering and Computer Science, University of Wisconsin-Milwaukee.
26. J. Culberson and R. Reckhow, “A unified approach to orthogonal polygon covering problems via dent diagrams,” Tech. Rep. 89–6, Dept. Computing Science, University of Alberta.
27. J. Culberson and R. Reckhow, “Covering polygons is hard,” in Proc. 29th Symp. Foundations of Computer Science, 1988, pp. 601-611; manuscript, 1989.
28. J. Czyzowicz et al. “The aquarium keeper's problem,” presented at the Symposium on Data Structures and Algorithms, 1991.
29. J. Czyzowicz, E. Rivera-Campo, N. Santoro, J. Urrutia, and J. Zaks, “Guarding rectangular art galleries,” Tech. Rep. 89–27, Dept. Computer Sci., University of Ottawa, 1989.
30. J. Czyzowicz, E. Rivera-Campo, and J. Urrutia, “Illuminating rectangles and triangles on the plane,” Tech. Rep. 89–50, Dept. Computer Sci., University of Ottawa, 1989.
31. J. Czyzowicz, E. Rivera-Campo, J. Urrutia, and J. Zaks, “Illuminating lines and circles on the plane,” Tech. Rep. 89–49, Dept. Computer Sci., University of Ottawa, 1989.
32. J. Czyzowicz, E. Rivera-Campo, J. Urrutia, and J. Zaks, “Guarding convex sets,” University du Quebec a Hull, Informatique RR 90/09-12. Extended abstract, Proc. 2nd Canadian Conf Computational Geometry (Ottawa), 1990, pp. 265–268.
33. J. Czyzowicz, E. Rivera-Campo, J. Urrutia, and J. Zaks, “Illuminating and guarding segments in the plane,” Universite du Quebec a Hull, Informatique RR 90/09-13.
34. L. Davis and M. Benedikt, “Computational models of space: Isovists and isovist fields,” Computer Graphics and Image Processing, vol. 11, pp. 49–72, 1979.
35. J. Dean, A. Lingas, and J.-R. Sack, “Recognizing polygons, or how to spy,” The Visual Computer, vol. 3, pp. 344—355, 1988.
36. H. Edelsbrunner, Algorithms in Combinatorial Geometry. Berlin: Springer-Verlag, 1987.
37. H. Edelsbrunner, J. O'Rourke, and E. Welzl, “Stationing guards in rectilinear art galleries,” Computer Vision, Graphics, and Image Processing, vol. 27, pp. 167–176, 1984.
38. K. Eo and C. Kyung, “Hybrid shadow testing scheme for ray tracing,” Computer-Aided Design, vol. 21, pp. 38–48, 1989.
39. H. ElGindy, “Hierarchical decomposition of polygons with applications,” Ph.D. thesis, McGill University, 1985.
40. H. ElGindy and D. Avis, “A linear algorithm for computing the visibility polygon from a point,” J. Algorithms, vol 2, pp. 186–197, 1981.
41. V. Estivill-Castro and V. Raman, “Visibility in finitely oriented polygons,” in Proc.2nd Canadian Conf. Computational Geometry (Ottawa), 1990, pp. 181–185.
42. H. Everett, “Visibility graph recognition,” Ph.D. thesis, Department of Computer Science, University of Toronto, 1989.
43. H. Everett and D. Corneil, “Recognizing visibility graphs of spiral polygons,” J. Algorithms, vol. 11, pp. 1–26, 1990.
44. H. Everett and D. Corneil, “Forbidden induced subgraphs of visibility graphs,” manuscript, 1990.
45. H. Everett and G. Toussaint, “On illuminating isothetic rectangles in the plane,” manuscript, 1990.
46. L. Fejes Toth, “Illumination of convex discs,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 29, pp. 355–360, 1977.
47. S. Fisk, “A short proof of Chvatal's watchman theorem,” J. Combinatorial Theory Series B, vol. 24, p. 374, 1978.
48. D. Franzblau and D. Kleitman, “An algorithm for covering polygons with rectangles,” Information and Control, vol. 63, pp. 164–189, 1984.
49. Z. Furedi and D. Kleitman, “The prison yard problem,” manuscript, MIT, 1990.
50. R. Garey, and D. Johnson, Computers and Intractability. New York: W. H. Freeman, 1979.
51. L. Gewali and S. Ntafos, “Minimum covers for grids and orthogonal polygons by periscope guards,” in Proc. 2nd Canadian Conf Computational Geometry (Ottawa), 1990, pp. 358–361.
52. S. Ghosh, “Approximation algorithms for art gallery problems,” in Proc. Canadian Information Processing Society Congress, 1987.
53. S. Ghosh, “On recognizing and characterizing visibility graphs of simple polygons,” in Lecture Notes in Computer Science no. 318, R. Karlsson and A. Lingas, Eds. New York: Springer-Verlag, 1988, pp. 96–104.
54. S. Ghosh, A. Maheshwari, S. Pal, S. Saluja, and C. Veni Madhavan, “Characterizing weak visibility polygons and related problems,” Tech. Rep. IISc-CSA-90-1, Indian Institute of Science, Bangalore, Department of Computer Science and Automation; also in Proc. 2nd Canadian Conf Computational Geometry (Ottawa), 1990, pp. 93–97.
55. S. Ghosh, A. Maheshwari, S. Pal, and C. Veni Madhavan, “An algorithm for recognizing palm polygons,” in Proc. 2nd Canadian Conf Computational Geometry (Ottawa), 1990, pp. 246–251.
56. S. Computer Ghosh and D. Mount, “An output sensitive algorithm for computing visibility graphs,” in Proc. IEEE Symp. Foundations of Science, 1987, pp. 11–19.
57. H.-D. Hecker and D. Herwig, “Some NP-hard polygon cover problems,” J. Inf Process. Cybern. EIK, vol. 25,101-108,1989.
58. J. Hershberger, “Finding the visibility graph of a polygon in time proportional to its size,” in Proc. 3rd Annual ACM Symp. Computational Geometry (Waterloo), 1987, pp. 11–20.
59. F. Hoffmann, “On the rectilinear art gallery problem,” in Proc.Int.Colloq. Automata, Languages, and Programming, 1990.
60. A. Horn and F. Valentine, “Some properties of L-sets in the plane,” Duke Mathematics J., vol. 16, pp. 131–140, 1949.
61. C. Icking and R. Klein, “The two guards problem,” in Proc. 7th ACM Symp. Computational Geometry (North Conway), 1991, pp. 166–175.
62. H. Imai and T. Asano, “Efficient algoirthms for geometric graph search problems,” SIAM J. Computing, vol. 15, pp. 478–494, 1986.
63. G. Jennings, “Line segment visibility: Necessity, sufficiency and complexity,” senior thesis, Williams College, 1990.
64. J. Kahn, M. Klawe, and D. Kleitman, “Traditional galleries require fewer watchmen,” SIAM J. Algebraic and Discrete Methods, vol. 4, pp. 194–206, 1983.
65. S. Kapoor and S. Maheshwari, “Efficient algorithms for Euclidean shortest path and visibility problems with polygonal obstacles,” in Proc. Fourth ACM Symp. Computational Geometry (Urbana-Champaign), 1988, pp. 172–182.
66. D. Kay and M. Guay, “Convexity and a certain property Pm,” Israel J. Mathematics, vol. 8, pp. 39–52, 1970.
67. Y. Ke, “Polygon visibility algorithms for weak visibility and link distance problems,” Ph.D. thesis, Johns Hopkins University, Baltimore, 1989.
68. J. M. Keil, “Decomposing a polygon into simpler components,” SIAM J. Computing, vol. 14, pp. 799–817, 1985.
69. J. M. Keil, “Minimally covering a horizontally convex orthogonal polygon,” in Proc. 2nd ACM Symp. Computational Geometry, 1986, pp. 43–51.
70. R. Kuc and M. Siegel, “Efficient represenation of reflecting structures for a sonar navigation model,” in Proc. 1987 IEEE Int. Conf. Robotics and Automation, pp. 1916–1923.
71. D. Lee and A. Lin, “Computational complexity of art gallery problems,” IEEE Trans. Inform. Theory, vol. 32, pp. 276–282, 1986.
72. D. Lee and F. Preparata, “An optimal algorithm for finding the kernel of a polygon,” J. Ass. Comput. Mach., vol. 26, pp. 415–421, 1979.
73. W. Lenhart and G. Jennings, “An art gallery theorem for line segments in the plane,” manuscript, Williams College, 1990.
74. A. Lingas, “The power of non-rectilinear holes,” in Proc. 9th Colloq. Automata, Languages, and Programming, 1982, pp. 369–383.
75. W. Liou, J. Tan, and R. Lee, “Minimum partitioning simple rectilinear polygons in 0(n log log n)-time,” in Proc. Fifth ACM Symp. Computational Geometry (Saarbruchen), 1989, pp. 344–353.
76. T. Lozano-Perez and M. Wesley, “An algorithm for planning collision-free paths among polyhedral obstacles,” Commun. Ass. Comput. Mach., vol. 22, pp. 560–570, 1979.
77. A. Lubiw, “Decomposing polygonal regions into convex quadrilaterals,” in Proc. 1st ACM Computational Geometry Symp. (Baltimore), 1985, pp. 97–106.
78. A. Lubiw, “Orderings and some combinatorial optimization problems with geometric applications,” Ph.D. thesis, University of Toronto, 1985.
79. W. Masek, “Some NP-complete set covering problems,” manuscript, MIT, 1978 (cited in [50, p. 232]).
80. M. McKenna, “Worst-case optimal hidden-surface removal,” ACM Trans. Graphics, vol. 6, pp. 19–28, 1987.
81. J. Mitchell, “An algorithmic approach to some problems in terrain navigation,” in Geometric Reasoning, D. Kapur and J. Mundy, Eds. Cambridge, MA: MIT Press, 1988.
82. R. Motwani, A. Raghunathan, and H. Saran, “Perfect graphs and orthogonally convex covers,” SIAM J. Disc. Math., vol. 2, pp. 371–392, 1989.
83. R. Motwani, A. Raghunathan, and H. Saran, “Covering orthogonal polygons with star polygons: The perfect graph approach,” J. Computer and System Sciences, vol. 40, pp. 19–48, 1989.
84. J. Munro, M. Overmars, and D. Wood, “Variations on visibility,” in Proc. Third ACM Symp. Computational Geometry, 1987, pp. 291–299.
85. C. Nash-Williams, “Unexplored and semi-explored territories in graph theory,” in New Directions in Graph Theory, F. Harary, Ed. New York: Academic Press, 1973.
86. B. Nilsson and D. Wood, “Optimum watchmen routes in spiral polygons,” in Proc. 2nd Canadian Conf Computational Geometry (Ottawa), 1990, pp. 269–272.
87. S. Ntafos, “The robber route problem,” Information Processing Lett., vol. 34, pp. 59–63, 1990.
88. S. Ntafos, “Watchman routes under limited visibility,” in Proc. 2nd Canadian Conf.Computational Geometry (Ottawa) 1990, pp. 89–92.
89. S. Ntafos and L. Gewali, “External watchman routes,” presented at the First Canadian Conference on Computational Geometry, manuscript, 1989.
90. J. O'Rourke, “The complexity of computing minimum convex covers for polygons,” in Proc. 20th Allerton Conf. (Monticello), 1982, pp. 75–84.
91. J. O'Rourke, “Galleries need fewer mobile guards: A variation on Chvatal's theorem,” Geometriae Dedicata, vol. 14, pp. 273–283, 1983.
92. J. O'Rourke, Art Gallery Theorems and Algorithms. Oxford: Oxford University Press, 1987.
93. J. O'Rourke, “Recovery of convexity from visibility graphs,” Tech. Rep. 90.4.6, Dept. Computer Science, Smith College, 1990.
94. J. O'Rourke and K. Supowit, “Some NP-hard polygon decomposition problems,” IEEE Trans. Inform. Theory, vol. 29, pp. 181–190, 1983.
95. M. Overmars and E. Welzl, “New methods for computing visibility graphs,” in Proc. Fourth ACM Symp. Computational Geometry (Urbana-Champaign), 1988, pp. 164–171.
96. G. Pesant, “Factorizations of covers,” M.Sc. thesis, School of Computer Science, McGill University, 1989.
97. A. Raghunathan, “Polygon Decomposition and Perfect Graphs,” Ph.D. thesis, Department of Computer Science, University of California at Berkeley, 1988.
98. G. Rawlins, “Explorations in restricted orientation geometry,” Ph.D. thesis, TR CS-87-57, Department of Computer Science, University of Waterloo, 1987.
99. R. Reckhow and J. Culberson, “Covering a simple orthogonal polygon with a minimum number of orthogonally convex polygons,” in Proc.3rd ACM Symp. Computational Geometry (Waterloo), 1987, pp. 268–277.
100. G. Roy and J. Owen, “SR: A PC-based CAD system for shadow and reflection studies in the built environment,” Computer-Aided Design, vol. 21, pp. 497–504, 1989.
101. J. Sack, “An 0(n log n) algorithm for decomposing simple rectilinear polygons into convex quadrilaterals,” in Proc.20th Allerton Conf Communication, Control, and Computing (Monticello), 1982, pp. 64–74.
102. J. Sack and S. Suri, “An optimal algorithm for detecting weak visibility of a polygon,” IEEE Trans. Comput., vol. 39, pp. 1213–1219, 1990.
103. J. Sack and G. Toussaint, “A linear-time algorithm for decomposing rectilinear star-shaped polygons into convex quadrilaterals,” in Proc. 19th Allerton Conf.Communication, Control, and Computing (Monticello), 1981, pp. 21–30.
104. J. Sack and G. Toussaint, “Guard placement in rectilinear polygons,” in Computational Morphology G. Toussaint, Ed. New York: North-Holland, 1988.
105. S. Schuierer, G. Rawlins, and D. Wood, “Toward a general theory of visibility,” in Proc.2nd Canadian Conf.Computational Geometry (Ottawa), 1990, pp. 354–357.
106. T. Shermer, “Hiding people in polygons,” Computing, vol. 42, pp. 109–131, 1989.
107. T. Shermer, “Visibility properties of polygons,” Ph.D. thesis, McGill University, 1989.
108. T. Shermer, “Covering and guarding polygons using Lk-sets”, Geometriae Dedicata, vol. 37, pp. 183–203, 1991.
109. T. Shermer, “On recognizing unions of two convex polygons and related problems,” Pattern Recognition Lett., to be published.
110. T. Shermer, “Several short results in the combinatorics of visibility,” CMPT TR 91–2, Simon Fraser University, June 1991.
111. T. Shermer, “A tight bound on the combinatorial edge guard problem,” manuscript in preparation.
112. T. Shermer and G. Mac Donald, “Isomorphism of spiral polygons,” manuscript in preparation.
113. W. Stamey and J. Marr, “Unions of two convex sets,” Canadian J. Mathematics, vol. 15, pp. 152–156, 1963.
114. J. Stenstrom and C. Connolly, “Building wire frames for multiple range views,” in Proc. 1986 IEEE Int. Conf. Robotics and Automation, 1986, pp. 615–620.
115. K. Sugihara, I. Suzuki, and M. Yamashita, “The searchlight scheduling problem,” SIAM J. Computing, to be published.
116. S. Suri, “Minimum link paths in polygons and related problems,” Ph.D. thesis, Johns Hopkins University, Baltimore, 1987.
117. R. Tamassia and I. Tollis, “A unified approach to visibility representations of planar graphs,” Discrete and Computational Geometry, vol. 1, pp. 321–341, 1986.
118. R. Taijan and C. Van Wyk, “An O(n log log n)algorithm for triangulating simple polygons,” SIAM J. Computing, vol. 17, pp. 143–178, 1988.
119. G. Toussaint, “Pattern recognition and geometrical complexity,” in Proc. 5th Int. Conf Pattern Recognition, 1980, pp. 1324–1347.
120. J. Urrutia and J. Zaks, “Illuminating convex sets,” Tech. Rep. 89–31, Dept. Computer Sci., University of Ottawa, 1989.
121. F. Valentine, “Local convexity and Lnsets,” Proc. Amer.Math. Soc., vol. 16, pp. 1305–1310, 1965.
122. V. Vinay and C. Veni Madhavan, “The structure graphs of simple polygons,” manuscript, Department of Computer Science and Automation, Indian Institute of Science, Bangalore 1990.
123. S. Wismath, “Characterizing bar Iine-of-sight graphs,” in Proc. 1st ACM Computational Geometry Symp. (Baltimore), 1985, pp. 147–152.
124. S. Xie, T. Calvert, and B. Bhattacharya, “Planning views for the incremental construction of body models,” in Proc. Int. Conf Pattern Recognition (Paris), 1986, pp. 154–157.
125. M. Yachida, “3-D data acquisition by multiple views,” in Robotics Research: Third Int. Symp., 1986, pp. 11–18.