Abstract
Given a simple polygon P, we consider the problem of finding a convex polygon Q contained in P that minimizes H(P,Q), where H denotes the Hausdorff distance. We call such a polygon Q a Hausdorff core of P. We describe polynomial-time approximations for both the minimization and decision versions of the Hausdorff core problem, and we provide an argument supporting the hardness of the problem.
Funding for this research was made possible by the NSERC strategic grant on Optimal Data Structures for Organization and Retrieval of Spatial Data.
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Agarwal, P.K., Arge, L., Murali, T.M., Varadarajan, K.R., Vitter, J.S.: I/O-efficient algorithms for contour-line extraction and planar graph blocking. In: Proc. SODA, pp. 117–126. SIAM, Philadelphia (1998)
Aggarwal, A., Klawe, M.M., Moran, S., Shor, P., Wilber, R.: Geometric applications of a matrix-searching algorithm. Algorithmica 2(1), 195–208 (1987)
Aggarwal, A., Park, J.: Notes on searching in multidimensional monotone arrays. In: Proc. SFCS, pp. 497–512. IEEE Computer Society Press, Los Alamitos (1988)
Bajaj, C.: The algebraic degree of geometric optimization problems. Disc. & Comp. Geom. 3, 177–191 (1988)
Bhattacharya, B.K., Mukhopadhyay, A.: On the minimum perimeter triangle enclosing a convex polygon. In: Akiyama, J., Kano, M. (eds.) JCDCG 2002. LNCS, vol. 2866, pp. 84–96. Springer, Heidelberg (2003)
Bose, P., Toussaint, G.: Computing the constrained Euclidean geodesic and link center of a simple polygon with applications. In: Proc. CGI, p. 102. IEEE, Los Alamitos (1996)
Chang, J.S., Yap, C.K.: A polynomial solution for the potato-peeling problem. Disc. & Comp. Geom. 1(1), 155–182 (1986)
Chassery, J.-M., Coeurjolly, D.: Optimal shape and inclusion. In: Mathematical Morphology: 40 Years On, vol. 30, pp. 229–248. Springer, Heidelberg (2005)
Dobkin, D.P., Snyder, L.: On a general method for maximizing and minimizing among certain geometric problems. In: Proc. SFCS, pp. 9–17 (1979)
Fekete, S.P., Mitchell, J.S.B., Weinbrecht, K.: On the continuous Fermat-Weber problem. Oper. Res. 53, 61–76 (2005)
Hakimi, S.L.: Location theory. In: Rosen, Michaels, Gross, Grossman, Shier (eds.) Handbook Disc. & Comb. Math. CRC Press, Boca Raton (2000)
Kirkpatrick, D.: Optimal search in planar subdivisions. SIAM J. Comp. 12(1), 28–35 (1983)
Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Lin. Alg. & App. 284(1–3), 193–228 (1998)
Lopez, M.A., Reisner, S.: Hausdorff approximation of convex polygons. Comp. Geom. Theory & App. 32(2), 139–158 (2005)
Melissaratos, E.A., Souvaine, D.L.: On solving geometric optimization problems using shortest paths. In: Proc. SoCG, pp. 350–359. ACM Press, New York (1990)
O’Rourke, J., Aggarwal, A., Maddila, S., Baldwin, M.: An optimal algorithm for finding minimal enclosing triangles. J. Alg. 7, 258–269 (1986)
Schwarz, C., Teich, J., Vainshtein, A., Welzl, E., Evans, B.L.: Minimal enclosing parallelogram with application. In: Proc. SoCG, pp. 434–435. ACM Press, New York (1995)
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Dorrigiv, R. et al. (2009). Finding a Hausdorff Core of a Polygon: On Convex Polygon Containment with Bounded Hausdorff Distance. In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_20
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DOI: https://doi.org/10.1007/978-3-642-03367-4_20
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