Abstract
In this paper, we first present a linear-time algorithm to find the smallest circle enclosing n given points in \(\mathfrak {R}^2\) with the constraint that the center of the smallest enclosing circle lies inside a given disk. We extend this result to \(\mathfrak {R}^3\) by computing constrained smallest enclosing sphere centered on a given sphere. We generalize the result for the case of points in \(\mathfrak {R}^d\) where center of the minimum enclosing ball lies inside a given ball. We show that similar problem of minimum intersecting/stabbing ball for set of hyper planes in \(\mathfrak {R}^d\) can also be solved using similar techniques. We also show how minimum intersecting disk with center constrained on a given disk can be computed to intersect a set of convex polygons. Lastly, we show that this technique is applicable when the center of minimum enclosing/intersecting ball lies in a convex region bounded by constant number of non-linear constraints with computability assumptions. We solve each of these problems in linear time complexity for fixed dimension.
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Barba, L., Bose, P., Langerman, S.: Optimal algorithms for constrained 1-center problems. In: Pardo, A., Viola, A. (eds.) LATIN 2014. LNCS, vol. 8392, pp. 84–95. Springer, Heidelberg (2014)
Bose, P., Toussaint, G.T.: Computing the constrained euclidean geodesic and link center of a simple polygon with application. Comput. Graph. Int. Conf. CGI 1996, 102–110 (1996)
Bose, P., Wang, Q.: Facility location constrained to a polygonal domain. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 153–164. Springer, Heidelberg (2002)
Chazelle, B., Matousek, J.: On linear-time deterministic algorithms for optimization problems in fixed dimension. J. Algorithms 21(3), 579–597 (1996)
Clarkson, K.L.: A randomized algorithm for closest-point queries. SIAM J. Comput. 17(4), 830–847 (1988)
Dyer, M.E.: On a multidimensional search technique and its application to the euclidean one-centre problem. SIAM J. Comput. 15(3), 725–738 (1986)
Hurtado, F., Sacristan, V., Toussaint, G.: Some constrained minimax and maximin location problems. Studies in Locational Analysis, 15:1735 (2000)
Jadhav, S., Mukhopadhyay, A., Bhattacharya, B.K.: An optimal algorithm for the intersection radius of a set of convex polygons. J. Algorithms 20(2), 244–267 (1996)
Matousek, J., Sharir, M., Welzl, E.: A subexponential bound for linear programming. Algorithmica 16(4/5), 498–516 (1996)
Megiddo, N.: Linear-time algorithms for linear programming in r\({}^{\text{3 }}\) and related problems. SIAM J. Comput. 12(4), 759–776 (1983)
Megiddo, N.: Linear programming in linear time when the dimension is fixed. J. ACM 31(1), 114–127 (1984)
Preparata, F.: Minimum spanning circle. In: Steps into Computational Geometry, Technical report, University Illinois, Urbana, IL (1977)
Roy, S., Karmakar, A., Das, S., Nandy, S.C.: Constrained minimum enclosing circle with center on a query line segment. Comput. Geom. 42(6–7), 632–638 (2009)
Shamos, M.I.: Computational Geometry. Ph.D. thesis, Department of Computer Science, Yale Universiy, New Haven, CT (1978)
Shamos, M.I., Hoey, D.: Closest-point problems. In: 16th Annual Symposium on Foundations of Computer Science, pp. 151–162 (1975)
Sharir, M., Welzl, E.: A combinatorial bound for linear programming and related problems. In: STACS 92, 9th Annual Symposium on Theoretical Aspects of Computer Science, pp. 569–579 (1992)
Sylvester, J.J.: A question in the geometry of situation. Q. J. Math. 1, 79 (1857)
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Das, S., Nandy, A., Sarvottamananda, S. (2016). Linear Time Algorithms for Euclidean 1-Center in \(\mathfrak {R}^d\) with Non-linear Convex Constraints. In: Govindarajan, S., Maheshwari, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2016. Lecture Notes in Computer Science(), vol 9602. Springer, Cham. https://doi.org/10.1007/978-3-319-29221-2_11
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DOI: https://doi.org/10.1007/978-3-319-29221-2_11
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