Abstract
We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of n points in \(\mathbb {R}^d\) each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e., a random sample including each point with its existence probability. We are interested in computing certain expected statistics of a SCH, including diameter, width, and combinatorial complexity. For diameter, we establish the first deterministic 1.633-approximation algorithm with a time complexity polynomial in both n and d. For width, two approximation algorithms are provided: a deterministic O(1)-approximation running in \(O(n^{d+1} \log n)\) time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact \(O(n^d)\)-time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest.
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References
Agarwal, P.K., Aronov, B., Har-Peled, S., Phillips, J.M., Yi, K., Zhang, W.: Nearest neighbor searching under uncertainty II. In: Proc. of the 32nd PODS. ACM (2013)
Agarwal, P.K., Cheng, S.W., Yi, K.: Range searching on uncertain data. ACM Transactions on Algorithms (TALG) 8(4), 43 (2012)
Agarwal, P.K., Har-Peled, S., Suri, S., Yıldız, H., Zhang, W.: Convex hulls under uncertainty. In: Schulz, A.S., Wagner, D. (eds.) ESA 2014. LNCS, vol. 8737, pp. 37–48. Springer, Heidelberg (2014). doi:10.1007/978-3-662-44777-2_4
Fink, M., Hershberger, J., Kumar, N., Suri, S.: Hyperplane separability and convexity of probabilistic point sets. In: Proc. of the 32nd SoCG. ACM (2016)
Huang, L., Li, J.: Approximating the expected values for combinatorial optimization problems over stochastic points. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 910–921. Springer, Heidelberg (2015). doi:10.1007/978-3-662-47672-7_74
Huang, L., Li, J., Phillips, J.M., Wang, H.: \(\epsilon \)-kernel coresets for stochastic points. arXiv preprint arXiv:1411.0194 (2014)
Jørgensen, A., Löffler, M., Phillips, J.M.: Geometric computations on indecisive points. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 536–547. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22300-6_45
Kamousi, P., Chan, T., Suri, S.: Stochastic minimum spanning trees in euclidean spaces. In: Proc. of the 27th SoCG, pp. 65–74. ACM (2011)
Kamousi, P., Chan, T.M., Suri, S.: Closest pair and the post office problem for stochastic points. Computational Geometry 47(2), 214–223 (2014)
Li, C., Fan, C., Luo, J., Zhong, F., Zhu, B.: Expected computations on color spanning sets. Journal of Combinatorial Optimization 29(3), 589–604 (2015)
Löffler, M., van Kreveld, M.: Largest and smallest convex hulls for imprecise points. Algorithmica 56(2), 235–269 (2010)
Suri, S., Verbeek, K.: On the most likely Voronoi Diagram and nearest neighbor searching. In: Ahn, H.-K., Shin, C.-S. (eds.) ISAAC 2014. LNCS, vol. 8889, pp. 338–350. Springer, Cham (2014). doi:10.1007/978-3-319-13075-0_27
Suri, S., Verbeek, K., Yıldız, H.: On the most likely convex hull of uncertain points. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 791–802. Springer, Heidelberg (2013). doi:10.1007/978-3-642-40450-4_67
Xue, J., Li, Y., Janardan, R.: On the separability of stochastic geometric objects, with applications. In: Proc. of the 32nd SoCG. ACM (2016)
Xue, J., Li, Y.: Colored stochastic dominance problems. arXiv preprint arXiv:1612.06954 (2016)
Xue, J., Li, Y.: Stochastic closest-pair problem and most-likely nearest-neighbor search in tree spaces. arXiv preprint arXiv:1612.04890 (2016)
Xue, J., Li, Y., Janardan, R.: On the expected diameter, width, and complexity of a stochastic convex-hull. arXiv preprint arXiv:1704.07028 (2017)
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Xue, J., Li, Y., Janardan, R. (2017). On the Expected Diameter, Width, and Complexity of a Stochastic Convex-Hull. In: Ellen, F., Kolokolova, A., Sack, JR. (eds) Algorithms and Data Structures. WADS 2017. Lecture Notes in Computer Science(), vol 10389. Springer, Cham. https://doi.org/10.1007/978-3-319-62127-2_49
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DOI: https://doi.org/10.1007/978-3-319-62127-2_49
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