Abstract
Let \(P\) be a set of n points in \(\mathbb {R}^2\). For a parameter \(\varepsilon \in (0,1)\), a subset \(C\subseteq P\) is an \(\varepsilon \)-kernel of \(P\) if the projection of the convex hull of \(C\) approximates that of \(P\) within \((1-\varepsilon )\)-factor in every direction. The set \(C\) is a weak \(\varepsilon \)-kernel of \(P\) if its directional width approximates that of \(P\) in every direction. Let \(\textsf{k}_{\varepsilon }(P)\) (resp. \(\textsf{k}^{\textsf{w}}_{\varepsilon }(P)\)) denote the minimum-size of an \(\varepsilon \)-kernel (resp. weak \(\varepsilon \)-kernel) of \(P\). We present an \(O(n\textsf{k}_{\varepsilon }(P)\log n)\)-time algorithm for computing an \(\varepsilon \)-kernel of \(P\) of size \(\textsf{k}_{\varepsilon }(P)\), and an \(O(n^2\log n)\)-time algorithm for computing a weak \(\varepsilon \)-kernel of \(P\) of size \(\textsf{k}^{\textsf{w}}_{\varepsilon }(P)\). We also present a fast algorithm for the Hausdorff variant of this problem. In addition, we introduce the notion of \(\varepsilon \)-core, a convex polygon lying inside , prove that it is a good approximation of the optimal \(\varepsilon \)-kernel, present an efficient algorithm for computing it, and use it to compute an \(\varepsilon \)-kernel of small size.
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Notes
Recall that for two sets \(A, B \in \mathbb {R}^2\), \(H(A,B) = \max \{h(A,B), h(B,A)\}\), where \(h(X,Y) = \max _{x\in X}\min _{y\in Y} \Vert x-y\Vert \).
By computing the union of arcs in \(\** \), we can decide, in \(O(n\log n)\) time, whether \(\** \) covers .
References
Agarwal, P.K., Har-Peled, S.: Computing instance-optimal kernels in two dimensions. Proc. 29th Int. Symp. Comput. Geom. 2023, pp. 4:1–4:15 (2023)
Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Approximating extent measures of points. J. Assoc. Comput. Mach 51(4), 606–635 (2004). https://doi.org/10.1145/1008731.1008736http://www.acm.org/jacm/
Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Geometric approximation via coresets. In: Goodman, J.E., Pollack, R., Welzl, E. (eds.) Combinatorial and Computational Geometry, pp. 1–30. Cambridge University Press, Cambridge, UK (2005)
Agarwal, P.K., Kumar, N., Sintos, S., Suri, S.: Efficient algorithms for \(k\)-regret minimizing sets. In: Iliopoulos, C.S., Pissis, S.P., Puglisi, S.J., Raman, R. (eds.) 16th Int. Symp. Exper. Alg., (SEA), pp. 7–1723 (2017). https://doi.org/10.4230/LIPIcs.SEA.2017.7. https://drops.dagstuhl.de/opus/volltexte/2017/7632/
Agarwal, P.K., Yu, H.: A space-optimal data-stream algorithm for coresets in the plane. In: Erickson, J. (ed.) Proc. 23rd Annu. Sympos. Comput. Geom. pp. 1–10. ACM, New York, NY, USA (2007). https://doi.org/10.1145/1247069.1247071
Agarwal, P.K., Har-Peled, S., Yu, H.: Robust shape fitting via peeling and grating coresets. Discret. Comput. Geom. 39(1–3), 38–58 (2008). https://doi.org/10.1007/s00454-007-9013-2
Blum, A., Har-Peled, S., Raichel, B.: Sparse approximation via generating point sets. ACM Trans. Algo. 15(3), 32–13216 (2019). https://doi.org/10.1145/3302249
Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite vc-dimension. Discrete Comput. Geom. 14(4), 463–479 (1995). https://doi.org/10.1007/BF02570718
Cao, W., Li, J., Wang, H., Wang, K., Wang, R., Wong, R.C., Zhan, W.: k-regret minimizing set: Efficient algorithms and hardness. In: 20th Int. Conf. Data. Theory, pp. 11–11119 (2017). https://doi.org/10.4230/LIPIcs.ICDT.2017.11
Chan, T.M., Har-Peled, S., Jones, M.: Optimal algorithms for geometric centers and depth. CoRR abs/1912.01639 (2021). ar**v:1912.01639 [cs.CG]
Chan, T.M.: Faster core-set constructions and data-stream algorithms in fixed dimensions. Comput. Geom. Theory Appl. 35(1–2), 20–35 (2006). https://doi.org/10.1016/j.comgeo.2005.10.002
Chazelle, B., Guibas, L.J.: Fractional cascading: II. applications. Algorithmica 1(2), 163–191 (1986). https://doi.org/10.1007/BF01840441
Chazelle, B., Guibas, L.J.: Visibility and intersection problems in plane geometry. Discret. Comput. Geom. 4, 551–581 (1989). https://doi.org/10.1007/BF02187747
Clarkson, K.L.: Algorithms for polytope covering and approximation. In: Dehne, F.K.H.A., Sack, J., Santoro, N., Whitesides, S. (eds.) Proc. 3th Workshop Algorithms Data Struct. Lect. Notes in Comp. Sci., vol. 709, pp. 246–252. Springer, Berlin (1993). https://doi.org/10.1007/3-540-57155-8_252
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge, MA (2009). http://mitpress.mit.edu/books/introduction-algorithms
Das, G., Goodrich, M.T.: On the complexity of optimization problems for 3-dimensional convex polyhedra and decision trees. Comput. Geom. 8, 123–137 (1997). https://doi.org/10.1016/S0925-7721(97)00006-0
de Berg, M., Cheong, O., van Kreveld, M.J., Overmars, M.H.: Computational Geometry: Algorithms and Applications, 3rd Edition. Springer, Berlin, Germany (2008). https://www.worldcat.org/oclc/227584184
Ghosh, S.K., Maheshwari,A.: An optimal algorithm for computing a minimum nested nonconvex polygon. Inform. Process. Lett. 36(6), 277–280 (1990). https://doi.org/10.1016/0020-0190(90)90038-Y
Guibas, L.J., Hershberger, J., Mitchell, J.S.B., Snoeyink, J.: Approximating polygons and subdivisions with minimum-link paths. Int. J. Comput. Geom. Appl. 3(04), 383–415 (1993). https://doi.org/10.1142/S0218195993000257
Har-Peled, S.: Geometric Approximation Algorithms. Math. Surveys & Monographs, vol. 173. Amer. Math. Soc., Boston, MA, USA (2011). https://doi.org/10.1090/surv/173. http://sarielhp.org/book/
Hershberger, J., Suri, S.: A pedestrian approach to ray shooting: Shoot a ray, take a walk. J. Algorithms 18(3), 403–431 (1995). https://doi.org/10.1006/jagm.1995.1017
Imai, H., Iri, M.: An optimal algorithm for approximating a piecewise linear function. J. Info. Process. 9(3), 159–162 (1986)
Klimenko, G., Raichel, B.: Fast and exact convex hull simplification. In: Bojanczyk, M., Chekuri, C. (eds.) Proc. 41th Conf. Found. Soft. Tech. Theoret. Comput. LIPIcs, vol. 213, pp. 26–12617. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Wadern, Germany (2021). https://doi.org/10.4230/LIPIcs.FSTTCS.2021.26
Lee, C.C., Lee, D.T.: On a circle-cover minimization problem. Inf. Process. Lett. 18(2), 109–115 (1984). https://doi.org/10.1016/0020-0190(84)90033-4
Mitchell, J.S., Polishchuk, V.: Minimum-perimeter enclosures. Inform. Process. Lett. 107(3–4), 120–124 (2008). https://doi.org/10.1016/j.ipl.2008.02.007
Mitchell, J.S., Suri, S.: Separation and approximation of polyhedral objects. Comput. Geom. Theory Appl. 5(2), 95–114 (1995). https://doi.org/10.1016/0925-7721(95)00006-U
Preparata, F.P., Shamos, M.I.: Computational Geometry - An Introduction. Springer, New York, NY (1985)
Schlag, M., Luccio, F., Maestrini, P., Lee, D.T., Wong, C.K.: A visibility problem in VLSI layout compaction. In: Preparata, F.P. (ed.) Advances in Computing Research, Volume 2: VLSI Theory, pp. 259–282. JAI Press Inc., Greenwich, CT (1984)
Wang, C.A., Chan, E.P.F.: Finding the minimum visible vertex distance between two non-intersecting simple polygons. In: Aggarwal, A. (ed.) Proc. 2nd Annu. Sympos. Comput. Geom., pp. 34–42. ACM, New York, NY, USA (1986). https://doi.org/10.1145/10515.10519
Wang, Y., Mathioudakis, M., Li, Y., Tan, K.: Minimum coresets for maxima representation of multidimensional data. In: Libkin, L., Pichler, R., Guagliardo, P. (eds.) Proc. 40th Symp. Principles Database Sys., pp. 138–152. ACM, New York, NY, USA (2021). https://doi.org/10.1145/3452021.3458322
Wang, C.A.: Finding minimal nested polygons. BIT Numerical Mathematics 31(2), 230–236 (1991). https://doi.org/10.1007/bf01931283
Yu, H., Agarwal, P.K., Poreddy, R., Varadarajan, K.R.: Practical methods for shape fitting and kinetic data structures using coresets. Algorithmica 52(3), 378–402 (2008). https://doi.org/10.1007/s00453-007-9067-9
Acknowledgements
We thank the reviewers for their helpful comments. Work by the first author on this paper was partially supported by NSF grants IIS-18-14493 and CCF-20-07556. Work by the second author on this paper was partially supported by an NSF AF awards CCF-1907400 and CCF-2317241.
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Agarwal, P.K., Har-Peled, S. Computing Instance-Optimal Kernels in Two Dimensions. Discrete Comput Geom (2024). https://doi.org/10.1007/s00454-024-00637-x
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DOI: https://doi.org/10.1007/s00454-024-00637-x