Abstract
Imagine that a set of objects is represented by points in space and that different types or classes of objects are represented by colors. We study the algorithmic problem of creating convex or Voronoi partitions of space with maximally diverse cells, using two classic diversity measures: the richness (number of different colors) and the Shannon index. The diversity of a partition is the sum of the diversity scores of its cells. Hence, we wish to compute either a diverse convex partition (DCP) or a diverse Voronoi partition (DVP), which maximizes the diversity score of the partition. Surprisingly, computing a DVP is NP-hard already in 1D and for only four colors, while DCP can easily be computed with dynamic programming. We show that DVP can be solved in polynomial time in 1D if a discrete set of candidate positions for the Voronoi sites is part of the input. These results apply to both the richness and the Shannon index. For richness, we also present a polynomial-time algorithm to compute a Voronoi partition whose diversity is at least \(1-\varepsilon \) times the optimal diversity. In 2D, we show that both DCP and DVP are NP-hard, for richness as diversity measure. The reductions use constantly many colors for DVP and polynomially many colors for DCP.
Research on the topic of this paper was initiated at the 3rd Workshop on Applied Geometric Algorithms (AGA 2018) in Langbroek, The Netherlands. Marc van Kreveld and Jérôme Urhausen were partially supported by the Dutch Research Council (NWO) under project no. 612.001.651. Bettina Speckmann was partially supported by the Dutch Research Council (NWO) under project no. 639.023.208.
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References
Bereg, S., Bose, P., Kirkpatrick, D.: Equitable subdivisions within polygonal regions. Comput. Geom. 34(1), 20–27 (2006)
Bereg, S., Díaz-Báñez, J.M., Lara, D., Pérez-Lantero, P., Seara, C., Urrutia, J.: On the coarseness of bicolored point sets. Comput. Geom. 46(1), 65–77 (2013)
Bereg, S., et al.: Balanced partitions of 3-colored geometric sets in the plane. Discret. Appl. Math. 181, 21–32 (2015)
Bespamyatnikh, S., Kirkpatrick, D., Snoeyink, J.: Generalizing ham sandwich cuts to equitable subdivisions. Discret. Comput. Geom. 24(4), 605–622 (2000)
Blagojević, P.V., Rote, G., Steinmeyer, J.K., Ziegler, G.M.: Convex equipartitions of colored point sets. Discret. Comput. Geom. 61(2), 355–363 (2019)
Borodin, A., Jain, A., Lee, H.C., Ye, Y.: Max-sum diversification, monotone submodular functions, and dynamic updates. ACM Trans. Algorithms (TALG) 13(3), 1–25 (2017)
Chierichetti, F., Kumar, R., Lattanzi, S., Vassilvitskii, S.: Fair clustering through fairlets. In: Advances in Neural Information Processing Systems, pp. 5029–5037 (2017)
Drosou, M., Jagadish, H., Pitoura, E., Stoyanovich, J.: Diversity in big data: a review. Big Data 5(2), 73–84 (2017)
Dumitrescu, A., Pach, J.: Partitioning colored point sets into monochromatic parts. Int. J. Comput. Geom. Appl. 12(05), 401–412 (2002)
Farahani, R.Z., SteadieSeifi, M., Asgari, N.: Multiple criteria facility location problems: a survey. Appl. Math. Model. 34(7), 1689–1709 (2010)
Hartvigsen, D.: Recognizing Voronoi diagrams with linear programming. ORSA J. Comput. 4(4), 369–374 (1992)
Holmsen, A.F., Kynčl, J., Valculescu, C.: Near equipartitions of colored point sets. Comput. Geom. 65, 35–42 (2017)
Kaneko, A., Kano, M.: Discrete geometry on red and blue points in the plane, a survey. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry, The Goodman-Pollack Festschrift, pp. 551–570. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-642-55566-4_25
Lacevic, B., Amaldi, E.: On population diversity measures in Euclidean space. In: IEEE Congress on Evolutionary Computation, pp. 1–8 (2010)
Magurran, A.E.: Ecological Diversity and its Measurement. Princeton University Press, Princeton (1988)
Majumder, S., Nandy, S.C., Bhattacharya, B.B.: Separating multi-color points on a plane with fewest axis-parallel lines. Fundamenta Informaticae 99(3), 315–324 (2010)
Tang, E.K., Suganthan, P.N., Yao, X.: An analysis of diversity measures. Mach. Learn. 65(1), 247–271 (2006)
Tuomisto, H.: A diversity of beta diversities: straightening up a concept gone awry. Part 1. Defining beta diversity as a function of alpha and gamma diversity. Ecography 33(1), 2–22 (2010)
Whittaker, R.H.: Vegetation of the Siskiyou mountains, Oregon and California. Ecol. Monogr. 30(3), 279–338 (1960)
Zheng, K., Wang, H., Qi, Z., Li, J., Gao, H.: A survey of query result diversification. Knowl. Inf. Syst. 51(1), 1–36 (2016). https://doi.org/10.1007/s10115-016-0990-4
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van Kreveld, M., Speckmann, B., Urhausen, J. (2021). Diverse Partitions of Colored Points. In: Lubiw, A., Salavatipour, M., He, M. (eds) Algorithms and Data Structures. WADS 2021. Lecture Notes in Computer Science(), vol 12808. Springer, Cham. https://doi.org/10.1007/978-3-030-83508-8_46
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