Diverse Partitions Of Colored Points

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 - epsilon 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.