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- W2600326758 abstract "Recently, the concept of coarseness was introduced as a measure of how blended a 2-colored point set S is. In the definition of this measure, a convex partition Π , that is, a partition of S into sets { S 1 , … , S k } of S whose convex hulls are pairwise disjoint, is considered. The discrepancy of Π , denoted by d ( S , Π ) , is the smallest (bichromatic) discrepancy of the elements of Π . The coarseness of S , denoted by C ( S ) , is then defined as the maximum of d ( S , Π ) over all convex partitions Π of S . Roughly speaking, the value of the coarseness is high when we can split S into blocks, each with large discrepancy. It has been conjectured that computing the coarseness is NP-hard. In this paper, we study how to compute the coarseness for two constrained cases: (1) when the k elements of Π are separated by k − 1 pairwise parallel lines (strips) and, (2) the case in which the cardinality of the partition is fixed and the elements of Π are covered by pairwise disjoint axis-aligned rectangles (boxes). For the first case we present an O ( n 2 log 2 n ) -time algorithm, and show that such a computation problem is 3SUM-hard; for the second, we show that computing the coarseness with k boxes is NP-hard, when k is part of the input. For k fixed, we show that the coarseness can be computed in O ( n 2 k − 1 ) time and propose more efficient algorithms for k = 2 , 3 , 4 ." @default.
- W2600326758 created "2017-04-07" @default.
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- W2600326758 date "2017-06-01" @default.
- W2600326758 modified "2023-10-17" @default.
- W2600326758 title "Computing the coarseness with strips or boxes" @default.
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- W2600326758 doi "https://doi.org/10.1016/j.dam.2017.02.022" @default.
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