On the coarseness of bicolored point sets

New results on the coarseness of bicolored point sets

“…Specifically, the approximate value of the coarseness that we provide is at least max C(S) 128 , C(S) 64 − disc(S) and at most C(S). With this result, we solve an open problem posted by Bereg et al [4].…”

“…Given a finite point set S in the plane and a coloring of S, computing the coarseness of S is believed to be NP-hard [4]. We also show for the first time that there exists a polynomial-time constant approximation algorithm.…”

“…. , k. The coarseness of S, denoted by C(S), is defined as the maximum of disc(Π) over all the convex partitions Π of S. This concept of coarseness was just recently defined by Bereg et al [4], as a parameter to measure how well blended a finite set R of red points and a finite set B of blue points are. The smaller C(R ∪ B) the more blended R and B [4].…”

Separability of Imprecise Points

“…Following that, Everett et al [11] solved this problem in O (nk log k + n log n) time, where k is the number of misclassified points. Chan [6] presented an O ((n + k 2 ) log n)-expected time algorithm for this problem, and Bereg et al [4] showed that the problem is 3SUM-hard. Cortés et al [7] presented an O (n 2 log n)-time algorithm for the weak separability problem for a strip.…”

“…of size O (r 2 log r) for a point set with m points can be computed in time O (m(r 2 log r) d ), where d is the exponent of the socalled shatter function of the underlying range space. For rectangles we have d = 4, so plugging in m= O (n(1/δ 1 ) log(1/δ 1 )) and r = 1/δ 2 gives a δ 2 -approximation of size O ((1/δ 2 ) 2 log(1/δ 2 )) in time O (n(1/δ 1 ) log(1/δ 1 )(1/δ 2 ) 8 log4 (1/δ 2 )). Since the size of the δ 2 -approximations is constant, the last step of the algorithm takes constant time.…”