Minimum width color spanning annulus

Color Spanning Annulus: Square, Rectangle and Equilateral Triangle

“…The color spanning annulus problem is comparatively new in the literature. Acharyya et al [4] presented two algorithms for finding the minimum width color spanning circular and axis-parallel square annulus. Both the algorithms run in O(n 4 log n) time using O(n) space.…”

“…The space complexity of all the algorithms is O(k). The algorithm for CSSA is an improvement of the existing result of [4] on this problem by a factor of n. Moreover, if k is constant, then the improvement factor is n log n.…”

“…Observation 1 [4] The points of distinct color lying on C in and C out are said to define an annulus A, and IN T (A) does not contain any point of color same as those defining the annulus A.…”

“…An axis-parallel color-spanning square annulus (CSSA) is a color-spanning annulus A bounded by two co-centric axis-parallel squares C out and C in . In [4], it is shown that either C in or C out of a minimum width square annulus (CSSA) has two points of different colors in its two boundaries. We prove a stronger claim.…”

“…If P ′ is not color spanning, then we can not have any CSSA contained in R. Hence we discard the horizontal strip defined by the pair p, q. Otherwise, for each point r ∈ P ′ (color(r) ∈ {color(p), color(q)}), we plot its distance in L ∞ metric from different points of line segment C. Each of these distance functions f (r) is a combination of line segments with slopes in {1, 0, −1} as described in [4,7] (see Fig. 1(b)).…”