Color Spanning Annulus: Square, Rectangle and Equilateral Triangle

Abstract

In this paper, we study different variations of minimum width color-spanning annulus problem among a set of points P = {p1, p2, . . . , pn} in IR 2 , where each point is assigned with a color in {1, 2, . . . , k}. We present algorithms for finding a minimum width color-spanning axis parallel square annulus (CSSA), minimum width color spanning axis parallel rectangular annulus (CSRA), and minimum width color-spanning equilateral triangular annulus of fixed orientation (CSET A). The time complexities of computing (i) a CSSA is O(n 3 + n 2 k log k) which is an improvement by a factor n over the existing result on this problem, (ii) that for a CSRA is O(n 4 log n), and for (iii) a CSET A is O(n 3 k). The space complexity of all the algorithms is O(k).