Minimum width color spanning annulus

Given a set P={p1,p2,…,pn} of n points in and each assigned with one of the given k distinct colors, we study the problem of finding the minimum width color spanning annulus of different shapes. Specifically, we consider the color spanning circular annulus (CSCA), axis-parallel square annulus (CSSA), axis parallel rectangular annulus (CSRA), and equilateral triangular annulus of fixed orientation (CSETA). The time complexities of the proposed algorithms for the respective problems are (i) O(n3log⁡n) for CSCA, (ii) O(n3+n2klog⁡k) for CSSA, (iii) O(n4) for CSRA, and (iv) O(n2k) for CSETA.