Color Spanning Annulus: Square, Rectangle and Equilateral Triangle

In this paper, we study different variations of minimum width color-spanning annulus problem among a set of points $P=\{p_1,p_2,\ldots,p_n\}$ in $I\!\!R^2$, where each point is assigned with a color in $\{1, 2, \ldots, k\}$. 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 $(CSETA)$. The time complexities of computing (i) a $CSSA$ is $O(n^3+n^2k\log k)$ 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 $CSETA$ is $O(n^3k)$. The space complexity of all the algorithms is $O(k)$.
Comment: 14 pages