Color Spanning Annulus: Square, Rectangle and Equilateral Triangle

TL;DR

This paper introduces efficient algorithms for finding minimum width color-spanning annuli of various shapes (square, rectangle, equilateral triangle) in 2D point sets, improving computational complexity over previous methods.

Contribution

It presents novel algorithms with improved time complexities for minimum width color-spanning annuli of square, rectangular, and equilateral triangular shapes.

Findings

Algorithm for CSSA runs in O(n^3 + n^2 k log k) time.

Algorithm for CSRA runs in O(n^4 log n) time.

Algorithm for CSETA runs in O(n^3 k) time.

Abstract

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 $O(k)$.