Efficiently approximating color-spanning balls

• Published in: Theoretical computer science Vol. 634; pp. 120 - 126
• Main Authors: Khanteimouri, Payam, Mohades, Ali, Abam, Mohammad Ali, Kazemi, Mohammad Reza
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 27.06.2016
• Subjects: Algorithms; Approximation; Approximation algorithms; Color-spanning objects; Complexity; Computation; Computing time; Hypercubes; Mathematical analysis; Run time (computers); Running in; Approximation algorithms; Color-spanning objects; Complexity
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2016.04.022

Suppose n colored points with k colors in Rd are given. The Smallest Color-Spanning Ball (SCSB) is the smallest ball containing at least one point of each color. As the computation of the SCSB in Lp metric (p≥1) is time-consuming, we focus on approximately computing the SCSB in near-linear time. Initially, we propose a 3-approximation algorithm running in O(nlog⁡n) time. This algorithm is then utilized to present a (1+ε)-approximation algorithm with the running time of O((1ε)dnlog⁡n). We improve the running time to O((1ε)dn) using randomized techniques. Afterward, spanning colors with two balls is studied. For a special case where d=1, there is an algorithm with O(n2) time. We demonstrate that for any ε>0 under the assumption that SETH is true, no approximation algorithm running in O(n2−ε) time exists for the problem even in one-dimensional space. Nevertheless, we consider the L∞ metric where a ball is an axis-parallel hypercube and present a (1+ε)-approximation algorithm running in O((1ε)2d(n2k)log2⁡n) time which is remarkable when k is large. This time can be reduced to O((1ε)n2klog⁡n) when d=1.