Efficiently approximating color-spanning balls

Khanteimouri, P ; Sharif University of Technology | 2016

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  1. Type of Document: Article
  2. DOI: 10.1016/j.tcs.2016.04.022
  3. Publisher: Elsevier , 2016
  4. Abstract:
  5. 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(n logn) time. This algorithm is then utilized to present a (1+ε)-approximation algorithm with the running time of O((1/ε)dn logn). We improve the running time to O((1/ε)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((1/ε)2d(n2k)log2n) time which is remarkable when k is large. This time can be reduced to O((1/ε)n2/k log n) when d=1
  6. Keywords:
  7. Approximation algorithms ; Color-spanning objects ; Complexity
  8. Source: Theoretical Computer Science ; Volume 634 , 2016 , Pages 120-126 ; 03043975 (ISSN)
  9. URL: http://www.sciencedirect.com/science/article/pii/S0304397516300615