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Covering Uncertain Points in a Tree

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Abstract

In this paper, we consider a coverage problem for uncertain points in a tree. Let T be a tree containing a set \({\mathcal {P}}\) of n (weighted) demand points, and the location of each demand point \(P_i\in {{\mathcal {P}}}\) is uncertain but is known to appear in one of \(m_i\) points on T each associated with a probability. Given a covering range\(\lambda \), the problem is to find a minimum number of points (called centers) on T to build facilities for serving (or covering) these demand points in the sense that for each uncertain point \(P_i\in {{\mathcal {P}}}\), the expected distance from \(P_i\) to at least one center is no more than \(\lambda \). The problem has not been studied before. We present an \(O(|T|+M\log ^2 M)\) time algorithm for the problem, where |T| is the number of vertices of T and M is the total number of locations of all uncertain points of \({{\mathcal {P}}}\), i.e., \(M=\sum _{P_i\in {{\mathcal {P}}}}m_i\). In addition, by using this algorithm, we solve a k-center problem on T for the uncertain points of \({{\mathcal {P}}}\).

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Authors and Affiliations

  1. Department of Computer Science, Utah State University, Logan, UT, 84322, USA

    Haitao Wang

  2. Department of Computer Science, University of Texas - Rio Grande Valley, Edinburg, TX, 78539, USA

    Jingru Zhang

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  1. Haitao Wang
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  2. Jingru Zhang
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Correspondence to Jingru Zhang.

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A preliminary version of this paper appeared in the Proceedings of the 15th Algorithms and Data Structures Symposium (WADS 2017). This research was supported in part by NSF under Grant CCF-1317143.

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Wang, H., Zhang, J. Covering Uncertain Points in a Tree. Algorithmica 81, 2346–2376 (2019). https://doi.org/10.1007/s00453-018-00537-6

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