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Some variations on constrained minimum enclosing circle problem

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Abstract

Given a set P of n points and a straight line L, we study three important variations of minimum enclosing circle problem as follows:

  1. (i)

    Computing k identical circles of minimum radius with centers on L, whose union covers all the points in P.

  2. (ii)

    Computing the minimum radius circle centered on L that can enclose at least k points of P.

  3. (iii)

    If each point in P is associated with one of the k given colors, then compute a minimum radius circle with center on L such that at least one point of each color lies inside it.

We propose three algorithms for Problem (i). The first one runs in O(nklogn) time and O(n) space. The second one is efficient where k≪n; it runs in O(nlogn+nk+k 2log3 n) time and O(nlogn) space. The third one is based on parametric search and it runs in O(nlogn+klog4 n) time. For Problem (ii), the time and space complexities of the proposed algorithm are O(nk) and O(n) respectively. For Problem (iii), our proposed algorithm runs in O(nlogn) time and O(n) space.

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Notes

  1. Note that, for each pair of consecutive points, there may not exist an element in the FV array.

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Authors and Affiliations

  1. School of Engineering, Tezpur University, Assam, 784028, India

    Arindam Karmakar

  2. Indian Statistical Institute, ACM Unit, Kolkata, 700108, India

    Sandip Das & Subhas C. Nandy

  3. School of Computing Science, Simon Fraser University, Burnaby, Canada, V5A 1S6

    Binay K. Bhattacharya

Authors
  1. Arindam Karmakar
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  2. Sandip Das
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  3. Subhas C. Nandy
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  4. Binay K. Bhattacharya
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Correspondence to Sandip Das.

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Karmakar, A., Das, S., Nandy, S.C. et al. Some variations on constrained minimum enclosing circle problem. J Comb Optim 25, 176–190 (2013). https://doi.org/10.1007/s10878-012-9452-4

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