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The two-center problem of uncertain points on a real line

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Abstract

Facility location problems on uncertain demand data have attracted significant attention recently. In this paper, we consider the two-center problem on uncertain points on a real line. The input is a set \(\mathcal {P}\) of n uncertain points on the line. Each uncertain point is represented by a probability density function that is a piecewise uniform distribution (i.e., a histogram) of complexity m. The goal is to find two points (centers) on the line so that the maximum expected distance of all uncertain points to their expected closest centers is minimized. A previous algorithm for the uncertain k-center problem can solve this problem in \(O(mn\log mn + n\log ^2n)\) time. In this paper, we propose a more efficient algorithm solving it in \(O(mn\log m+n\log n)\) time. Besides, we give an algorithm of the same time complexity for the discrete case where each uncertain point follows a discrete distribution.

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References

  • Agarwal P, Sharir M (1994) Planar geometric location problems. Algorithmica 11:185–195

    Article  MathSciNet  Google Scholar 

  • Alipour S (2020) Approximation algorithms for probabilistic \(k\)-center clustering. In: Proceedings of the IEEE international conference on data mining (ICDM), pp 1–11

  • Alipour S, Jafari A (2018) Improvements on the \(k\)-center problem for uncertain data. In: Proceedings of the 37th ACM SIGMOD-SIGACT-SIGAI symposium on principles of database systems, pp 425–433

  • Averbakh I, Bereg S (2005) Facility location problems with uncertainty on the plane. Discrete Optim. 2:3–34

    Article  MathSciNet  MATH  Google Scholar 

  • Averbakh I, Berman O (1997) Minimax regret \(p\)-center location on a network with demand uncertainty. Locat. Sci. 5:247–254

    Article  MATH  Google Scholar 

  • Banik A, Bhattacharya B, Das S, Kameda T, Song Z (2016) The \(p\)-center problem in tree networks revisited. In: Proceedings of the 15th Scandinavian symposium and workshops on algorithm theory (SWAT), pp 6:1–6:15

  • Ben-Moshe B, Bhattacharya B, Shi Q (2006) An optimal algorithm for the continuous/discrete weighted \(2\)-center problem in trees. In: Latin American symposium on theoretical informatics, pp 166–177

  • Bhattacharya B, Shi Q (2007) Optimal algorithms for the weighted \(p\)-center problems on the real line for small \(p\). In: Proceedings of the 10th international workshop on algorithms and data structures, pp 529–540

  • Chan T (1999) More planar two-center algorithms. Comput Geom Theory Appl 13:189–198

    Article  MathSciNet  MATH  Google Scholar 

  • Chau M, Cheng R, Kao B, Ngai J (2006) Uncertain data mining: An example in clustering location data. In: Proceedings of the 10th Pacific-Asia conference on advances in knowledge discovery and data mining (PAKDD), pp 199–204

  • Chazelle B, Guibas L (1986a) Fractional cascading: I. A data structuring technique. Algorithmica 1(1):133–162

  • Chazelle B, Guibas L (1986b) Fractional cascading: II. Applieacations. Algorithmica 1(1):163–191

  • Chen D, Wang H (2013) A note on searching line arrangements and applications. Inf Process Lett 113:518–521

    Article  MathSciNet  MATH  Google Scholar 

  • Chen D, Li J, Wang H (2015) Efficient algorithms for the one-dimensional \(k\)-center problem. Theor Comput Sci 592:135–142

    Article  MathSciNet  MATH  Google Scholar 

  • Cole R (1987) Slowing down sorting networks to obtain faster sorting algorithms. J ACM 34(1):200–208

    Article  MathSciNet  MATH  Google Scholar 

  • Eppstein D (1997) Faster construction of planar two-centers. In: Proceedings of the 8th Annual ACM-SIAM symposium on discrete algorithms, pp 131–138

  • Foul A (2006) A \(1\)-center problem on the plane with uniformly distributed demand points. Oper Res Lett 34(3):264–268

    Article  MathSciNet  MATH  Google Scholar 

  • Frederickson G (1991) Parametric search and locating supply centers in trees. In: Proceedings of the 2nd international workshop on algorithms and data structures (WADS), pp 299–319

  • Huang L, Li J (2017) Stochastic \(k\)-center and \(j\)-flat-center problems. In: Proceedings of the 28th ACM-SIAM symposium on discrete algorithms (SODA), pp 110–129

  • Jaromczyk J, Kowaluk M (1994) An efficient algorithm for the Euclidean two-center problem. In: Proceedings of the 10th annual symposium on computational geometry (SoCG), pp 303–311

  • Keikha V, Aghamolaei S, Mohades A, Ghodsi M (2021) Clustering geometrically-modeled points in the aggregated uncertainty model. CoRR arXiv:2111.13989

  • Megiddo N (1983) Applying parallel computation algorithms in the design of serial algorithms. J ACM 30(4):852–865

    Article  MathSciNet  MATH  Google Scholar 

  • Megiddo N, Supowit K (1984) On the complexity of some common geometric location problems. SIAM J Comput 13:182–196

    Article  MathSciNet  MATH  Google Scholar 

  • Megiddo N, Tamir A, Zemel E, Chandrasekaran R (1981) An \(o(n \log ^2 n)\) algorithm for the \(k\)-th longest path in a tree with applications to location problems. SIAM J Comput 10:328–337

    Article  MathSciNet  MATH  Google Scholar 

  • Nguyen Q, Zhang J (2021) Line-constrained \(l_\infty \) one-center problem on uncertain points. In: Proceedings of the 3rd international conference on advanced information science and system, vol 71, pp 1–5

  • Sharir M (1997) A near-linear algorithm for the planar 2-center problem. Discrete Comput Geom 18:125–134

    Article  MathSciNet  MATH  Google Scholar 

  • Snoeyink J (2007) Maximum independent set for intervals by divide and conquer with pruning. Networks 49:158–159

    Article  MathSciNet  MATH  Google Scholar 

  • Wang H (2014) Minmax regret 1-facility location on uncertain path networks. Eur J Oper Res 239:636–643

    Article  MathSciNet  MATH  Google Scholar 

  • Wang H (2020) On the planar two-center problem and circular hulls. In: Proceedings of the 36th international symposium on computational geometry (SoCG), vol 164, pp 68:1–68:14

  • Wang H, Zhang J (2015) One-dimensional \(k\)-center on uncertain data. Theor Comput Sci 602:114–124

    Article  MathSciNet  MATH  Google Scholar 

  • Wang H, Zhang J (2016) A note on computing the center of uncertain data on the real line. Oper Res Lett 44:370–373

    Article  MathSciNet  MATH  Google Scholar 

  • Wang H, Zhang J (2017) Computing the center of uncertain points on tree networks. Algorithmica 78(1):232–254

    Article  MathSciNet  MATH  Google Scholar 

  • Wang H, Zhang J (2021) An \(o(n\log n)\)-time algorithm for the \(k\)-center problem in trees. SIAM J Comput (SICOMP) 50(2):602–635

    Article  MATH  Google Scholar 

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  1. Cleveland State University, Cleveland, OH, 44115, USA

    Haitao Xu & Jingru Zhang

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

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Xu, H., Zhang, J. The two-center problem of uncertain points on a real line. J Comb Optim 45, 68 (2023). https://doi.org/10.1007/s10878-023-00996-w

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