On the long edges in the shortest tour throughN random points

Abstract

Consider the shortest tour throughn pointsX 1,...,X n independently uniformly distributed over [0,1]2. Then we show that for some universal constantK, the number of edges of length at leastun −1/2 is at mostKnxp(−u)2/K)with overwhelmingprobability.

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References

  1. [1]

    J. J. Bartholdi, andL. K. Platzman: Heuristic based on space filling curves for combinatorial problems in Euclidean space,Management Science 34 (1988), 281–305.

    Google Scholar 

  2. [2]

    J. Beardwood, J. H. Halton, andJ. V. Hammersley: The shortest path through many points,Proc. Cambridge Philosophial Society 55 299–327.

  3. [3]

    W. Rhee, andM. Talagrand: A sharp deviation inequality for the stochastic traveling salesman problem,Annals of Probability 17 (1989), 1–8.

    Google Scholar 

  4. [4]

    J. M. Steele: Complete convergence of short paths and Karp's algorithm for the TSP,Math. Oper. Res. 6 (1981), 374–378.

    Google Scholar 

  5. [5]

    J. M. Steele: Seedlings in the Theory of Shortest Paths,Disorder in Physical Systems: A Volume in Honor of J. M. Hammerslet (G. Grimmett and D. Welsh eds.), Cambridge University Press, 277–306, Loddon, 1990.

    Google Scholar 

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This research is in part supported by an NSF grant.

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Rhee, W.T., Talagrand, M. On the long edges in the shortest tour throughN random points. Combinatorica 12, 323–330 (1992). https://doi.org/10.1007/BF01285821

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AMS subject classification code (1991)

  • 60 D 05