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The shortest path through many points

  • Jillian Beardwood (a1), J. H. Halton (a2) and J. M. Hammersley (a3)
Abstract

We prove that the length of the shortest closed path through n points in a bounded plane region of area v is ‘almost always’ asymptotically proportional to √(nv) for large n; and we extend this result to bounded Lebesgue sets in k–dimensional Euclidean space. The constants of proportionality depend only upon the dimensionality of the space, and are independent of the shape of the region. We give numerical bounds for these constants for various values of k; and we estimate the constant in the particular case k = 2. The results are relevant to the travelling-salesman problem, Steiner's street network problem, and the Loberman—Weinberger wiring problem. They have possible generalizations in the direction of Plateau's problem and Douglas' problem.

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(1)Borůvka, O.On a minimal problem. Práce Moravské Pridovedecké Spolecnosti 3 (1926).
(2)Courant, R. and Robbins, H.What is mathematics? (Oxford, 1941).
(3)Courant, R. and Schiffer, M.Dirichlet's principle, conformed mapping, and minimal surfaces. (New York, 1950).
(4)Dantzig, G., Fulkerson, R. and Johnson, S.Solution of a large-scale traveling salesman problem. J. Oper. Res. Soc. Amer. 2 (1954), 393410.
(5)Douglas, J.Minimal surfaces of higher topological structure. Ann. Math., Princeton, 40 (1939), 205–98.
(6)Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.Tables of integral transforms. Vol. I. (New York, 1954).
(7)Flood, M. M.The traveling salesman problem. J. Oper. Res. Soc. Amer. 4 (1956), 6175.
(8)Ghosh, H. W.Expected travel among random points. Bull. Calcutta Statist. Ass. 2 (1948), 83–7.
(9)Halmos, P. R.Measure theory. (New York, 1950).
(10)Hardy, G. H.Divergent series. (Oxford, 1949).
(11)Heller, I.On the traveling salesman's problem. Part I. Basic Facts. George Washington Univ. Logistic Res. Project. (1954).
(12)Heller, I. On the traveling salesman's problem. Proc. Symp. Linear Programming. (Washington, 1955), 643–65.
(13)Holmes, H. S.Telegraph hand delivery. P.O. Telecomm. J. 9 (1957), 6671.
(14)Jahnke, E. and Emde, F.Tables of functions with formulae and curves. 4th ed. (New York, 1945).
(15)Jessen, R. J.Statistical investigation of a sample farm survey. Bull. Iowa St. Coll. Agric. Res. 304 (1942).
(16)Kruskal, J. B.On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Amer. Math. Soc. 7 (1956), 4850.
(17)Loberman, H. and Weinberger, A.Formal processes for connecting terminals with a minimum total wire length. J. Ass. Comp. Machy. 4 (1957), 428–37.
(18)Mahalanobis, P. C.A sample survey of the acreage under jute in Bengal. Sankhyā, 4 (1940), 511–31.
(19)Marks, E. S.A lower bound for the expected travel among m random points. Ann. Math. Statist. 19 (1948), 419–22.
(20)Morton, G. and Land, A. H.A contribution to the travelling-salesman problem. J. Roy. Statist. Soc. (B), 17 (1955), 185–94.
(21)Motzkin, T. S. The assignment problem. Proc. Symp. Appl. Math., Vol. 6., Numerical Analysis. (New York, 1956), 109–25.
(22)Oxford Atlas (Revised ed.) (Oxford, 1952).
(23)Tompkins, C. Permutation problems. Proc. Symp. Appl. Math., Vol. 6. Numerical Analysis (New York, 1956), 195211.
(24)Verblunsky, S.On the shortest path through a number of points. Proc. Amer. Math. Soc. (2) 6 (1951), 904–13.
(25)von Neumann, J.Functional operators, Vol. I. Annals of Math. Studies, 21 (Princeton, 1950).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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