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The shortest path through many points

Published online by Cambridge University Press:  24 October 2008

Jillian Beardwood ,
J. H. Halton  and
J. M. Hammersley
Jillian Beardwood
Affiliation:
St Hugh's CollegeOxford
J. H. Halton
Affiliation:
Balliol CollegeOxford
J. M. Hammersley
Affiliation:
Trinity CollegeOxford
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 55 , Issue 4 , October 1959 , pp. 299 - 327
Copyright
Copyright © Cambridge Philosophical Society 1959

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