Journal ArticleDOI
The shortest path through many points
Jillian Beardwood,John H. Halton,J. M. Hammersley +2 more
- Vol. 55, Iss: 4, pp 299-327
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TLDR
In this paper, it was shown 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 this result was extended to bounded Lebesgue sets in k-dimensional Euclidean space.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.read more
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Journal IssueDOI
The snake for visualizing and for counting clusters in multivariate data
Adam Petrie,Thomas R. Willemain +1 more
TL;DR: The ‘snake’ is introduced, a new tool for the visualization and exploration of a multivariate dataset that traces the local structure of a datacloud, so this visualization is most useful for detecting density fluctuations.
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Two new Probability inequalities and Concentration Results
TL;DR: In this article, two probability inequalities which generalize and strengthen Martingale inequalities for combinatorial problems like the TSP, MWST, graph coloring, bin-packing, sub-graph counts and Johnson-Lindenstrauss random projection theorem are presented.
Book ChapterDOI
Stochastic integer programming: The distribution problem
TL;DR: A brief summary is given of recent insights into the distribution problem for structured stochastic integer programming problems, as surveyed during the Gargnano conference.
Journal ArticleDOI
Information in the Traveling Salesman Problem
TL;DR: This work finds that information increases as the Simulated Annealing temperature decreases, and proposes the use of the Shannon information content of the probability distribution function of inter–city step lengths to improve the standard algorithm.
Enhancing livability with feeder transit services: Formulation and solutions to first/last mile connectivity problem
TL;DR: Chandra et al. as discussed by the authors proposed a novel street connectivity indicator (C.I) to predict transit performance by identifying the role that street network connectivity plays in influencing the service quality of demand responsive feeder transit services.
References
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Journal ArticleDOI
On the shortest spanning subtree of a graph and the traveling salesman problem
TL;DR: Kurosh and Levitzki as discussed by the authors, on the radical of a general ring and three problems concerning nil rings, Bull Amer Math Soc vol 49 (1943) pp 913-919 10 -, On the structure of algebraic algebras and related rings.
Book ChapterDOI
Solution of a Large-Scale Traveling-Salesman Problem
TL;DR: The RAND Corporation in the early 1950s contained Arrow, Bellman, Dantzig, Flood, Ford, Fulkerson, Gale, Johnson, Nash, Orchard-Hays, Robinson, Shapley, Simon, Wagner, and other household names as discussed by the authors.
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