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
Reads0
Chats0
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
Citations
More filters
Posted Content
On the quadratic random matching problem in two-dimensional domains
TL;DR: In this paper, the average minimum cost of a bipartite matching, with respect to the squared Euclidean distance, between two samples of n i.i.d. random points on a bounded Lipschitz domain in the plane is investigated.
Journal ArticleDOI
Cost distributions in large combinatorial optimisation problems
Neil Burgess,M A Moore +1 more
TL;DR: The authors have examined the cost distribution of locally optimal solutions in certain combinatorial optimisation problems and found it to be peaked about a value characteristics of the algorithm involved, with a width that decreases with the system size N.
Dissertation
Random graphs for structure discovery in high-dimensional data.
TL;DR: A novel geometric probability approach to the problem of estimating intrinsic dimension and entropy of manifold data, based on asymptotic properties of graphs such as Minimal Spanning Trees or k-Nearest Neighbor graphs is developed, and statistical consistency of the obtained estimators for the wide class of Riemann submanifolds of an Euclidean space is proved.
Dynamic Vehicle Routing for Robotic Systems Planning optimal routes for multiple vehicles performing different tasks is discussed; fundamental limits on achievable performance are established for tasks that are generated by exogenous processes.
TL;DR: This paper surveys recent concepts and algorithms for dynamic vehicle routing (DVR), that is, for the automatic planning of optimal multivehicle routes to perform tasks that are generated over time by an exogenous process.
Optimal transport methods for combinatorial optimization over two random point sets
Michael Goldman,Dario Trevisan +1 more
TL;DR: In this article , the authors investigated the minimum cost of a wide class of combinatorial optimization problems over random bipartite geometric graphs, where the edge cost between two points is given by a $p$-th power of their Euclidean distance.
References
More filters
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.
Journal ArticleDOI