The shortest path through many points

Beardwood, Jillian; Halton, John H.; Hammersley, J. M.

doi:10.1017/s0305004100034095

Cited by 749 publications

(520 citation statements)

References 14 publications

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“…If one follows the usual subadditivity argument (such as in [2,8]), one can go one step further and show that ELMsT(N.)>--/3MSTX/n---kl for a positive constant k r Also, adapting a classical argument given in [2] for the TSP, one can show that ELMsT(N n) < flMSTVn-+ k 2 for a positive constant k 2. Let us present these arguments (adapted from [4]), and combine the results in one lemma.…”

Section: Background Resultsmentioning

confidence: 99%

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Rate of convergence for the Euclidean minimum spanning tree limit law

Jaillet

1993

Operations Research Letters

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Section: Background Resultsmentioning

confidence: 99%

“…Questions about rates of convergence for these limit laws have been raised many times in the literature (see for example [2,5,8,9]). There are in fact two issues concerning information on rates of convergence (let P be a generic symbol representing any of the problems pre-cited):…”

Section: Introductionmentioning

confidence: 99%

Rate of convergence for the Euclidean minimum spanning tree limit law

Jaillet

1993

Operations Research Letters

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“…One can prove the second assertion of Theorem 1, by a variation of the approximation argument that Beardwood, Halton and Hammersley (1959) used in analysis of the traveling salesman problem. At this point the argument is routine, so we will just give a sketch.…”

Section: Completion Of the Argumentmentioning

confidence: 99%

“…. , X n ) that is of a kind that goes back to Beardwood, Halton and Hammersley (1959) for the traveling salesman problem.…”

Section: Introductionmentioning

confidence: 99%

Euclidean Networks with a Backbone and a Limit Theorem for Minimum Spanning Caterpillars

Jevtic

Steele

2015

Mathematics of OR

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Abstract. A caterpillar network (or graph) G is a tree with the property that removal of the leaf edges of G leaves one with a path. Here we focus on minimum weight spanning caterpillars (MSCs) where the vertices are points in the Euclidean plane and the costs of the path edges and the leaf edges are multiples of their corresponding Euclidean lengths. The flexibility in choosing the weight for path edges versus the weight for leaf edges gives some useful flexibility in modeling. In particular, one can accommodate problems motivated by communications theory such as the "last mile problem." Geometric and probabilistic inequalities are developed that lead to a limit theorem that is analogous to the well-known Beardwood, Halton Hammersley theorem for the length of the shortest tour through a random sample.

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“…Let ETSP(n) be a random variable returning the length of the Euclidean TSP tour through n points, independently and uniformly sampled from a compact set Q of unit area; in Beardwood et al (1959) it is shown that there exists a constant 2 such that, almost surely,…”

Section: B the Euclidean Traveling Salesman Problemmentioning

confidence: 99%

UAV Routing and Coordination in Stochastic, Dynamic Environments

Enright¹,

Pavone

Savla

2014

Handbook of Unmanned Aerial Vehicles

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Recent years have witnessed great advancements in the science and technology for unmanned aerial vehicles (UAVs), e.g., in terms of autonomy, sensing, and networking capabilities. This chapter surveys algorithms on task assignment and scheduling for one or multiple UAVs in a dynamic environment, in which targets arrive at random locations at random times, and remain active until one of the UAVs flies to the target's location and performs an on-site task. The objective is to minimize some measure of the targets' activity, e.g., the average amount of time during which a target remains active. The chapter focuses on a technical approach that relies upon methods from queueing theory, combinatorial optimization, and stochastic geometry. The main advantage of this approach is its ability to provide analytical estimates of the performance of the UAV system on a given problem, thus providing insight into how performance is affected by design and environmental parameters, such as the number of UAVs and the target distribution. In addition, the approach provides provable guarantees on the system's performance with respect to an ideal optimum. To illustrate this approach, a variety of scenarios are considered, ranging from the simplest case where one UAV moves along continuous paths and has unlimited sensing capabilities, to the case where the motion of the UAV is subject to curvature constraints, and finally to the case where the UAV has a finite sensor footprint. Finally, the problem of cooperative routing algorithms for multiple UAVs is considered, within the same queueing-theoretical framework, and with a focus on control decentralization.

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The shortest path through many points

Cited by 749 publications

References 14 publications

Rate of convergence for the Euclidean minimum spanning tree limit law

Rate of convergence for the Euclidean minimum spanning tree limit law

Euclidean Networks with a Backbone and a Limit Theorem for Minimum Spanning Caterpillars

UAV Routing and Coordination in Stochastic, Dynamic Environments

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