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Percolation and best-choice problem for powers of paths

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  22 June 2017

Fabricio Siqueira Benevides  and
Małgorzata Sulkowska
Fabricio Siqueira Benevides*
Affiliation:
Federal University of Ceará
Małgorzata Sulkowska*
Affiliation:
Wrocław University of Science and Technology
*
* Postal address: Department of Mathematics (Bloco 914), Federal University of Ceará, Av. Humberto Monte, s/n, 60.455-760, Fortaleza, Ceará, Brazil. Email address: fabricio@mat.ufc.br
** Postal address: Department of Computer Science, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, Grunwaldzki Sq. 13, 50-377 Wrocław, Poland. Email address: malgorzata.sulkowska@pwr.edu.pl
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Abstract

The vertices of the kth power of a directed path with n vertices are exposed one by one to a selector in some random order. At any time the selector can see the graph induced by the vertices that have already appeared. The selector's aim is to choose online the maximal vertex (i.e. the vertex with no outgoing edges). We give upper and lower bounds for the asymptotic behaviour of pn,kn1/(k+1), where pn,k is the probability of success under the optimal algorithm. In order to derive the upper bound, we consider a model in which the selector obtains some extra information about the edges that have already appeared. We give the exact asymptotics of the probability of success under the optimal algorithm in this case. In order to derive the lower bound, we analyse a site percolation process on a sequence of the kth powers of a directed path with n vertices.

Keywords

Site percolationsecretary problemsequence of graphsgraph powerpath

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 343 - 362
Copyright
Copyright © Applied Probability Trust 2017 

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