The Annals of Applied Probability

Search cost for a nearly optimal path in a binary tree

Robin Pemantle

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Consider a binary tree, to the vertices of which are assigned independent Bernoulli random variables with mean p≤1/2. How many of these Bernoullis one must look at in order to find a path of length n from the root which maximizes, up to a factor of 1−ɛ, the sum of the Bernoullis along the path? In the case p=1/2 (the critical value for nontriviality), it is shown to take Θ(ɛ−1n) steps. In the case p<1/2, the number of steps is shown to be at least n⋅exp(const ɛ−1/2). This last result matches the known upper bound from [Algorithmica 22 (1998) 388–412] in a certain family of subcases.

Article information

Ann. Appl. Probab., Volume 19, Number 4 (2009), 1273-1291.

First available in Project Euclid: 27 July 2009

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Zentralblatt MATH identifier

Primary: 68W40: Analysis of algorithms [See also 68Q25] 68Q25: Analysis of algorithms and problem complexity [See also 68W40]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60C05: Combinatorial probability

Branching random walk minimal displacement maximal displacement optimal path algorithm computational complexity


Pemantle, Robin. Search cost for a nearly optimal path in a binary tree. Ann. Appl. Probab. 19 (2009), no. 4, 1273--1291. doi:10.1214/08-AAP585.

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