DMTCS Proceedings, Discrete Random Walks, DRW'03

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DMTCS Conference vol AC (2003), pp. 345-358


Discrete Random Walks, DRW'03

Cyril Banderier and Christian Krattenthaler (eds.)

DMTCS Conference Volume AC (2003), pp. 345-358

author: Nisheeth Vishnoi
title: Non Uniform Random Walks
keywords: Non uniform random walk
abstract: Given
∈ [0,1)
for each
1 < i < n,
a particle performs the following random walk on
If the particle is at
, it chooses a point uniformly at random (u.a.r.) from
If the current position of the particle is
), with probability
it decides to go back, in which case it chooses a point u.a.r. from
. With probability
it decides to go forward, in which case it chooses a point u.a.r. from
. The particle moves to the selected point.
What is the expected time taken by the particle to reach 1 if it starts the walk at
Apart from being a natural variant of the classical one dimensional random walk, variants and special cases of this problem arise in Theoretical Computer Science [Linial, Fagin, Karp, Vishnoi].
In this paper we study this problem and observe interesting properties of this walk. First we show that the expected number of times the particle visits
(before getting absorbed at 1) is the same when the walk is started at
for all
j > i.
Then we show that for the following parameterized family of
= (n-i) / (n-i+ α · (i-1)) , 1<i<n
does not depend on
the expected number of times the particle visits
is the same when the walk is started at
for all
Using these observations we obtain the expected absorption time for this family of
's. As
varies from infinity to 1, this time goes from
Θ(log n)
Θ (n).

Finally we study the behavior of the expected convergence time as a function of
. It remains an open question to determine whether this quantity increases when all
's are increased. We give some preliminary results to this effect.
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reference: Nisheeth Vishnoi (2003), Non Uniform Random Walks, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 345-358
bibtex: For a corresponding BibTeX entry, please consider our BibTeX-file.
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