### Non Uniform Random Walks

*Nisheeth Vishnoi*

#### Abstract

Given εi ∈ [0,1) for each 1 < i < n, a particle performs the following random walk on {1,2,...,n}:

If the particle is at n, it chooses a point uniformly at random (u.a.r.) from {1,...,n-1}. If the current position of the particle is m (1<m<n), with probability εm it decides to go back, in which case it chooses a point u.a.r. from {m+1,...,n}. With probability 1-εm it decides to go forward, in which case it chooses a point u.a.r. from {1,...,m-1}. 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 n?

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 i (before getting absorbed at 1) is the same when the walk is started at j, for all j > i. Then we show that for the following parameterized family of ε's: εi = (n-i) / (n-i+ α · (i-1)) , 1<i<n where α does not depend on i, the expected number of times the particle visits i is the same when the walk is started at j, for all j<i. Using these observations we obtain the expected absorption time for this family of ε's. As α varies from infinity to 1, this time goes from Θ(log n) to Θ (n).

Finally we study the behavior of the expected convergence time as a function of ε. It remains an open question to determine whether this quantity increases when all ε's are increased. We give some preliminary results to this effect.

If the particle is at n, it chooses a point uniformly at random (u.a.r.) from {1,...,n-1}. If the current position of the particle is m (1<m<n), with probability εm it decides to go back, in which case it chooses a point u.a.r. from {m+1,...,n}. With probability 1-εm it decides to go forward, in which case it chooses a point u.a.r. from {1,...,m-1}. 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 n?

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 i (before getting absorbed at 1) is the same when the walk is started at j, for all j > i. Then we show that for the following parameterized family of ε's: εi = (n-i) / (n-i+ α · (i-1)) , 1<i<n where α does not depend on i, the expected number of times the particle visits i is the same when the walk is started at j, for all j<i. Using these observations we obtain the expected absorption time for this family of ε's. As α varies from infinity to 1, this time goes from Θ(log n) to Θ (n).

Finally we study the behavior of the expected convergence time as a function of ε. It remains an open question to determine whether this quantity increases when all ε's are increased. We give some preliminary results to this effect.

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