A tight upper bound on the cover time for random walks on graphs

1995 ◽  
Vol 6 (1) ◽  
pp. 51-54 ◽  
Author(s):  
Uriel Feige
Keyword(s):  
Random Walks ◽  
Upper Bound ◽  
Cover Time ◽  
Random Walks On Graphs

scholarly journals On the cover time of random walks on graphs

1989 ◽  
Vol 2 (1) ◽  
pp. 121-128 ◽  
Author(s):  
Jeff D. Kahn ◽  
Nathan Linial ◽  
Noam Nisan ◽  
Michael E. Saks
Keyword(s):  
Random Walks ◽  
Cover Time ◽  
Random Walks On Graphs

Many Random Walks Are Faster Than One

2011 ◽  
Vol 20 (4) ◽  
pp. 481-502 ◽  
Author(s):  
NOGA ALON ◽  
CHEN AVIN ◽  
MICHAL KOUCKÝ ◽  
GADY KOZMA ◽  
ZVI LOTKER ◽  
...  
Keyword(s):  
Distributed Computing ◽  
Random Walks ◽  
Efficient Algorithms ◽  
Large Collection ◽  
Cover Time ◽  
Expected Time ◽  
Probabilistic Distribution ◽  
Random Walks On Graphs ◽  
Speed Up ◽  
Time Required

We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time – the expected time required to visit every node in a graph at least once – and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probabilistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s–t connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds.

scholarly journals Coalescent random walks on graphs

2007 ◽  
Vol 202 (1) ◽  
pp. 144-154 ◽  
Author(s):  
Jianjun Paul Tian ◽  
Zhenqiu Liu
Keyword(s):  
Random Walks ◽  
Random Walks On Graphs

The ruin problem and cover times of asymmetric random walks and Brownian motions

2000 ◽  
Vol 32 (01) ◽  
pp. 177-192 ◽  
Author(s):  
K. S. Chong ◽  
Richard Cowan ◽  
Lars Holst
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Cover Time ◽  
Brownian Motions ◽  
Brownian Motion With Drift ◽  
And Brownian Motion ◽  
Cover Times ◽  
Ruin Problem

A simple asymmetric random walk on the integers is stopped when its range is of a given length. When and where is it stopped? Analogous questions can be stated for a Brownian motion. Such problems are studied using results for the classical ruin problem, yielding results for the cover time and the range, both for asymmetric random walks and Brownian motion with drift.

On a Result of Aleliunas et al. Concerning Random Walks on Graphs

1990 ◽  
Vol 4 (4) ◽  
pp. 489-492 ◽  
Author(s):  
José Luis Palacios
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Proof ◽  
Hitting Times ◽  
Initial State ◽  
Random Walks On Graphs ◽  
Minimum Number

Aleliunas et al. [3] proved that for a random walk on a connected raph G = (V, E) on N vertices, the expected minimum number of steps to visit all vertices is bounded by 2|E|(N - 1), regardless of the initial state. We give here a simple proof of that result through an equality involving hitting times of vertices that can be extended to an inequality for hitting times of edges, thus obtaining a bound for the expected minimum number of steps to visit all edges exactly once in each direction.

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