Lecture 6 : Random Walks versus Independent Sampling

25.1.2 Random Walks and Their Properties

The previous lecture showed that, for self-reducible problems, the problem of estimating the size of the set of feasible solutions is equivalent to the problem of sampling nearly uniformly from that set. This lecture explores the applications of that result by developing techniques for sampling from a uniform distribution. Specifically, this lecture introduces the concept of Markov Chain Monte ...

Reusing Random Walks in Monte Carlo Methods for Linear Systems

In this paper, we present an approach of reusing random walks in Monte Carlo methods for linear systems. The fundamental idea is, during the Monte Carlo sampling process, the random walks generated to estimate one unknown element can also be effectively reused to estimate the other unknowns in the solution vector. As a result, when the random walks are reused, a single random walk can contribut...

Lecture 8: Random Walks in Undirected Graphs

We introduce the notion of a random walk in an undirected graph. What happens after we take T random steps in the walk? We show how to analyze walks and their convergence properties.

Introductory lecture notes on MARKOV CHAINS AND RANDOM WALKS

On sampling generating sets of finite groups

The problem of determining the structure of Nk(G) is of interest in several contexts. The counting problem goes back to Philip Hall, who expressed Nk(G) as a Möbius type summation of Nk(H) over all maximal subgroups H ⊂ G (see [23]). Recently the counting problem has been studied for large simple groups where remarkable progress has been made (see [25, 27]). In this paper we analyze Nk for solv...