High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! The paper is a joint work by Martin Dyer, Alan M. Frieze and Ravindran Kannan. The main result of the paper is a randomized algorithm for finding an ? approximation to the volume of a convex body K in n-dimensional Euclidean space by assume the existence of a membership oracle. The algorithm takes time bounded by a polynomial in n, the dimension of K and 1 / ?. The algorithm is a sophisticated usage of the so-called Markov chain Monte Carlo (MCMC) method. The basic scheme of the algorithm is a nearly uniform sampling from within K by placing a grid consisting n-dimensional cubes and doing a random walk over these cubes. By using the theory of rapidly mixing Markov chains, they show that it takes a polynomial time for the random walk to settle down to being a nearly uniform distribution.

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