Efficient volume computation in high dimension is an important question both theoretically and practically. The first polynomial time randomized algorithm to compute the volume of a convex body in Rn was given by Dyer, Frieze and Kannan in their pathbreaking paper [3]. A very high power of the dimension n (about 26) occurred in the running time bound of this algorithm, but subsequent improvements brought the exponent down to 5 [7]. In this paper, we further improve the running time to O\u00a4(n4) (where the asterisk means that we suppress factors that are logarithmic in n). The algorithm uses O\u00a4(n) points for its computations, and in this sense it is nearly optimal, since any algorithm must use \u00ad(n) points.<\/p>\n","protected":false},"excerpt":{"rendered":"
Efficient volume computation in high dimension is an important question both theoretically and practically. The first polynomial time randomized algorithm to compute the volume of a convex body in Rn was given by Dyer, Frieze and Kannan in their pathbreaking paper [3]. 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