The Solution of Linear Probabilistic Recurrence Relations

Abstract

Linear probabilistic divide-and-conquer recurrence relations arise when analyzing the running time of divide-and-conquer randomized algorithms. We consider first the problem of finding the expected value of the random process T(x) , described as the output of a randomized recursive algorithm T . On input x , T generates a sample (h1,···,hk) from a given probability distribution on [0,1]k and recurses by returning g(x) + Σi=1kciT(hi x) until a constant is returned when x becomes less than a given number. Under some minor assumptions on the problem parameters, we present a closed-form asymptotic solution of the expected value of T(x) . We show that E[T(x)] = Θ( xp + xp∈t1x(g(u)/ up+1 ) du) , where p is the nonnegative unique solution of the equation Σi=1kciE[hip] = 1 . This generalizes the result in [1] where we considered the deterministic version of the recurrence. Then, following [2], we argue that the solution holds under a broad class of perturbations including floors and ceilings that usually accompany the recurrences that arise when analyzing randomized divide-and-conquer algorithms.