A Generalization of Self-Improving Algorithms

Ailon et al. [SICOMP'11] proposed self-improving algorithms for sorting and Delaunay triangulation (DT) when the input instances x(1),..., x(n) follow some unknown product distribution. That is, x(i) is drawn independently from a fixed unknown distribution D-i. After spending O(n(1+epsilon)) time in a learning phase, the subsequent expected running time is O((n + H)/epsilon), where H is an element of{H-S, H-DT}, and H-S and H-DT are the entropies of the distributions of the sorting and DT output, respectively. In this article, we allow dependence among the x(i) 's under the group product distribution. There is a hidden partition of [1, n] into groups; the x(i) 's in the kth group are fixed unknown functions of the same hidden variable u(k); and the u(k) 's are drawn from an unknown product distribution. We describe self-improving algorithms for sorting and DT under this model when the functions that map u(k) to x(i) 's are well-behaved. After an O(poly(n))-time training phase, we achieve O(n + H-S) and O(n alpha (n)+H-DT) expected running times for sorting and DT, respectively, where alpha (center dot) is the inverse Ackermann function.