A Generalization of Self-Improving Algorithms

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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+ε ) time in a learning phase, the subsequent expected running time is O (( n + H )/ε), where H ∊ { 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 k th 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 α ( n ) + H DT ) expected running times for sorting and DT, respectively, where α (⋅) is the inverse Ackermann function.