6 research outputs found
Extensions of Self-Improving Sorters
Ailon et al. (SICOMP 2011) proposed a self-improving sorter that tunes its performance to the unknown input distribution in a training phase. The distribution of the input numbers x_1,x_2,...,x_n must be of the product type, that is, each x_i is drawn independently from an arbitrary distribution D_i, and the D_i\u27s are independent of each other. We study two extensions that relax this requirement. The first extension models hidden classes in the input. We consider the case that numbers in the same class are governed by linear functions of the same hidden random parameter. The second extension considers a hidden mixture of product distributions
Self-Improving Voronoi Construction for a Hidden Mixture of Product Distributions
We propose a self-improving algorithm for computing Voronoi diagrams under a given convex distance function with constant description complexity. The n input points are drawn from a hidden mixture of product distributions; we are only given an upper bound m = o(?n) on the number of distributions in the mixture, and the property that for each distribution, an input instance is drawn from it with a probability of ?(1/n). For any ? ? (0,1), after spending O(mn log^O(1)(mn) + m^? n^(1+?) log(mn)) time in a training phase, our algorithm achieves an O(1/? n log m + 1/? n 2^O(log^* n) + 1/? H) expected running time with probability at least 1 - O(1/n), where H is the entropy of the distribution of the Voronoi diagram output. The expectation is taken over the input distribution and the randomized decisions of the algorithm. For the Euclidean metric, the expected running time improves to O(1/? n log m + 1/? H)
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 follow
some unknown \emph{product distribution}. That is, comes from a fixed
unknown distribution , and the 's are drawn independently.
After spending time in a learning phase, the subsequent
expected running time is , where , and and are the
entropies of the distributions of the sorting and DT output, respectively. In
this paper, we allow dependence among the 's under the \emph{group product
distribution}. There is a hidden partition of into groups; the 's
in the -th group are fixed unknown functions of the same hidden variable
; and the '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 to 's are well-behaved. After an
-time training phase, we achieve and
expected running times for sorting and DT,
respectively, where is the inverse Ackermann function