A dynamic data structure for top-k queries on uncertain data

In an uncertain data set s = (S, p,f) where S is the ground set consisting of n elements, p : S -> [0, 1] a probability function, and f : S -> R a score function, each element i epsilon S with score f(i) appears independently with probability p(i). The top-k query on s asks for the set of k elements that has the maximum probability of appearing to be the k elements with the highest scores in a random instance of s. Computing the top-k answer on a fixed S is known to be easy. In this paper, we consider the dynamic problem, that is, how to maintain the top-k query answer when 8 changes, including element insertions and deletions in the ground set S, changes in the probability function p and in the score function f. We present a fully dynamic data structure that handles an update in O(k log n) time, and answers a top-j query in O(log n + j) time for any j <= k. The structure has O(n) size and can be constructed in O(n log k) time. As a building block of our dynamic structure, we present an algorithm for the all-top-k problem, that is, computing the top-j answers for all j = 1, ..., k, which may be of independent interest. (C) 2008 Elsevier B.V. All rights reserved.