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∈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 S 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(klogn) time, and answers a top-j query in O(logn+j) time for any j≤k. The structure has O(n) size and can be constructed in O(nlogk) 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.