Recursive Stratified Sampling: A New Framework for Query Evaluation on Uncertain Graphs

Rong-Hua Li, Jeffrey Xu Yu, Rui Mao, Tan Jin

Abstract: Uncertain graph management has been recognized as an important research topic in recent years. In this paper, we first introduce two types of query evaluation problems on uncertain graphs, named expectation query evaluation and threshold query evaluation. Most previous solutions for these problems are based on naive Monte-Carlo ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">NMC </i> ) sampling, which typically result in large variances. To reduce the variance of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">NMC</i> , we propose two efficient estimators, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RSS-I</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RSS-II</i> estimators, based on the idea of recursive stratified sampling ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RSS</i> ). To further reduce the variances of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RSS-I</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RSS-II</i> , we propose a recursive <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">cut-set</i> based stratified sampling estimator for a particular kind of query evaluation problem. We show that all the proposed estimators are unbiased and their variances are significantly smaller than that of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">NMC</i> . Moreover, the time complexity of all the proposed estimators are the same as that of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">NMC</i> under a mild assumption. In addition, we develop an elegant graph simplification technique to further improve the accuracy and running time of our estimators. We also apply the proposed estimators to three different uncertain graph query evaluation problems. Finally, we conduct extensive experiments to evaluate the proposed estimators, and the results show the accuracy, efficiency, and scalability of our estimators.