A top-k query retrieves the k tuples with highest scores according to a user preference, defined as a scoring function. It is difficult for a user to precisely specify the scoring function. Instead, obtaining the distribution on scoring functions, i.e., the preference distribution, has been extensively explored in many fields. Motivated by this, we introduce the uniform (r,k)-hit (UrkHit) problem. Given a preference distribution, UrkHit aims to select a representative set of r tuples to maximize the probability of containing a tuple attractive to the user. We say a tuple attracts a user, if it is a top-k tuple for the scoring function adopted by the user. Further, we generalize UrkHit and propose the (r,k)-hit (rkHit) problem with an additional penalty function to model the user satisfaction with the tuple ranked i-th. rkHit aims to maximize the expected user satisfaction with the representative set. In 2D space, we design an exact algorithm 2DH for rkHit, indicating rkHit is in P for d=2. We show that rkHit is NP-hard when d\ge3. In 3D space, assuming a uniform preference distribution, we propose a (1-1/e)-approximation algorithm 3DH based on space partitioning. In addition, we propose an approximate algorithm MDH suitable for any dimension and distribution, which creatively combines the ideas of sampling and clustering. It relaxes the approximation guarantee slightly. Comprehensive experiments demonstrate the efficiency and effectiveness of our algorithms.

rkHit: Representative Query with Uncertain Preference

Selecting the best items in a dataset is a common task in data exploration. However, the concept of “best” lies in the eyes of the beholder: Different users may consider different attributes more important and, hence, arrive at different rankings. Nevertheless, one can remove “dominated” items and create a “representative” subset of the data, comprising the “best items” in it. A Pareto-optimal representative is guaranteed to contain the best item of each possible ranking, but it can be a large portion of data. A much smaller representative can be found if we relax the requirement of including the best item for each user and instead just limit the users’ “regret.” Existing work defines regret as the loss in score by limiting consideration to the representative instead of the full dataset, for any chosen ranking function. However, the score is often not a meaningful number, and users may not understand its absolute value. Sometimes small ranges in score can include large fractions of the dataset. In contrast, users do understand the notion of rank ordering. Therefore, we consider items’ positions in the ranked list in defining the regret and propose the rank-regret representative as the minimal subset of the data containing at least one of the top-k of any possible ranking function. This problem is polynomial time solvable in two-dimensional space but is NP-hard on three or more dimensions. We design a suite of algorithms to fulfill different purposes, such as whether relaxation is permitted on k, the result size, or both, whether a distribution is known, whether theoretical guarantees or practical efficiency is important, and so on. Experiments on real datasets demonstrate that we can efficiently find small subsets with small rank-regrets.

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Jianwen Zhao

Papers

On Finding Rank Regret Representatives