On the semantics of top-k ranking for objects with uncertain data

The goal of top‐ $k$ ranking for objects is to rank the objects so that the best $k$ of them can be determined. In this paper we consider an object to be an entity which consists of a number of attributes whose roles in the object are determined by an aggregation function. The problem of top‐ranking in this case is conceptually simple for data that are complete and certain – the aggregation value of an object represents its strength and therefore its rank. For uncertain data, the semantic basis of top‐ $k$ objects becomes unclear. In this paper, we formulate a semantics of top‐ $k$ ranking for objects modeled by uncertain data, where the values of an object’s attributes are expressed by probability distributions and constrained by some stated conditions. Under this setting, we present a theory of top‐ $k$ ranking for objects so that their strengths can be determined in the presence of uncertain data. We present our theory in three stages. The first deals with discrete domains, which is extended to include continuous domains. We show that top‐ $k$ ranking for objects in this context is closely related to high‐dimensional space studied in mathematics. In particular, the computation of the volumes of a high‐dimensional polyhedron represented by a system of linear inequations is a special case of top‐ $k$ ranking under our theory. We further extend this theory to add weights to objects’ positions and aggregation values in determining ranking results. We show that a number of previous proposals for top‐ $k$ ranking are special cases of our theory.