Top-$k$ Preferences in High Dimensions

Abstract

Given a set of objects $\mathcal {O}$ , each with $d$ numeric attributes, a top- $k$ preference scores these objects using a linear combination of their attribute values, where the weight on each attribute reflects the interest in this attribute. Given a query preference $q$ , a top- $k$ query finds the $k$ objects in $\mathcal {O}$ with highest scores with respect to $q$ . Given a query object $o$ and a set of preferences $\mathcal {Q}$ , a reverse top- $k$ query finds all preferences $q\in \mathcal {Q}$ for which $o$ becomes one of the top $k$ objects with respect to $q$ . Previous solutions to these problems are effective only in low dimensions. In this paper, we develop a solution for much higher dimensions (up to high tens), if many preferences exhibit sparsity —i.e., each specifies non-zero weights for only a handful (say $5$ - $7$ ) of attributes (though the subsets of such attributes and their weights can vary greatly). Our idea is to select carefully a set of low-dimensional core subspaces to “cover” the sparse preferences in a workload. These subspaces allow us to index them more effectively than the full-dimensional space. Being multi-dimensional, each subspace covers many possible preferences; furthermore, multiple subspaces can jointly cover a preference, thereby expanding the coverage beyond each subspace’s dimensionality. Experimental evaluation validates our solution’s effectiveness and advantages over previous solutions.