Rank quantization

We study the problem of aggregating and summarizing partial orders, on a large scale. Our motivation is two-fold: to discover elements at similar preference levels and to reduce the number of bits needed to store an element's position in a full ranking.We proceed in two steps: first, we find a total order by linearizing the rankings induced by the multiple partial orders and removing potentially inconsistent pairwise preferences. Next, given a total order, we introduce and formalize the rank quantization problem, which intuitively aims to bucketize the total order in a manner that mostly preserves the relations appearing in the partial orders. We show an exact quadratic-time quantization algorithm, as well as a greedy 2/3-approximation algorithm whose running is substantially faster on sparse instances. As an application, we aggregate rankings of top-10 search results over millions of search engine queries, approximately reproducing and then efficiently encoding the underlying static ranks used by the engine. We evaluate the performance of our algorithms on a web dataset of 12 million(2^{23.5}) unique pages and show that we can quantize the pages' static ranks using as few as eight bits, with only a minor degradation in search quality.