Top-K Color Queries for Document Retrieval

TL;DR

The paper introduces an optimal data structure for top-K color queries in arrays, enabling efficient retrieval of the highest-priority colors within a range, with applications to document retrieval.

Contribution

It presents a novel, optimal data structure for top-K color queries that is both space-efficient and provides worst-case optimal query time.

Findings

Queries answered in O(K) time

Data structure uses O(N log σ) bits

Applications to document retrieval problems

Abstract

In this paper we describe a new efficient (in fact optimal) data structure for the {\em top-$K$ color problem}. Each element of an array $A$ is assigned a color $c$ with priority $p(c)$. For a query range $[a,b]$ and a value $K$, we have to report $K$ colors with the highest priorities among all colors that occur in $A[a..b]$, sorted in reverse order by their priorities. We show that such queries can be answered in $O(K)$ time using an $O(N\log \sigma)$ bits data structure, where $N$ is the number of elements in the array and $\sigma$ is the number of colors. Thus our data structure is asymptotically optimal with respect to the worst-case query time and space. As an immediate application of our results, we obtain optimal time solutions for several document retrieval problems. The method of the paper could be also of independent interest.