Hierarchical Categories in Colored Searching

TL;DR

This paper introduces hierarchical variants of colored range counting, enabling efficient querying of categories with internal structure, and provides data structures and complexity bounds for these problems.

Contribution

It defines hierarchical range counting and sub-category colored range counting, analyzes their complexity, and develops efficient data structures for tree hierarchies, extending CRC to structured categories.

Findings

Special cases are equivalent to known problems.

Efficient data structures are proposed for tree hierarchies.

Conditional lower bounds are established for DAG hierarchies.

Abstract

In colored range counting (CRC), the input is a set of points where each point is assigned a ``color'' (or a ``category'') and the goal is to store them in a data structure such that the number of distinct categories inside a given query range can be counted efficiently. CRC has strong motivations as it allows data structure to deal with categorical data. However, colors (i.e., the categories) in the CRC problem do not have any internal structure, whereas this is not the case for many datasets in practice where hierarchical categories exists or where a single input belongs to multiple categories. Motivated by these, we consider variants of the problem where such structures can be represented. We define two variants of the problem called hierarchical range counting (HCC) and sub-category colored range counting (SCRC) and consider hierarchical structures that can either be a DAG or a tree. We show that the two problems on some special trees are in fact equivalent to other well-known problems in the literature. Based on these, we also give efficient data structures when the underlying hierarchy can be represented as a tree. We show a conditional lower bound for the general case when the existing hierarchy can be any DAG, through reductions from the orthogonal vectors problem.