Subtree Mode and Applications

TL;DR

This paper introduces an optimal algorithm for the Subtree Mode problem in leaf-colored trees, enabling efficient computation of the most frequent leaf colors for all nodes, with applications in text analytics and biology.

Contribution

The paper presents a time-optimal $O(N)$ algorithm for the Subtree Mode problem in leaf-colored trees, including adaptations for node-colored trees and top-$k$ frequent colors, with proven efficiency and practical validation.

Findings

Algorithm is faster than baselines by at least an order of magnitude.

The solution is highly space-efficient and scalable to billion-node trees.

Effective in applications like pattern mining, sequence search, and biological data analysis.

Abstract

The mode of a collection of values (i.e., the most frequent value in the collection) is a key summary statistic. Finding the mode in a given range of an array of values is thus of great importance, and constructing a data structure to solve this problem is in fact the well-known Range Mode problem. In this work, we introduce the Subtree Mode (SM) problem, the analogous problem in a leaf-colored tree, where the task is to compute the most frequent color in the leaves of the subtree of a given node. SM is motivated by several applications in domains such as text analytics and biology, where the data are hierarchical and can thus be represented as a (leaf-colored) tree. Our central contribution is a time-optimal algorithm for SM that computes the answer for every node of an input $N$-node tree in $O(N)$ time. We further show how our solution can be adapted for node-colored trees, or for computing the $k$ most frequent colors, for any given $k=O(1)$, in the optimal $O(N)$ time. Moreover, we prove that a similarly fast solution for when the input is a sink-colored directed acyclic graph instead of a leaf-colored tree is highly unlikely. Our experiments on real datasets with trees of up to $7.3$ billion nodes demonstrate that our algorithm is faster than baselines by at least one order of magnitude and much more space efficient. They also show that it is effective in pattern mining, sequence-to-database search, and biology applications.