Optimal top dag compression

TL;DR

This paper presents a linear-time method for constructing a top dag compression of ordered node-labelled trees that achieves optimal size bounds and logarithmic height, improving upon previous methods.

Contribution

It introduces a new top dag construction algorithm with optimal size bounds and logarithmic height for ordered node-labelled trees.

Findings

Constructs top dags in linear time with optimal size bounds.

Achieves a height of O(log n) for the top dag.

Improves previous bounds on top dag size.

Abstract

It is shown that for a given ordered node-labelled tree of size $n$ and with $s$ many different node labels, one can construct in linear time a top dag of height $O(\log n)$ and size $O(n / \log_\sigma n) \cap O(d \cdot \log n)$, where $\sigma = \max\{ 2, s\}$ and $d$ is the size of the minimal dag. The size bound $O(n / \log_\sigma n)$ is optimal and improves on previous bounds.