Compaction for two models of logarithmic-depth trees: Analysis and Experiments

TL;DR

This paper analyzes the compaction ratios of recursive and plane binary increasing trees, showing they reduce to O(n / log n) nodes, and presents experimental validation with a prototype implementation.

Contribution

It provides a new generic approach to analyze compaction ratios in these tree models and compares results with classical models.

Findings

Compacted recursive trees have O(n / log n) nodes.

Compacted plane binary increasing trees also have O(n / log n) nodes.

Experimental results support theoretical analysis.

Abstract

We are interested in the quantitative analysis of the compaction ratio for two classical families of trees: recursive trees and plane binary increasing trees. These families are typical representatives of tree models with a small depth. Once a tree of size $n$ is compacted by keeping only one occurrence of all fringe subtrees appearing in the tree the resulting graph contains only $O(n / \ln n)$ nodes. This result must be compared to classical results of compaction in the families of simply generated trees, where the analogous result states that the compacted structure is of size of order $n / \sqrt{\ln n}$. The result about the plane binary increasing trees has already been proved, but we propose a new and generic approach to get the result. Finally, an experimental study is presented, based on a prototype implementation of compacted binary search trees that are modeled by plane binary increasing trees.