Fast Algorithms for Constructing Maximum Entropy Summary Trees

TL;DR

This paper develops efficient algorithms for constructing maximum entropy summary trees, significantly improving their practicality by reducing computational complexity for large trees with real weights.

Contribution

It presents the first polynomial-time algorithm for maximum entropy summary trees with real weights and improves existing algorithms' speed.

Findings

Achieved an O(k^2 n + n log n) algorithm for maximum entropy summary trees.

Enhanced approximation algorithm to run in O(n + (k^4/eps?) log(k/eps?)) time.

Optimized greedy algorithm to run in O(kn + n log n) time.

Abstract

Karloff? and Shirley recently proposed summary trees as a new way to visualize large rooted trees (Eurovis 2013) and gave algorithms for generating a maximum-entropy k-node summary tree of an input n-node rooted tree. However, the algorithm generating optimal summary trees was only pseudo-polynomial (and worked only for integral weights); the authors left open existence of a olynomial-time algorithm. In addition, the authors provided an additive approximation algorithm and a greedy heuristic, both working on real weights. This paper shows how to construct maximum entropy k-node summary trees in time O(k^2 n + n log n) for real weights (indeed, as small as the time bound for the greedy heuristic given previously); how to speed up the approximation algorithm so that it runs in time O(n + (k^4/eps?) log(k/eps?)), and how to speed up the greedy algorithm so as to run in time O(kn + n log n). Altogether, these results make summary trees a much more practical tool than before.