Fast Algorithms for Constructing Maximum Entropy Summary Trees

Abstract

AbstractKarloff 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 polynomial-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 + nlogn) 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/ε) log(k/ε)), and how to speed up the greedy algorithm so as to run in time O(kn + n logn). Altogether, these results make summary trees a much more practical tool than before.KeywordsGreedy AlgorithmGreedy HeuristicIntegral WeightOpen ExistenceSummary TreeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.