Fast algorithms for constructing maximum entropy summary trees

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 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(k² 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 + (k⁴/ε) log(k/ε)), 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.