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 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.
A full version of the paper is available at http://arxiv.org/abs/1404.5660
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Cole, R., Karloff, H. (2014). Fast Algorithms for Constructing Maximum Entropy Summary Trees. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43948-7_28
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DOI: https://doi.org/10.1007/978-3-662-43948-7_28
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