### 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 + 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}/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.

Original language | English (US) |
---|---|

Title of host publication | Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings |

Publisher | Springer Verlag |

Pages | 332-343 |

Number of pages | 12 |

Volume | 8572 LNCS |

Edition | PART 1 |

ISBN (Print) | 9783662439470 |

DOIs | |

State | Published - 2014 |

Event | 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014 - Copenhagen, Denmark Duration: Jul 8 2014 → Jul 11 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Number | PART 1 |

Volume | 8572 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014 |
---|---|

Country | Denmark |

City | Copenhagen |

Period | 7/8/14 → 7/11/14 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings*(PART 1 ed., Vol. 8572 LNCS, pp. 332-343). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8572 LNCS, No. PART 1). Springer Verlag. https://doi.org/10.1007/978-3-662-43948-7_28

**Fast algorithms for constructing maximum entropy summary trees.** / Cole, Richard; Karloff, Howard.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings.*PART 1 edn, vol. 8572 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), no. PART 1, vol. 8572 LNCS, Springer Verlag, pp. 332-343, 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014, Copenhagen, Denmark, 7/8/14. https://doi.org/10.1007/978-3-662-43948-7_28

}

TY - GEN

T1 - Fast algorithms for constructing maximum entropy summary trees

AU - Cole, Richard

AU - Karloff, Howard

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84904205687&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904205687&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-43948-7_28

DO - 10.1007/978-3-662-43948-7_28

M3 - Conference contribution

AN - SCOPUS:84904205687

SN - 9783662439470

VL - 8572 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 332

EP - 343

BT - Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings

PB - Springer Verlag

ER -