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DOI: 10.4230/LIPIcs.MFCS.2021.6

An Approximation Algorithm for the Matrix Tree Multiplication Problem

Authors: Mahmoud Abo-Khamis, Ryan Curtin, Sungjin Im, Benjamin Moseley, Hung Ngo, Kirk Pruhs, and Alireza Samadian

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract

We consider the Matrix Tree Multiplication problem. This problem is a generalization of the classic Matrix Chain Multiplication problem covered in the dynamic programming chapter of many introductory algorithms textbooks. An instance of the Matrix Tree Multiplication problem consists of a rooted tree with a matrix associated with each edge. The output is, for each leaf in the tree, the product of the matrices on the chain/path from the root to that leaf. Matrix multiplications that are shared between various chains need only be computed once, potentially being shared between different root to leaf chains. Algorithms are evaluated by the number of scalar multiplications performed. Our main result is a linear time algorithm for which the number of scalar multiplications performed is at most 15 times the optimal number of scalar multiplications.

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Mahmoud Abo-Khamis, Ryan Curtin, Sungjin Im, Benjamin Moseley, Hung Ngo, Kirk Pruhs, and Alireza Samadian. An Approximation Algorithm for the Matrix Tree Multiplication Problem. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{abokhamis_et_al:LIPIcs.MFCS.2021.6,
  author =	{Abo-Khamis, Mahmoud and Curtin, Ryan and Im, Sungjin and Moseley, Benjamin and Ngo, Hung and Pruhs, Kirk and Samadian, Alireza},
  title =	{{An Approximation Algorithm for the Matrix Tree Multiplication Problem}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{6:1--6:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.6},
  URN =		{urn:nbn:de:0030-drops-144464},
  doi =		{10.4230/LIPIcs.MFCS.2021.6},
  annote =	{Keywords: Matrix Multiplication, Approximation Algorithm}
}

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An Approximation Algorithm for the Matrix Tree Multiplication Problem

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