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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.

Cite as

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)

```@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},
URL =		{https://drops.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}
}```
Document
Track A: Algorithms, Complexity and Games
Relational Algorithms for k-Means Clustering

Authors: Benjamin Moseley, Kirk Pruhs, Alireza Samadian, and Yuyan Wang

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
This paper gives a k-means approximation algorithm that is efficient in the relational algorithms model. This is an algorithm that operates directly on a relational database without performing a join to convert it to a matrix whose rows represent the data points. The running time is potentially exponentially smaller than N, the number of data points to be clustered that the relational database represents. Few relational algorithms are known and this paper offers techniques for designing relational algorithms as well as characterizing their limitations. We show that given two data points as cluster centers, if we cluster points according to their closest centers, it is NP-Hard to approximate the number of points in the clusters on a general relational input. This is trivial for conventional data inputs and this result exemplifies that standard algorithmic techniques may not be directly applied when designing an efficient relational algorithm. This paper then introduces a new method that leverages rejection sampling and the k-means++ algorithm to construct a O(1)-approximate k-means solution.

Cite as

Benjamin Moseley, Kirk Pruhs, Alireza Samadian, and Yuyan Wang. Relational Algorithms for k-Means Clustering. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 97:1-97:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

```@InProceedings{moseley_et_al:LIPIcs.ICALP.2021.97,
author =	{Moseley, Benjamin and Pruhs, Kirk and Samadian, Alireza and Wang, Yuyan},
title =	{{Relational Algorithms for k-Means Clustering}},
booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
pages =	{97:1--97:21},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-195-5},
ISSN =	{1868-8969},
year =	{2021},
volume =	{198},
editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.97},
URN =		{urn:nbn:de:0030-drops-141668},
doi =		{10.4230/LIPIcs.ICALP.2021.97},
annote =	{Keywords: k-means, clustering, approximation, big-data, databases}
}```