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Minimum Many-to-Many Matchings for Computing the Distance Between Two Sequences

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Abstract

Motivated by a problem in music theory of measuring the distance between chords, scales, and rhythms we consider algorithms for obtaining a minimum-weight many-to-many matching between two sets of points on the real line. Given sets \(A\) and \(B\), we seek to find the best rigid translation of \(B\) and a many-to-many matching that minimizes the sum of the squares of the distances between matched points. We provide discrete algorithms that solve this continuous optimization problem, and discuss other related matters.

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Acknowledgments

This work was initiated at the 2nd Bellairs Winter Workshop on Mathematics and Music, co-organized by Dmitri Tymoczko and Godfried Toussaint, held on February 6-12, 2010. We thank the other participants of that workshop, Fernando Benadon, Adrian Childs, Richard Cohn, Rachel Hall, John Halle, Jay Rahn, Bill Sethares, and Steve Taylor, for providing a stimulating research environment

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Authors and Affiliations

  1. School of Computing, Queen’s University, Kingston, ON, Canada

    Mustafa Mohamad & David Rappaport

  2. Department of Computer Science, New York University Abu Dhabi, Abu Dhabi, United Arab Emirates

    Godfried Toussaint

  3. School of Computer Science, McGill University, Montreal, QC, Canada

    Godfried Toussaint

Authors
  1. Mustafa Mohamad
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  2. David Rappaport
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  3. Godfried Toussaint
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Corresponding author

Correspondence to David Rappaport.

Additional information

D. Rappaport: Research supported by NSERC Discovery Grant 388-329.

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Mohamad, M., Rappaport, D. & Toussaint, G. Minimum Many-to-Many Matchings for Computing the Distance Between Two Sequences. Graphs and Combinatorics 31, 1637–1648 (2015). https://doi.org/10.1007/s00373-014-1467-4

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  • DOI: https://doi.org/10.1007/s00373-014-1467-4

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