Partial-Matching RMS Distance Under Translation: Combinatorics and Algorithms

Rinat Ben-Avraham, Matthias Henze*, Rafel Jaume, Balázs Keszegh, Orit E. Raz, Micha Sharir, Igor Tubis

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


We consider the problem of minimizing the RMS distance (sum of squared distances between pairs of points) under translation between two point sets A and B, in the plane, with m= | B| ≪ n= | A| , in the partial-matching setup, in which each point in B is matched to a distinct point in A. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision DB , A of the plane and derive improved bounds on its complexity. Specifically, we show that this complexity is O(n2m3.5(eln m+ e) m) , so it is only quadratic in |A|. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is minimized. In addition, we show how to compute a local minimum of the partial-matching RMS distance under translation, in polynomial time.

Original languageAmerican English
Pages (from-to)2400-2421
Number of pages22
Issue number8
StatePublished - 1 Aug 2018
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2017, Springer Science+Business Media New York.


  • Convex subdivision
  • Local minimum
  • Partial matching
  • RMS distance
  • Shape matching


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