, Volume 80, Issue 8, pp 2400–2421 | Cite as

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


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 \(m=|B|\ll n=|A|\), 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 \({\mathcal {D}}_{B,A}\) of the plane and derive improved bounds on its complexity. Specifically, we show that this complexity is \(O(n^2m^{3.5} (e \ln m+e)^m)\), 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.


Shape matching Partial matching RMS distance Convex subdivision Local minimum 



Rinat Ben-Avraham was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation. Matthias Henze was supported by ESF EUROCORES programme EuroGIGA-VORONOI, (DFG): RO 2338/5-1. Rafel Jaume was supported by “Obra Social La Caixa” and the DAAD. Balázs Keszegh was supported by Hungarian National Science Fund (OTKA), under Grants PD 108406, NN 102029 (EUROGIGA project GraDR 10-EuroGIGA-OP-003), NK 78439, by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the DAAD. Orit E. Raz was supported by Grant 892/13 from the Israel Science Foundation. Micha Sharir was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation, by Grant 892/13 from the Israel Science Foundation, by the Israeli Centers for Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Igor Tubis was supported by the Deutsch Institute. A preliminary version of this paper has appeared in [5].


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.Institut für MathematikFreie Universität BerlinBerlinGermany
  3. 3.Institut für InformatikFreie Universität BerlinBerlinGermany
  4. 4.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary

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