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Spatial Separation of Molecular Conformers and Clusters
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Published on: January 9, 2014

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Error Resilient Space Partitioning.

Orr Dunkelman1, Zeev Geyzel2, Chaya Keller3

  • 1Computer Science Department, University of Haifa, Haifa, Israel.

Discrete & Computational Geometry
|April 7, 2026
PubMed
Summary
This summary is machine-generated.

This study introduces error-resilient space partitioning for noisy data, using colored tiles to ensure similar measurements round to the same value. It characterizes tradeoffs between the number of colors (k) and minimum distance (t) in d-dimensional Euclidean space.

Keywords:
Consistent hashingError resilienceRoundingSpace partitioningSparse partitionsSphere packingTiling

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Last Updated: Apr 8, 2026

Spatial Separation of Molecular Conformers and Clusters
10:37

Spatial Separation of Molecular Conformers and Clusters

Published on: January 9, 2014

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Area of Science:

  • Discrete geometry
  • Computational geometry
  • Error-resilient data processing

Background:

  • Partitioning Euclidean space (ℝᵈ) is crucial for data representation and analysis.
  • Rounding noisy measurements requires robust methods to maintain data integrity.
  • Existing partitioning methods may not adequately handle measurement inaccuracies.

Purpose of the Study:

  • Introduce a novel space partitioning method for error-resilient rounding of noisy measurements.
  • Investigate the relationship between the number of colors (k) and minimum separation distance (t) in partitioned space.
  • Characterize the trade-offs between k and t for different dimensions (d).

Main Methods:

  • Partitioning d-dimensional Euclidean space into bounded-size tiles.
  • Assigning one of k colors to tiles such that tiles of the same color are at least distance t apart.
  • Utilizing techniques such as isoperimetric inequalities, Brunn-Minkowski theorem, and sphere packing bounds.

Main Results:

  • Established that k = d + 1 colors are necessary and sufficient for non-zero separation (t > 0) in ℝᵈ.
  • Derived numerous upper and lower bounds for t as a function of k.
  • Obtained sharp asymptotic bounds on t for d = 3, 4, 8, 24 as k approaches infinity.

Conclusions:

  • The proposed colored partitioning scheme enables error-resilient rounding of noisy data.
  • The study provides a quantitative understanding of the achievable trade-offs between color count and separation distance.
  • Results have implications for data discretization and robust measurement processing in various dimensions.