Error Resilient Space Partitioning

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.