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Lower Density Selection Schemes via Small Universal Hitting Sets with Short Remaining Path Length.

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Summary

Universal hitting sets (UHS) ensure every long sequence contains a word from the set. This study links efficient local schemes to UHS size, providing new bounds and a near-optimal local scheme.

Keywords:
de Bruijn graphdepathing setminimizerssequence sketchuniversal hitting set

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Area of Science:

  • Theoretical Computer Science
  • Combinatorics on Words
  • Data Compression

Background:

  • Universal Hitting Sets (UHS) are unavoidable sets of words crucial for sequence analysis.
  • Efficient UHS are linked to low-density minimizer schemes, essential for data compression and pattern matching.
  • Local schemes generalize minimizer schemes, offering potential for greater efficiency.

Purpose of the Study:

  • To establish the relationship between efficient local schemes and the minimum length of strings hit by Universal Hitting Sets.
  • To derive bounds for the remaining path length of the Mykkeltveit UHS.
  • To develop a novel local scheme with improved density.

Main Methods:

  • Theoretical analysis of Universal Hitting Sets and their properties.
  • Development and analysis of generalized local schemes.
  • Algorithmic construction and density evaluation of new local schemes.

Main Results:

  • A clear link is established between efficient local schemes and the minimum length required for UHS coverage.
  • New bounds are provided for the remaining path length of the Mykkeltveit UHS.
  • A new local scheme is introduced with the lowest known density, approaching the theoretical lower bound.

Conclusions:

  • Efficient local schemes offer a powerful framework for understanding and constructing compact Universal Hitting Sets.
  • The findings advance the theory of sequence covering and have implications for data compression and string algorithms.
  • The newly developed local scheme represents a significant improvement in efficiency for practical applications.