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Arslan showed that computing all-pairs Hamming distances is easily reducible to arithmetic 0–1 matrix multiplication (IPL 2018). We provide a reverse, linear-time reduction of arithmetic 0–1 matrix multiplication to computing all-pairs distances in a Hamming space. On the other hand, we present a fast randomized algorithm for approximate all-pairs distances in a Hamming space. By combining it with our reduction, we obtain also a fast randomized algorithm for approximate 0–1 matrix multiplication. Finally, we present an output-sensitive randomized algorithm for a minimum spanning tree of a set of points in a generalized Hamming space, the lower is the cost of the minimum spanning tree the faster is our algorithm.(A preliminary version of this article appeared in arXiv.org.)

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Approximate All-Pairs Hamming Distances and 0–1 Matrix Multiplication

  • 会议论文

  • 发表日期：

    2026年

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  • Mirosław Kowaluk,
  • Andrzej Lingas,
  • Mia Persson

摘要

Arslan showed that computing all-pairs Hamming distances is easily reducible to arithmetic 0–1 matrix multiplication (IPL 2018). We provide a reverse, linear-time reduction of arithmetic 0–1 matrix multiplication to computing all-pairs distances in a Hamming space. On the other hand, we present a fast randomized algorithm for approximate all-pairs distances in a Hamming space. By combining it with our reduction, we obtain also a fast randomized algorithm for approximate 0–1 matrix multiplication. Finally, we present an output-sensitive randomized algorithm for a minimum spanning tree of a set of points in a generalized Hamming space, the lower is the cost of the minimum spanning tree the faster is our algorithm.(A preliminary version of this article appeared in arXiv.org.)