Approximate all-pairs Hamming distances and 0-1 matrix multiplication
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915251597869056 |
|---|---|
| author | Kowaluk, Miroslaw Lingas, Andrzej Persson, Mia |
| author_facet | Kowaluk, Miroslaw Lingas, Andrzej Persson, Mia |
| contents | 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. Next, 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. Finally, we provide $(2+ε)$- approximation
algorithms for the $\ell$-center clustering and minimum-diameter
$\ell$-clustering problems in a Hamming space $\{0,1\}^d$ that are
substantially faster than the known $2$-approximation ones when both
$\ell$ and $d$ are super-logarithmic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14723 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximate all-pairs Hamming distances and 0-1 matrix multiplication Kowaluk, Miroslaw Lingas, Andrzej Persson, Mia Data Structures and Algorithms Computational Complexity F.2.2 F.2.2 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. Next, 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. Finally, we provide $(2+ε)$- approximation algorithms for the $\ell$-center clustering and minimum-diameter $\ell$-clustering problems in a Hamming space $\{0,1\}^d$ that are substantially faster than the known $2$-approximation ones when both $\ell$ and $d$ are super-logarithmic. |
| title | Approximate all-pairs Hamming distances and 0-1 matrix multiplication |
| topic | Data Structures and Algorithms Computational Complexity F.2.2 F.2.2 |
| url | https://arxiv.org/abs/2504.14723 |