Abstract
Given n vectors \(x_0, x_1, \ldots , x_{n-1}\) in \(\{0,1\}^{m}\), how to find two vectors whose pairwise Hamming distance is minimum? This problem is known as the Closest Pair Problem. If these vectors are generated uniformly at random except two of them are correlated with Pearson-correlation coefficient \(\rho \), then the problem is called the Light Bulb Problem. In this work, we propose a novel coding-based scheme for the Close Pair Problem. We design both randomized and deterministic algorithms, which achieve the best known running time when the minimum distance is very small compared to the length of input vectors. When applied to the Light Bulb Problem, our algorithms yields state-of-the-art deterministic running time when the Pearson-correlation coefficient \(\rho \) is very large.
A full version of the paper is available at https://arxiv.org/abs/1802.09104.
N. Xie, S. Xu and Y. Xu—Research supported in part by NSF grant 1423034.
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- 1.
Subqubic fast matrix multiplication algorithms are practical only for Strassen-based ones [11, 22]. Even though the recent breakthrough results [20, 35, 42] achieve asymptotically faster than Strassen’s algorithm [36], however, these algorithms are all based on Coppersmith-Winograd’s algorithm [15], and to the best of our knowledge, there are no practical implementations of these trilinear based algorithms.
- 2.
Actually, we only need to “check” when the two adjacent decoded vectors are identical.
- 3.
The Hamming bound, also known as the sphere packing bound, specifies an upper bound on the number codewords a code can have given the block length and the minimum distance of the code.
- 4.
The GV bound is known to be attainable by the random codes.
- 5.
All these results are due to the fact that Valiant’s algorithms are much more robust to weak correlations than other algorithms. Our algorithms therefore do not give improved bounds for these learning problems in the general settings.
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Acknowledgements
We would like to thank Karthik C.S. and the anonymous referees for their valuable comments.
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Xie, N., Xu, S., Xu, Y. (2018). A New Algorithm for Finding Closest Pair of Vectors (Extended Abstract). In: Fomin, F., Podolskii, V. (eds) Computer Science – Theory and Applications. CSR 2018. Lecture Notes in Computer Science(), vol 10846. Springer, Cham. https://doi.org/10.1007/978-3-319-90530-3_27
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