Math on CuriousCoding

A near-tight lower bound on minimizer density

Tue, 25 Jun 2024 00:00:00 +0200

Table of Contents

Succinct background
- Definitions
- Lower bounds
A new lower bound
Discussion
Post scriptum
Acknowledgement

The results of this post are now published in Bioinformatics: DOI, PDF:

Kille, Bryce, Ragnar Groot Koerkamp, Drake McAdams, Alan Liu, and Todd J Treangen. 2024. “A near-tight Lower Bound on the Density of Forward Sampling Schemes.” Edited by Yann Ponty. Bioinformatics, December. https://doi.org/10.1093/bioinformatics/btae736.

This content has also been absorbed into my thesis chapter on minimizers.

Perfect NtHash for Robust Minimizers

Sun, 31 Dec 2023 00:00:00 +0100

Table of Contents

NtHash
Minimizers
- Robust minimizers
Is NtHash injective on kmers?
- Searching for a collision
- Proving perfection
Alternatives
SmHasher results

NtHash Link to heading

NtHash (Mohamadi et al. 2016) is a rolling hash suitable for hashing any kind of text, but made for DNA originally. For a string of length $k$ it is a $64$ bit value computed as:

A Combinatorial Identity

Sun, 16 Oct 2022 00:00:00 +0200

Some notes regarding the identity

\begin{equation} \sum_{k=0}^n \binom{2k}k \binom{2n-2k}{n-k} = 4^n \end{equation}

Gould has two derivations:
- The first, from Jensens equality, (18) in (Jensen 1902; Shijie 1303).
- A second via the Chu-Vandermonde convolution:
  
  \begin{equation} \sum_{k=0}^n \binom{x}k \binom{y}{n-k} = \binom{x+y}n \end{equation}
  
  using $x=y=-\frac 12$ and using the $-\frac 12$-transform:
  
  \begin{equation} \binom{-1/2}{n} = (-1)^n\binom{2n}{n}\frac 1 {2^{2n}} \end{equation}
Duarte and de Oliveira (2012) has a combinatorial proof.

References Link to heading

Duarte, Rui, and António Guedes de Oliveira. 2012. “New Developments of an Old Identity.” https://doi.org/10.48550/ARXIV.1203.5424.

Jensen, J. L. W. V. 1902. “Sur Une Identité D’abel et Sur D’autres Formules Analogues.” Acta Mathematica 26 (0): 307–18. https://doi.org/10.1007/bf02415499.

Shijie, Zhu. 1303. Jade Mirror of the Four Unknowns.