Method on CuriousCoding

Notes on bidirectional anchors

Mon, 15 Jan 2024 00:00:00 +0100

Table of Contents

Paper overview
Remarks on the paper
Thoughts

\[ \newcommand{\A}{\mathcal{A}_\ell} \newcommand{\T}{\mathcal{T}_\ell} \]

These are some notes on Bidirectional String Anchors (Loukides, Pissis, and Sweering 2023), also called bd-anchors.

Resources:

Loukides and Pissis (2021): preceding conference paper with subset of content.
Loukides, Pissis, and Sweering (2023): The paper discussed here.
Ayad, Loukides, and Pissis (2023): follow-up/second paper containing
- a faster average-case \(O(n)\) construction algorithm;
- a more memory efficient construction algorithms for the index.
https://github.com/solonas13/bd-anchors: code for first paper
https://github.com/lorrainea/BDA-index: code for follow-up paper

The remainder of this post is split into an overview of the paper, Remarks on the paper, and further Thoughts.

Notes on SsHash

Mon, 15 Jan 2024 00:00:00 +0100

Table of Contents

Paper summary
Remarks
Ideas

\[\newcommand{\S}{\mathcal{S}}\]

Paper summary Link to heading

Intro Link to heading

SsHash (Pibiri 2022) is a datastructure for indexing kmers. Given a set of kmers \(\S\), it supports two operations:

\(Lookup(g)\): return the unique id \(i\in [|\S|]\) of the kmer \(g\).
\(Access(i)\): return the kmer corresponding to id \(i\).

It also supports streaming queries, looking up all kmers from a longer string consecutively, by expoiting the overlap between them.

BBHash: some ideas

Mon, 04 Sep 2023 00:00:00 +0200

Table of Contents

Possible speedup?

BBHash Limasset et al. (2017) uses multiple layers to create a minimal perfect hashing functions (MPHF), that hashes some input set into \([n]\).

(See also my note on PTHash (Pibiri and Trani 2021).)

Simply said, it maps the \(n\) elements into \([\gamma \cdot n]\) using hashing function \(h_0\). The \(k_0\) elements that have collisions are mapped into \([\gamma \cdot k_0]\) using \(h_1\). Then, the \(k_1\) elements with collisions are mapped into \([\gamma \cdot k_1]\), and so on.

BitPAl bitpacking algorithm

Sun, 03 Sep 2023 00:00:00 +0200

Table of Contents

Problem
Input
Example
Discussion
Found the bug
Outlook

The supplement (download) of the Loving, Hernandez, and Benson (2014) paper introduces a \(15\) operation version of Myers (1999) bitpacking algorithm, which uses \(16\) operations when modified for edit distance.

I tried implementing it, but it seems to have a bug that I will describe below. The fix is here.

Problem Link to heading

To recap, this algorithm solves the unit-cost edit distance problem by using bitpacking to compute a \(1\times w\) at a time. As input, it takes

BWT and FM-index

Tue, 18 Oct 2022 00:00:00 +0200

Table of Contents

Burrows-Wheeler Transformation (BWT)
Bi-directional BWT

These are some notes about the Burrows-Wheeler Transform (BWT), FM-index, and variants.

See my post on the linear time suffix array construction algorithm for notation and terminology.

At the bottom you can find a visualization.
This page has an interactive demo.

Source code for visualizations is this GitHub repo.

Burrows-Wheeler Transformation (BWT) Link to heading

The BWT of a string \(S\) is generated as follows:

Linear-time suffix array construction

Thu, 13 Oct 2022 00:00:00 +0200

Table of Contents

Notation
Small and Large suffixes
Building the suffix array from a smaller one
Visualization

These are some notes about linear time suffix array (SA) construction algorithms (SACA’s).

At the bottom you can find a visualization.
This page has an interactive demo.

History of suffix array construction algorithms:

1990 first algorithm: Manber and Myers (1993)
2002 small/large suffixes, explained below: Ko and Aluru (2005)
2009 recursion only on LMS suffixes: Nong, Zhang, and Chan (2009)

These slides from Stanford are a nice reference for the last algorithm.

AStarix

Fri, 12 Nov 2021 13:05:00 +0100

Papers

AStarix is a method for aligning sequences (reads) to graphs:

Input

A reference sequence or graph
Alignment costs \((\Delta_{match}, \Delta_{subst}, \Delta_{del}, \Delta_{ins})\) for a match, substitution, insertion and deletion
Sequence(s) to align

Output

An optimal alignment of each input sequence

The input is a reference graph (automaton really) \(G_r = (V_r, E_r)\) with edges \(E_r \subseteq V_r\times V_r\times \Sigma\) that indicate the transitions between states.

Neighbour joining

Fri, 12 Nov 2021 11:57:00 +0100

Neighbour joining (NJ, paper) is a phylogeny reconstruction method. It differs from UPGMA in the way it computes the distances between clusters.

This algorithm first assumes that the phylogeny is a star graph. Then it finds the pair of vertices that when merged and split out gives the minimal total edge length \(S_{ij}\) of the new almost-star graph. (See eq. (4) and figure 2a and 2b in the paper.) \[ S_{i,j} = \frac1{2(n-2)} \sum_{k\not\in \{i,j\}}(d(i, k)+d(j,k)) + \frac 12 d(i,j)+\frac 1{n-2} \sum_{k<l,\, k, l\not\in\{i,j\}}d(k,l). \] After subtracting the sum of all pairwise distances (which is a constant) and multiplying by \(2(n-2)\), we obtain the familiar \[ Q(i, j) = (n-2) d(i, j) - \sum_{k=1}^n d(i, k) - \sum_{k=1}^n d(j, k). \] Thus, we merge the two vertices that minimize \(Q\). The distance from the merging of vertices \(i\) and \(j\) to each other vertex \(k\) is \(d_{(i-j)k} = (d_{i,k} + d_{j,k})/2\).

UPGMA

Thu, 28 Oct 2021 11:56:00 +0200

Unweighted pair group method with arithmetic mean (UPGMA) is a phylogeny reconstruction method.

Input: Matrix of pairwise distances
Output: Phylogeny
Algorithm: Repeatedly merge the nearest two clusters. The distance between clusters is the average of all pairwise distances between them. When merging two clusters, the distances of the new cluster are the weighted averages of distances from the two clusters being merged.
Complexity: \(O(n^3)\) naive, \(O(n^2 \ln n)\) using heap.