Notes on bidirectional anchors

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

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.

Paper overview

Minimizers: First recall the definition of minimizers, introduced independently by Roberts et al. (2004) and Schleimer, Wilkerson, and Aiken (2003): Given a $(k+w-1)$-mer, consider the \(w\) contained $k$-mers. The $k$-mer with minimal hash (for some given hash function \(h\)) is the minimizer.

Density: By definition, minimizers sample at least one in every \(w\) positions, so trivially have a density of at least \(1/w\). Schleimer, Wilkerson, and Aiken (2003) proves that on random strings, any minimizer scheme has a density at least \((1.5+1/2w)/(w+1)\geq 1.5/(w+1)\). Random minimizers (i.e. using a good hash) have a density of \(2/(w+1) + o(1/w)\), as proven in both papers, which is only \(33\%\) above the lower bound. (NO_ITEM_DATA:improved-minimizers-2) corrects this and adds the further restriction that this only holds when \(k > (3+\epsilon) \log_\sigma (w+1)\). I.e. for \(w=24\) and \(\sigma=4\), \(k\) should be at least \(7\). See also (NO_ITEM_DATA:improved-minimizers-1). It is unclear to me whether this bound is tight, i.e. whether the constant \(3\) could be improved. (TODO think about this. For \(k>2\log_\sigma (w+1)\) we obtain \(2/(w+1)+o(1)\) which may or may not be good enough. Experiments needed.)

Bd-anchors: Whereas minimizers take a minimum over contained substrings, bd-anchors take a minimum over all rotations of a string. This means that the minimum is over \(\ell\) elements instead of just \(w\) elements.

Example: Take text BACADE and \(l=4\).

• The minimal rotation of BACA starts at the A in position 3 in the text (0-based).
• The minimal rotation of ACAD starts at the A in position 1 in the text.
• The minimal rotation of CADE starts at the A in position 3 in the text.

Thus, the anchors are at position \(1\) and \(3\). Note that unlike minimizers, anchors are not monotone!

Reduced bd-anchors: One issue with bd-anchors is that they are not very stable when the minimal rotation starts close to the end of the string. To prevent this, reduced bd-anchors take the position of the minimum rotation that does not start in the last \(r\) positions. Typically \(r=C\log_\sigma(\ell)\). Reduced bd-anchors are very similar to minimizers for \(k=r+1\).

Construction algorithm

• There is a straightforward \(O(n\ell)\) construction that finds the minimal rotation of each $ℓ$-mer in \(O(\ell)\) time.
• The main data structure used in Theorem 2 for linear time construction is sadly just a block box. Some explanation would have been nice, but I suppose it’s not practical anyway.
• Ayad, Loukides, and Pissis (2023) presents an average-case \(O(n)\) linear time construction algorithm for \(r=4\log_\sigma(\ell)\) reduced bd-anchors based on minimizers:
  1. A $r$-reduced bd-anchor is an \(k=r+1\) lexicographic minimizer.
  2. Hence these \(k=r+1\) minimizers are a superset of bd-anchors.
     • When is such a minimizer not a bd-anchor? Only when there are exact duplicates within an $ℓ$-mer?
  3. Then, a fairly involved procedure is used to find the minimal reduced bd-anchor of each $ℓ$-mer.
     • Rotations corresponding to contained minimizers are compared in \(O(1)\) using LCP (longest common prefix) queries. This requires the suffix array of the input text, which does not seem practical.
     • A simpler argument, that does not require the decomposition into fragments, seems to be that each minimizer is in \(w\leq \ell\) windows, and hence contributes less than \(\ell\) comparisons. Total expected runtime: \(w \cdot O(n/l) \cdot O(1) = O(n)\).

Theoretical density

• Lemma 5: When \(\ell \leq \sigma\) (or rather, \(\log \ell = O(\log \sigma)\)), \(|\A| = O(n/\ell)\) for random strings.
  • The big-\(O\) makes this result not very useful in practice.
• Lemma 6: The density of reduced bd-anchors with \(r=4\log_\sigma(\ell)\) is \(O(n/\ell)\), and in particular it’s bounded by \(2/(\ell-r) + o(1/\ell)\).
  • Remark: similar to minimizers, it seems that \(r=(3+\epsilon) \log_\sigma (\ell)\) is actually sufficient here. And indeed \(r=3\log_\sigma(\ell)\) is used in evaluations.
  • Remark 2: The \(2/(\ell-r)\) bound seems tight for large alphabets.
• Theorem 4: Either \(|\A|\) or \(|\A^r|\) is in \(O(n/\ell)\).
  • It seems to me that \(|\A^r|\) is always \(O(n/\ell)\) and there is no real benefit to falling back to \(|\A|\) itself?
    • This is indeed how Lemma 2.8 is presented in Ayad, Loukides, and Pissis (2023).

Empirical density

• Surprisingly, Miniception (NO_ITEM_DATA:improved-minimizers-2) did not do better than random minimizers. The miniception paper quite consistently does better with densities around \(1.7\) (as also mentioned in 6.3). I’m very curious why.
• Fig 2:
  • Random minimizers (with robust winnowing) consistently have density \(2/(w+1)\).
  • Bd-anchors are much better when \(k\geq w\).
  • In most cases, reduced anchors are better. For DNA, there are some cases where reduced anchors are much worse.
  • Cases with the most gain such as \((w,k) = (8, 128)\) are not used in practice.

Text indexing

• Given the bd-anchors, build two compacted tries (represented as suffix arrays and LCP arrays):
  • \(\T^R\) on suffixes of \(T\) starting at bd-anchors;
  • \(\T^L\) on reversed prefixes of \(T\) ending at bd-anchors.
• For both tries, store for each node the interval of leaf nodes corresponding to it.
• Given a query of length at least \(\ell\):
  1. Find a bd-anchor.
  2. Search the reverse-prefix of length \(\alpha\) before the bd-anchor in \(\T^L\). Find the interval of nodes.
  3. Search the suffix of length \(\beta = \ell -\alpha\) in \(\T^R\). Find the interval of nodes.
  4. Do a 2D range query to determine if the intervals intersect.
• As expected, this 2D range query is slow in practice.
• Faster alternative in practice:
  1. Search the longer of the reverse-prefix and suffix in its trie.
  2. For each match, verify if the rest of the query matches as well using simple string comparison.

Remarks on the paper

Some remarks on the paper content itself. Some typos were found by Giulio.

Theory

• \(\A\) is ill-defined. It is defined as the set of bd-anchors over all contained $ℓ$-mers. But all those bd-anchors are integers between \(1\) and \(\ell\), implying \(|\A|\leq \ell\) which is not intended. Some identification from the position in the $ℓ$-mer back into the text \(T\) should be made.
  • The same imprecision is present in Ayad, Loukides, and Pissis (2023).

Density proofs

• Proof of Lemma 5, first line: First \(\A\) should be \(|\A|\). Second \(\A\) should be \(\mathbb{E}(|\A|)\).
• Equation 8: \(p^{red} \geq r\) is missing.

Evaluation

• Fig 2: Many of the parameters shown are not very practical. I have not seen parameters like \(k=128\), \(w=8\) in practice.
• The bottom row of table 3a is wrong. Shown are values of \(32/l\), not \(n/l = 20/l\).
• According to https://github.com/solonas13/bd-anchors, for chromosome 1 of length \(n=230\ 000\ 000\), \(|\A|=900\ 000\) positions are sampled, or \(3.6\cdot n/l\). That’s not at all close to \(< 2n/l\) hinted at by table 3. Probably \(r=0\) was used instead of the recommended \(r=4\log_\sigma(n)\).

Remarks on Ayad, Loukides, and Pissis (2023)

• section 5 implementations: Minimizers have a very similar density as bd-anchors when \(k\sim r\). Excluding minimizers based on density is not a good reason.
• All figures: I assume all log scales are base 2? Quite unclear.

Thoughts

Here are some further thoughts and ideas and analysis.

Naive construction algorithm: The following seems like a much simpler construction algorithm. Maybe I am missing something that makes this not \(O(n)\) expected runtime?

1. Assume the input is a random string.
2. Set \(r = (3+\epsilon)\log_\sigma(\ell)\). Find the \(k=r+1\) lexicographic minimizers in \(O(n)\) time. Each minimizer fits in a constant number of machine words.
3. By choice of \(r\), \((1-o(1))\) of the $ℓ$-mers have a unique minimizer.
4. W.h.p., the minimizer is also the reduced bd-anchor, so simply (over-) report the set of minimizers as bd-anchors.
5. To prevent over-reporting, equal minimizers can be detected on-the-fly while computing them, and suffixes/rotations can be compared as needed.

The presented construction algorithm is kinda slow: From fig 5a in the 2nd paper, computing bd-anchors takes around 20 seconds for 200MB of DNA. 10MB/s is slow when processing up to TBs of data. NtHash (Mohamadi et al. 2016) goes up to 200MB/s and can still be a bottleneck. (But I understand the implementation wasn’t optimized yet.)

Density on long random strings: Neither of the papers makes a claim about observed density on long random strings. So I wrote a small tool to do some experiments: github:me/anchors.

Results:

• For large alphabet size \(\sigma = 256\), the density is very close to \(2/(\ell + 1 - r)\) and minimized for \(r=0\).
• For small alphabet \(\sigma = 4\), \(r=0\) has high density due to minimizers wrapping around. The optimal \(r\) seems to grow sort of logarithmic in \(\ell\), and has density only around \(10\%\) above the expected lower bound. The link has a table.

Density of bd-anchors:

• Each bd-anchor ends up in two suffix arrays, so one could argue that the density of this scheme should be doubled.
• In the faster text index, only the longer of the prefix and suffix is actually queried in the sparse suffix array. This means that actually we can store for each (reduced) bd-anchor only the longer of the prefix and suffix. This should reduce the size of the data structure quite a bit. (Around 30% would be my guesstimate.)

References