This repo has been archived for two reasons. First, the naming of the repo "lv89" is inaccurate as Ukkonen first found the algorithm implemented here (see also #4). Second, the implementation is inefficient. When I have time, I may reimplement the algorithm along the direction of miniwfa.
# Compile the test program for lv89 and edlib
git clone https://github.com/lh3/lv89
cd lv89
make
./ed-test -c test/t2-t.fa test/t2-q.fa
# Compile with WFA2
git clone https://github.com/lh3/lv89
cd lv89
git clone https://github.com/smarco/WFA2-lib
(cd WFA2-lib && make)
make WFA2_ROOT=WFA2-libThis repo provides a reasonably fast yet lightweight implementation of the Landau-Vishkin/Myers86 algorithm to compute the edit distance between two strings, in a formulation inspired by WFA2. This repo doesn't aim at the fastest implementation. It is usually slower than edlib and WFA2 but is simpler.
The basic LV89 algorithm does global alignment. It takes O(nd) time and O(d) space to compute the distance, where n is the length of the longer sequence and d is their edit distance. Deriving the base alignment requires O(d2) space, which could be significant even when this repo uses only 2 bits per entry.
To alleviate this issue, we employed a divide-and-conquer approach inspired by Hirschberg's algorithm. We split the alignment into ceil(d/q) blocks where each block, except the last one, has edit distance exactly q. We then perform base alignment inside each block. The new algorithm takes O(n(d+q)) in time and O(d*ceil(d/q)+q2) in space. When q equals 1000, the space can be reduced by tens of folds in practice in comparison to the basic algorithm. I later realized that this is somewhat similar to Eizenga and Paten (2022), though the other algorithm does not explicitly segment the alignment.
We only use two pairs of sequences for evaluation. The first pair consists of the two haplotypes of NA19240 around the C4A/C4B gene. They are 100-150kb in length with an edit distance of 26k. The second pair consists of GRCh38 and CHM13 around MHC. They are about 5Mb in length with an edit distance of 123kb. These sequences can be found via Zenodo.
| Method | CIGAR | CMD option | Time MHC (s) | RAM MHC (MB) | Time C4 (s) | RAM C4 (kB) |
|---|---|---|---|---|---|---|
| lv89 | N | 102 | 16 | 0.36 | 1632 | |
| edlib | N | -e | 36 | 25 | 0.24 | 1848 |
| WFA2 | N | -w | 38 | 47 | 1.52 | 4424 |
| lv89 | Y | -c | 156 | 3659 | 0.68 | 22440 |
| lv89 | Y | -cs1000 | 151 | 191 | 0.67 | 2652 |
| edlib | Y | -ec | 99 | 47 | 0.65 | 3232 |
| WFA2-low | Y | -wcm1 | 146 | 1207 | 4.50 | 109132 |
| WFA2-med | Y | -wcm2 | 145 | 1310 | 4.51 | 109132 |
| WFA2-high | Y | -wcm3 | 74 | 58937 | 2.51 | 3123828 |
edlib is the overall winner. It is fast and uses the least memory. WFA2 is slower on the C4 sequences probably because lv89 prunes diagonals that are unlikely to yield better alignments (this is not a heuristic). WFA2 also uses more memory to generate base alignment. Nonetheless, WFA2 is the only library here that supports affine gap penalties, which will be crucial to many biological applications.
Ukkonen found the first O(nd) algorithm to compute edit distances. The
work was established in 1983 and published in 1985. Landau and Vishkin
cited Ukkonen's work but presented it in a different formulation which is
closer to the modern formulation implemented here. Landau and Vishkin submitted
their paper in 1986 and published in 1989. Meanwhile, Myers
independently published another O(nd) algorithm in 1986.
His formulation is almost identical to Landau-Vishkin's. Each of the three
papers also described additional algorithms (for example, Myers (1986) gave a
linear-space algorithm to find base alignment, inspired by Hirschberg (1975);
Landau and Vishkin (1989) introduced suffix trees to speed up search).
Nonetheless, these papers are more often remembered as the first batch of
papers on the O(nd) algorithm.
Edlib implements Myers' bit-parallel algorithm published in 1999. The basic bit-parallel algorithm is O(mn/w) in time, where w is the number of bits in an integer. This algorithm is easy to implement. The full algorithm has an O(nd/w) expected (not worse-case) time complexity. Implementing this algorithm is much more challenging. Edlib additionally supports base alignment in linear space. It is an engineering feat and the best overall library for edit distance.
The search for an O(nd) algorithm with linear or affine gap penalties took much longer. To the best of my knowledge, Xin et al first found this algorithm but apparently they have never published the result in a peer-reviewed journal. WFA is the first published algorithm. It also comes with a highly efficient implementation, beating all global alignment algorithms by a large margin. WFA is the best overall library for global and semi-global alignment under affine gap penalties.