Radix Tree and Trie-Based String Distances.
seqtrie
Background
seqtrie is a collection of radix tree algorithms for finding similar strings, with a focus on biological sequence data. Here, “similar” usually means close under Hamming distance, Levenshtein distance, or semi-global alignment.
A radix tree, also called a compressed trie, is an efficient data structure for searching large sets of sequences. It stores shared prefixes once, so a search can skip whole branches instead of checking every sequence one by one.
For bounded-distance searches, this can be sub-linear in the number of stored sequences because the tree can prune whole branches.
Basic usage
One common job is a similarity join: find every pair of sequences within a fixed distance. The example below has 133,033 TCR beta CDR3 DNA sequences (Nolan et al. 2020).
data(covid_cdr3)
results <- dist_search(covid_cdr3, max_distance = 3,
nthreads = 8, tree_class = "StarTree")
Trie-based algorithms are a good fit for this kind of search. The tree structure lets seqtrie rule out large groups of targets without checking every sequence against every other sequence.
Tree classes
seqtrie has three tree classes.
- RadixTree (
radix_tree()): the classic radix tree and the most general-purpose class. It supports insertion, deletion, prefix lookup, Hamming search, Levenshtein search, custom costs, single-gap search, and anchored search. - RadixForest (
radix_forest()): many radix trees, split by sequence length. This can be faster for Hamming and Levenshtein searches, especially when length is informative. - StarTree (
star_tree()): a specialized DNA similarity join class. It is a modified version of the Starcode all-pairs search algorithm from Zorita, Cuscó, and Filion (2015), adapted to operate over a radix trie rather than being a direct reimplementation. It supports fixed global/Levenshtein joins, fixed anchored joins (mode = "anchored"), and fixed Hamming joins (mode = "hamming"). For small edit-distance DNA self-joins it typically outperforms the other two classes (see the benchmark below).
dist_search() is a convenience wrapper around these classes. It can search for similarity between query and target, or within query when target is NULL.
Below is a benchmark applying dist_search to the covid_cdr3 dataset.
Understanding Radix Trees
Here is a small radix tree. We insert a few strings, erase one, and plot the result.
The useful part is the shared prefix. cargo, cart, carburetor, and carbuncle all start with car, so the tree stores car once. After that, it only stores the pieces where the strings split.
tree <- radix_tree()
insert(tree, c("cargo", "cart", "carburetor", "carbuncle", "bar", "zebra"))
erase(tree, "zebra")
# plot_tree requires igraph and ggplot2
set.seed(1); plot_tree(tree)
Hamming distance search
Hamming distance is similar to Levenshtein distance, but it does not allow insertions or deletions. Sequences must be the same length. Because of this restriction, Hamming distance is generally a lot faster.
Use max_fraction to set the distance threshold relative to query length. For example, max_fraction = 0.06 keeps matches within roughly 6% Hamming distance.
results <- align_search(tree, covid_cdr3, max_fraction = 0.035,
mode = "hamming", nthreads = 2)
results <- align_search(tree, covid_cdr3, max_fraction = 0.06,
mode = "hamming", nthreads = 2)
results <- align_search(tree, covid_cdr3, max_fraction = 0.15,
mode = "hamming", nthreads = 2)
Anchored alignment searches
An anchored alignment is a form of semi-global alignment. The query is anchored to the beginning of both the query and target sequences, but the end of either sequence can be unaligned.
tree <- radix_tree()
insert(tree, "CARTON")
insert(tree, "CAR")
insert(tree, "CARBON")
align_search(tree, "CART", max_distance = 0, mode = "anchored")
## query target distance query_size target_size
## 1 CART CAR 0 3 3
## 2 CART CARTON 0 4 4
With max_distance = 0, the query "CART" finds "CAR" and "CARTON" but not "CARBON". Anchored search also returns where the alignment ends in the query and target. This is useful when reads have variable length but start from the same genomic position or primer site.
Custom distance searches and affine gap alignment
seqtrie supports custom substitution costs and affine gap penalties.
tree <- radix_tree()
insert(tree, covid_cdr3)
# Define a custom substitution matrix. Use generate_cost_matrix for convenience.
cost_mat <- generate_cost_matrix("ACGT", match = 0, mismatch = 5)
print(cost_mat)
## A C G T
## A 0 5 5 5
## C 5 0 5 5
## G 5 5 0 5
## T 5 5 5 0
# Set gap penalties via parameters (not in the matrix):
# - Linear gaps: set gap_cost only
# - Affine gaps: set both gap_cost and gap_open_cost
# Linear example
results_linear <- align_search(tree, covid_cdr3, max_distance = 8,
mode = "global",
cost_matrix = cost_mat,
gap_cost = 2,
nthreads = 2)
# Affine example
results_affine <- align_search(tree, covid_cdr3, max_distance = 8,
mode = "global",
cost_matrix = cost_mat,
gap_cost = 2,
gap_open_cost = 5,
nthreads = 2)
results_linear[results_linear$query != results_linear$target, , drop = FALSE]
## query
## 130 TGTGCCAGCAGTGGGGGGAACACTGAAGCTTTCTTT
## 159 TGTGCCAGCAGCTCTAGCGGGGTGAGCACAGATACGCAGTATTTT
## 165 TGTGCCAGCAGCTCTACAGGGGTGAGCACAGATACGCAGTATTTT
## 203 TGTGCCAGTAGTCTCGGACAGGGCCTCTCCTACGAGCAGTACTTC
## 204 TGTGCCAGTAGTCTCGGACAGGGCCTCTCCTACGAGCAGTACTTC
## 258 TGTGCCAGCAGTTATTCGAACACCGGGGAGCTGTTTTTT
## 286 TGTGCCAGTAGTGTCGGACAGGGCCTCTCCTACGAGCAGTACTTC
## 287 TGTGCCAGTAGTGTCGGACAGGGCCTCTCCTACGAGCAGTACTTC
## 292 TGTGCCAGCAGCTCGGGACTGTCCTACGAGCAGTACTTC
## 293 TGTGCCAGCAGCTCGGGACTGTCCTACGAGCAGTACTTC
## 313 TGTGCCAGTAGTATGGGACAGGGGCTATCCTACGAGCAGTACTTC
## 329 TGTGCCAGTAGTATTGGACAGGGGACATCCTACGAGCAGTACTTC
## 330 TGTGCCAGTAGTATTGGACAGGGGACATCCTACGAGCAGTACTTC
## 342 TGTGCCAGCAGCTTCGCTAGCAATGAGCAGTTCTTC
## 345 TGTGCCAGCAGCTACAGGGGCTCCTACGAGCAGTACTTC
## 428 TGTGCCAGCAGCCCTTCGGGAAATGAGCAGTTCTTC
## 542 TGTGCCAGCAGCTTCGGGTACAATGAGCAGTTCTTC
## 544 TGTGCCAGCAGCTTCGGGTACAATGAGCAGTTCTTC
## 560 TGTGCCAGCAGCCCTACGGGGGGTGAAGCTTTCTTT
## 567 TGTGCCAGCAGTTACCGGGGAAGCACTGAAGCTTTCTTT
## 615 TGTGCCAGCAGTTTATACGGAGAGACCCAGTACTTC
## 635 TGTGCCAGTCAGCTTATCAACACCGGGGAGCTGTTTTTT
## 659 TGTGCCAGCAGCTCCTTTACCTACGAGCAGTACTTC
## 667 TGTGCCAGTAGTATTGGACAGGGACTCTCCTACGAGCAGTACTTC
## 687 TGTGCCAGCAGTCCGGGGAACACTGAAGCTTTCTTT
## 688 TGTGCCAGCAGTCCGGGGAACACTGAAGCTTTCTTT
## 728 TGTGCCAGCGGGACTAGCGGGGGGGCCCTAGATACGCAGTATTTT
## 760 TGTGCCAGCAGTTTTAGGGGCTACGAGCAGTACTTC
## 782 TGTGCCAGCAGTCTTAGTGGAGAGACCCAGTACTTC
## 794 TGTGCCAGCAGTTTTGCGGGTTACGAGCAGTACTTC
## 818 TGTGCCAGCAGCTTATATACCTACGAGCAGTACTTC
## 919 TGTGCCAGCAGCTCGGATTCCTACGAGCAGTACTTC
## 929 TGTGCCAGCAGCTCGGGGACCTCCTACGAGCAGTACTTC
## 930 TGTGCCAGCAGCTCGGGGACCTCCTACGAGCAGTACTTC
## 971 TGTGCCAGTAGTCTCGGGACAGGGTTCTCCTACGAGCAGTACTTC
## 1019 TGTGCCAGTAGTAAGGGACAGGGCCTCTCCTACGAGCAGTACTTC
## 1031 TGTGCCAGCAGGGCTAGCGGGGGGGCTCCAGATACGCAGTATTTT
## 1035 TGTGCCAGCAGCCTCGGGGGTGAAGCTTTCTTT
## target distance
## 130 TGTGCCAGCAGTCCGGGGAACACTGAAGCTTTCTTT 8
## 159 TGTGCCAGCAGCTCTACAGGGGTGAGCACAGATACGCAGTATTTT 4
## 165 TGTGCCAGCAGCTCTAGCGGGGTGAGCACAGATACGCAGTATTTT 4
## 203 TGTGCCAGTAGTCTCGGGACAGGGTTCTCCTACGAGCAGTACTTC 8
## 204 TGTGCCAGTAGTGTCGGACAGGGCCTCTCCTACGAGCAGTACTTC 4
## 258 TGTGCCAGTCAGCTTATCAACACCGGGGAGCTGTTTTTT 8
## 286 TGTGCCAGTAGTAAGGGACAGGGCCTCTCCTACGAGCAGTACTTC 8
## 287 TGTGCCAGTAGTCTCGGACAGGGCCTCTCCTACGAGCAGTACTTC 4
## 292 TGTGCCAGCAGCTCGGGGACCTCCTACGAGCAGTACTTC 8
## 293 TGTGCCAGCAGCTCGGATTCCTACGAGCAGTACTTC 6
## 313 TGTGCCAGTAGTATTGGACAGGGGACATCCTACGAGCAGTACTTC 8
## 329 TGTGCCAGTAGTATTGGACAGGGACTCTCCTACGAGCAGTACTTC 8
## 330 TGTGCCAGTAGTATGGGACAGGGGCTATCCTACGAGCAGTACTTC 8
## 342 TGTGCCAGCAGCTTCGGGTACAATGAGCAGTTCTTC 8
## 345 TGTGCCAGCAGCTCGGGGACCTCCTACGAGCAGTACTTC 8
## 428 TGTGCCAGCAGCTTCGGGTACAATGAGCAGTTCTTC 8
## 542 TGTGCCAGCAGCCCTTCGGGAAATGAGCAGTTCTTC 8
## 544 TGTGCCAGCAGCTTCGCTAGCAATGAGCAGTTCTTC 8
## 560 TGTGCCAGCAGCCTCGGGGGTGAAGCTTTCTTT 6
## 567 TGTGCCAGCAGTCCGGGGAACACTGAAGCTTTCTTT 6
## 615 TGTGCCAGCAGTCTTAGTGGAGAGACCCAGTACTTC 8
## 635 TGTGCCAGCAGTTATTCGAACACCGGGGAGCTGTTTTTT 8
## 659 TGTGCCAGCAGCTTATATACCTACGAGCAGTACTTC 8
## 667 TGTGCCAGTAGTATTGGACAGGGGACATCCTACGAGCAGTACTTC 8
## 687 TGTGCCAGCAGTTACCGGGGAAGCACTGAAGCTTTCTTT 6
## 688 TGTGCCAGCAGTGGGGGGAACACTGAAGCTTTCTTT 8
## 728 TGTGCCAGCAGGGCTAGCGGGGGGGCTCCAGATACGCAGTATTTT 8
## 760 TGTGCCAGCAGTTTTGCGGGTTACGAGCAGTACTTC 8
## 782 TGTGCCAGCAGTTTATACGGAGAGACCCAGTACTTC 8
## 794 TGTGCCAGCAGTTTTAGGGGCTACGAGCAGTACTTC 8
## 818 TGTGCCAGCAGCTCCTTTACCTACGAGCAGTACTTC 8
## 919 TGTGCCAGCAGCTCGGGACTGTCCTACGAGCAGTACTTC 6
## 929 TGTGCCAGCAGCTACAGGGGCTCCTACGAGCAGTACTTC 8
## 930 TGTGCCAGCAGCTCGGGACTGTCCTACGAGCAGTACTTC 8
## 971 TGTGCCAGTAGTCTCGGACAGGGCCTCTCCTACGAGCAGTACTTC 8
## 1019 TGTGCCAGTAGTGTCGGACAGGGCCTCTCCTACGAGCAGTACTTC 8
## 1031 TGTGCCAGCGGGACTAGCGGGGGGGCCCTAGATACGCAGTATTTT 8
## 1035 TGTGCCAGCAGCCCTACGGGGGGTGAAGCTTTCTTT 6
StarTree for fixed DNA self-joins
star_tree is a fixed DNA-only search structure based on the Starcode all-pairs search strategy described by Zorita, Cuscó, and Filion (2015). It is a modified version of that strategy, adapted to operate over a radix trie rather than being a direct reimplementation.
This class follows two Starcode ideas:
- A lossless prefilter: a k-mer hash check can exclude the vast majority of negative matches up front, allowing only plausible matches to search the tree.
- A caching strategy: sorted queries with shared prefixes can reuse part of the tree search state instead of starting from scratch.
The Starcode algorithm remains one of the fastest methods for self-similarity joins at small edit distances on DNA.
Unlike the other two classes, the sequences and search parameters are supplied at construction time, the self-join runs immediately, and the object does not support later insertion or deletion.
st <- star_tree(c("ACGT", "ACGA", "AAAA", "AAAT"),
max_distance = 1,
mismatch_cost = 1,
gap_cost = 1,
nthreads = 2)
result(st)
## query target distance
## 1 AAAT AAAA 1
## 2 ACGT ACGA 1
# Search another query set using the same fixed costs and threshold.
align_search(st, c("ACGT", "AAAC"))
## query target distance
## 1 AAAC AAAA 1
## 2 AAAC AAAT 1
## 3 ACGT ACGA 1
## 4 ACGT ACGT 0
# The same path is available through dist_search().
dist_search(c("ACGT", "ACGA", "AAAA", "AAAT"),
max_distance = 1,
tree_class = "StarTree")
## query target distance
## 1 AAAT AAAA 1
## 2 ACGT ACGA 1
StarTree supports global/Levenshtein-style, anchored, and Hamming DNA alignment with A, C, G, T, and N. It does not support custom substitution matrices, affine gaps, or per-query distance thresholds. Alignment parameters are fixed when the tree is built; align_search() only accepts query, nthreads, and show_progress for star_tree objects.
StarTree anchored mode for fixed anchored self-joins
star_tree(..., mode = "anchored") is the fixed-tree equivalent for anchored alignment. It is intended for DNA sets where every sequence starts at the same biological position but may end at different positions. The construction-time self-join returns each unordered pair once, and additional query sets can be searched against the same fixed targets.
ast <- star_tree(c("ACGT", "ACG", "AAAA", "AA"),
max_distance = 1,
mode = "anchored",
mismatch_cost = 1,
gap_cost = 1,
nthreads = 2)
result(ast)
## query target distance query_size target_size
## 1 AAAA AA 0 2 2
## 2 ACG AA 1 1 2
## 3 ACGT AA 1 1 2
## 4 ACGT ACG 0 3 3
align_search(ast, c("ACGT", "AA"))
## query target distance query_size target_size
## 1 AA AA 0 2 2
## 2 AA AAAA 0 2 2
## 3 AA ACG 1 2 1
## 4 AA ACGT 1 2 1
## 5 ACGT AA 1 1 2
## 6 ACGT ACG 0 3 3
## 7 ACGT ACGT 0 4 4
dist_search(c("ACGT", "ACG", "AAAA", "AA"),
max_distance = 1,
mode = "anchored",
tree_class = "StarTree")
## query target distance query_size target_size
## 1 AAAA AA 0 2 2
## 2 ACG AA 1 1 2
## 3 ACGT AA 1 1 2
## 4 ACGT ACG 0 3 3
Anchored StarTree mode supports scalar mismatch and linear gap costs through mismatch_cost and gap_cost. It does not support custom substitution matrices, affine gaps, or per-query distance thresholds.
StarTree Hamming mode for fixed same-length joins
star_tree(..., mode = "hamming") is a substitution-only fixed join: only sequences of the same length can match, and max_distance is the maximum number of mismatched positions. Because no insertions or deletions are considered, mismatch_cost and gap_cost do not apply, and it is typically much faster than global mode on the same data.
hst <- star_tree(c("ACGT", "ACGA", "TCGT", "ACG"),
max_distance = 1,
mode = "hamming",
nthreads = 2)
result(hst)
## query target distance
## 1 ACGT ACGA 1
## 2 TCGT ACGT 1
align_search(hst, c("ACGT", "TTGT"))
## query target distance
## 1 ACGT ACGA 1
## 2 ACGT ACGT 0
## 3 ACGT TCGT 1
## 4 TTGT TCGT 1
dist_search(c("ACGT", "ACGA", "TCGT", "ACG"),
max_distance = 1,
mode = "hamming",
tree_class = "StarTree")
## query target distance
## 1 ACGT ACGA 1
## 2 TCGT ACGT 1
N is treated as a regular base that mismatches every base, including another N, so two aligned N positions still count as a mismatch.
References
- Radix tree. Wikipedia. https://en.wikipedia.org/wiki/Radix_tree
- Nolan, Sean; Vignali, Marissa; Klinger, Mark; Dines, Jennifer N.; et al. “A large-scale database of T-cell receptor beta (TCRB) sequences and binding associations from natural and synthetic exposure to SARS-CoV-2.” Research Square. 2020 Aug 4:rs.3.rs-51964. https://doi.org/10.21203/rs.3.rs-51964/v1
- Schulz, Klaus U.; and Mihov, Stoyan. “Fast string correction with Levenshtein automata.” International Journal on Document Analysis and Recognition 5, no. 1 (2002): 67-85. https://doi.org/10.1007/s10032-002-0082-8
- Li, Heng; and Homer, Nils. “A survey of sequence alignment algorithms for next-generation sequencing.” Briefings in Bioinformatics 11, no. 5 (2010): 473-483. https://doi.org/10.1093/bib/bbq015
- Hanov, Steve. “Fast and Easy Levenshtein distance using a Trie.” 2011. https://stevehanov.ca/blog/index.php?id=114
- Zorita, Eduard; Cuscó, Pol; and Filion, Guillaume J. “Starcode: sequence clustering based on all-pairs search.” Bioinformatics 31, no. 12 (2015): 1913-1919. https://doi.org/10.1093/bioinformatics/btv053
- Buse-Dragomir, Alexandru; Popescu, Paul Stefan; and Mihaescu, Marian Cristian. “Spell Checker Application Based on Levenshtein Automaton.” In Intelligent Data Engineering and Automated Learning - IDEAL 2021, Lecture Notes in Computer Science 13113, 45-53. Springer, Cham, 2021. https://doi.org/10.1007/978-3-030-91608-4_5