Indexing labeled sequences
 Published
 Accepted
 Received
 Academic Editor
 Rahul Shah
 Subject Areas
 Bioinformatics, Computational Biology, Algorithms and Analysis of Algorithms
 Keywords
 Data structures, Text indexing, Burrows–Wheeler transform, Wavelet Tree, V(D)J recombination
 Copyright
 © 2018 Rocher et al.
 Licence
 This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction and adaptation in any medium and for any purpose provided that it is properly attributed. For attribution, the original author(s), title, publication source (PeerJ Computer Science) and either DOI or URL of the article must be cited.
 Cite this article
 2018) Indexing labeled sequences. PeerJ Computer Science 4:e148 https://doi.org/10.7717/peerjcs.148 (
Abstract
Background
Labels are a way to add some information on a text, such as functional annotations such as genes on a DNA sequences. V(D)J recombinations are DNA recombinations involving two or three short genes in lymphocytes. Sequencing this short region (500 bp or less) produces labeled sequences and brings insight in the lymphocyte repertoire for oncohematology or immunology studies.
Methods
We present two indexes for a text with nonoverlapping labels. They store the text in a Burrows–Wheeler transform (BWT) and a compressed label sequence in a Wavelet Tree. The label sequence is taken in the order of the text (TLindex) or in the order of the BWT (TL_{BW}index). Both indexes need a space related to the entropy of the labeled text.
Results
These indexes allow efficient text–label queries to count and find labeled patterns. The TL_{BW}index has an overhead on simple label queries but is very efficient on combined pattern–label queries. We implemented the indexes in C++ and compared them against a baseline solution on pseudorandom as well as on V(D)J labeled texts.
Discussion
New indexes such as the ones we proposed improve the way we index and query labeled texts as, for instance, lymphocyte repertoire for hematological and immunological studies.
Introduction
Labels are a way to add some information on a text, as the semantics of words on an English sentence or functional annotations such as genes on a DNA sequences. Can we build an index that saves a labeled text like ACGCC…TTGA (of size 96), which have the label L_{1} in the positions 12–41 and the label L_{2} in the positions 56–96? We consider here the case where a same label can be given to different (but similar) patterns and can occur several times in the text. We introduce two indexes which store a labeled text and answers to position–label association queries. Those indexes share some ideas with the RLFMI (Mäkinen & Navarro, 2004) which uses a Burrows–Wheeler transform (BWT) and a Wavelet Tree (WT). Using a somewhat similar organization, we index a labeled text. The following sections present the TL and TL_{BW}indexes (text–label indexes) and their associated queries. The last section presents experimental results on simulated and real genomic data.
Let T = t_{0} t_{1} … t_{n − 1} be a text of length n over an alphabet of size σ. The text may be composed of several sequences, each sequence ending with the symbol $. Let L = {L_{0}, L_{1} … L_{l − 1}} be a set of labels. A labeled text (T, A) is here a text with nonoverlapping labels: a letter should have at most one label. Each position i of the text is labeled by exactly one label a_{i} ∈ L ∪ {ε}, where the special ε label is put on every letter without label. A = a_{0} a_{1} … a_{n − 1} is the label string. Figure 1 shows the text which will be used all over this article.
Given a bit vector B = B[0]B[1] … B[n − 1], where each B[i] is either 0 or 1, we define as B[i, j] the vector B[i]B[i + 1] … B[j]. Let us call rank(b, i, B) the number of times the bit b ∈ {0, 1} appears in the prefix B[0, i] and select (b, j, B) the position i of the jth appearance of the bit b in B. Such a bit vector B can be stored in nH_{0}(B) + o(n) bits to support rank and select in O(1), where H_{0}(B) is the zeroth order entropy of B (Raman, Raman & Rao, 2002).
The BWT (Burrows & Wheeler, 1994) is a reversible algorithm which reorganizes the letters of a text. The transformed text, BWT(T), is the concatenation of the last letters of the lexicographically sorted rotations of the text. BWT(T) is easier to compress and can be stored using nH_{k}(T) + o(n) bits, where H_{k} is the kth order empirical entropy of the text T. The FMindex (Ferragina & Manzini, 2000) uses the BWT, a table C, where C[α] gives the number of letters lexicographically smaller than α, and a function Occ(α, i) giving the number of occurrences of α in BWT(T)[0, i]. The FMindex allows to efficiently search a pattern using backward search.
A WT (Grossi, Gupta & Vitter, 2003) is a binary tree storing a text, where each symbol from the alphabet corresponds to a leaf. The root is a bit vector where every position corresponds to the element it has to index. Any position marked 0 (respectively 1) corresponds to an element whose leaf is on the left (respectively right) descendant of the node. The process is repeated recursively until the leaves. For a text A of length a in an alphabet of size l, the construction of a balanced WT needs $O\left(a\lceil \mathrm{log}l/\sqrt{\mathrm{log}a}\rceil \right)$ time (Munro, Nekrich & Vitter, 2016) and requires nH_{0}(A) + o(a log l) bits when bit vectors are zeroorder compressed (Navarro & Mäkinen, 2007).
The usual fulltext indexes such as the FMindex (Ferragina & Manzini, 2000) or the LZindex (Kärkkäinen & Ukkonen, 1996) do not index labeled texts. Some recent researches allow bidimensional range queries: Arroyuelo et al. (2015) represent an XML tree structure as a compressed bit vector and allow queries on structured patterns.
We focus here on nonoverlapping labels and look for ways to efficiently store such labels along with sequences as well as to query them. The basic idea is that consecutive labels in A are often the same: we will thus store compressed label sequences.
Materials and Methods
TLindex: indexing labels over a text
Given a labeled text (T, A), we define the TLindex as, using a FMindex built on a BWT U to index the text, a bit vector B_{A} marking the positions in the text where the labels change, and a WT W_{A} indexing a compressed label sequence (Fig. 2A).
BWT U. Let U = u_{0} u_{1} … u_{n − 1} be the BWT of T = t_{0} t_{1} … t_{n − 1}. As usually done, the FMindex samples every log^{1+ε} n values of a suffix array to retrieve the text positions of any occurrence (Navarro & Mäkinen, 2007).
Bit vector B_{A}. Let B_{A} a compressed bit vector of length n such that B_{A}[0] = 1, and, for i ≥ 1, B_{A}[i] = 0 if a_{i} = a_{i − 1}, and otherwise B_{A}[i] = 1.
Wavelet Tree W_{A}. Let A′ = 〈a_{i}  B_{A}[i] = 1〉. A′ = a′_{0} a′_{1} … a′ _{a − 1} is called the compressed label sequence. It is a subsequence of A, of length a, containing only the changing labels according to the positions in A. The compressed label sequence A′ is stored using a WT W_{A}. W_{A} is defined as (Q, q_{0}), where Q is the set of nodes of W_{A} and q_{0} ∈ Q is the root node. Each node q ∈ Q is q = (q.val, q.label, q.left, q.right, q.parent), where q.label ∈ L ∪ {ε} is the label of q and q.parent is the parent node of q (q_{0}.parent being the null value ⊥). Both q.left and q.right are child nodes in Q ∪ {⊥}. A leaf is a node q where q.right and q.left are ⊥. Each leaf q is linked to a label q.label ∈ L ∪ {ε}. The l + 1 leaves are exactly the labels of L ∪ {ε} and we define leaf (q.label) = q. On a leaf q, we have q.val = ⊥. Let q be a nonleaf node: we have q.label = q.val is the bit vector rooted at q in the WT. We explain in “Shaping the WT for a Label Hierarchy” how W_{A} can be further shaped depending on a label hierarchy.
W_{A} is part of the index. This WT is used to answer efficiently bidimensional range queries where the labels and the label positions are the two dimensions (Mäkinen & Navarro, 2007). A balanced WT has a height of log l, with l, the number of leaves. The accessor W〈i〉 returns a′_{i}, in O(log l) time. This is a classical query within a WT. Given a label L_{x} ∈ L, the function selectW〈L_{x}, i〉 gives the position of the ith L_{x} label in A′ in O(log l) time. The accessor W^{−1}〈L_{x}〉 gives the list of positions in A′ where a′_{i} = L_{x}. It runs in O(log l × occ) time, with occ the number of occurrences of L_{x} in A′.
TL_{BW}index: indexing labels in the order of the BWT
The BWT tends to store text repetitions consecutively. As those repetitions may have the same labels, it would be interesting that the labels benefit from the BWT reordering. Hence, labels can also be stored in the order of U.
Given a labeled text (T, A), the TL_{BW}index is defined as (U, B_{D}, W_{D}) (Fig. 2B). The BWT U is built in the same way as the TLindex. Let D = d_{0} d_{1} … d_{n − 1} the labels in the order of U. The bit vector B_{D} of size n is such that B_{D}[0] = 1, and, for i ≥ 1, B_{D}[i] = 0 if d_{i} = d_{i − 1}, and otherwise B_{D}[i] = 1. Let D′ = 〈d_{i}  B_{D}[i] = 1〉. D′ = d′_{0}, d′_{1}…d′_{d − 1} is a compressed label sequence of length d, subsequence of D. The WT W_{D} now indexes the compressed label sequence D′. The TL_{BW}index will be slower to build than the TLindex as it needs D. On the other side, as it is aware of the order of letters in the BWT, it will be able to support faster text/label queries.
Queries
The indexes allow the following classical queries.
label(t_{i}) (also called access(a_{i}))—Which label is on the letter t_{i}? This query is done is O(log l) time in the TLindex, and in O(log^{1+ε} n + log l) time in the TL_{BW}index since it has to convert positions in U order (see Fig. 3).
findP(P) P—Which are the occurrences of a pattern P? It is solved with the FMindex alone (Ferragina & Manzini, 2000). This query runs in O(P + occ × log^{1+ε} n) time in both indexes, where occ is the number of occurrences of P in T.
findL(L_{x})—Which text positions are labeled L_{x}? The query runs in O(y × log l) time in the TLindex, and in O(y(log l + log^{1+ε} n)) time in the TL_{BW}index, where Y = W^{−1}〈L_{x}〉 and y = Y. See Fig. 4.
The three previous queries are well known in text indexes. The two next queries search for a pattern and a label at the same time.
countPL(P, L_{x})—How many text positions are labeled L_{x} and begin an occurrence of a pattern P? As in the findL(L_{x}) query, the occurrences of P are found in U, in the positions from i to j.

–
TLindex: We translate all these occ occurrences to have the corresponding positions in the text. For each of them, we run the query label(t_{i}). The total time is O(P + occ (log n^{1+ε} + log l)).

–
TL_{BW}index: See Algorithm 1. i and j correspond to the positions i′ = rank(1, i, B_{D}) and j′ = rank(1, j, B_{D}) in the root q_{0} of W_{D}. We then use an accessor customized from the rangeLocate function of (Mäkinen & Navarro, 2007), simulating a twodimensional range query on [L_{x}, L_{x}] × [i′, j′] in W_{D} to give the set of positions Z = {z  a′_{z} = L_{x} and i′ ≤ z ≤ j′ } in O(Z × log l) time. This accessor first traverses the WT from the root to the leaf L_{x} and then traverses it back to the root to find the positions in q_{0}.val which match the label in the given range. For every position found, we find the corresponding positions in B_{D} and expand them to the following 0bits in B_{D}. This query runs in O(P + Z × log l) time.

findPL(P, L_{x})—Which text positions are labeled L_{x} and begin an occurrence of a pattern P?

–
TLindex: This query is exactly the same as countPL(P, L_{x}).

–
TL_{BW}index: We use the countPL(P, L_{x}) query detailed in Algorithm 1, replacing the counter cnt with a list Y holding the matching positions in U. The positions are converted at the end in the text order. This query runs in O(P + Z × log l + y × log^{1+ε} n) time, with Y = y.

(i,j) from findP(P) ▹ starting and ending positions of occurrences of P in U 
i′ = rank(1, i, B_{D}) 
j′ = rank(1, j, B_{D}) 
C = path(L_{x}) ▹ bit vector representing the path from the root to leaf(L_{x}) 
node = q_{0} 
for p in 0 to C − 2 do ▹ loop corresponding to rangeLocate in (Mäkinen & Navarro, 2007) 
i′ = rank(C[p], i′ − 1, node.val) 
j′ = rank(C[p], j′, node.val) − 1 
node = (C[p] == 0)? node.left : node.right 
if i′ > j′ then return 0 
cnt = 0 
for k in i′ to j′ do 
k′ = selectW〈L_{x}, rank(C[C − 1], k, node.val)〉 ▹ ith L_{x} label in A′ 
i″ = select(1, k′, B_{D}) 
j″ = select(1, k′ + 1, B_{D}) 
cnt = cnt + j″ − i″ ▹ positions [i″, j″ − 1] in U taken into account 
return cnt 
As Z ≤ y ≤ occ, the countPL() and the findPL() queries may thus be faster on the TL_{BW}index, Z depending of the compression B_{D} can do on Y.
Note that the countPL(P, L_{x}) query could be faster if the WT was directly built on the labels, without the intermediate bit vector B_{D} (B_{A}): the answer would be known while reaching the leaf of the WT in O(P + log l) time. We chose to favor the execution time of the findPL(P, L_{x}) query, as well as the size of the structure (when A can be compressed).
The findPL(P, L_{x}) can vary to find the patterns P which have the label L_{x} on the ith position of the pattern, or in any pattern’s position. Adapting this queries is easy in the TLindex as we find the positions of the pattern in the BWT, translate them in the text order and then find the label of the ith position following each of them. In the TL_{BW}index, we find the patterns’ positions in the BWT, access to the ith letter (we need to sample the BWT to read the patterns in the forward way), and find the label as usual. To have the label in any pattern’s position, in the TLindex we need to find a label L_{x} between the first and last letter of the pattern (with only two access in the WT) but in the TL_{BW}index, we look for the label of all the pattern’s letters.
Construction and space
We recall that the text (T, A) is of length n and is labeled with l unique labels. As defined above, the indexes store U in nH_{k}(T) + o(n) bits. The TLindex stores the bit vector with rank and select capabilities in nH_{0}(B_{A}) + o(n) bits. The size of W_{A} depends on the compressed label sequence A′, of length a. W_{A} takes a H_{0}(A′) + o(a log l) bits. Similarly, the TL_{BW}index stores B_{D} in nH_{0}(B_{D}) + o(n) bits and W_{D} takes d H_{0}(D′) + o(d log l) bits, where d is the length of D′. The BWT can be built in linear time while using little space (Belazzougui et al., 2016; Munro, Navarro & Nekrich, 2017). B_{A} is built while reading A in O(n) time. To make B_{D}, we need to read the labels in the order of the original data file in O(n) time. To make W_{A}, we find the occurrence of each label, corresponding to a 1bit in B_{A}, in O(a) time. Then we form the shape of W_{A} in O(l). The labels corresponding to a 1bit are extracted to make the WT’s root q_{0}. For each node containing at least two labels, we separate them by following the shape previously calculated, in $O\left(a\lceil \mathrm{log}l/\sqrt{\mathrm{log}a}\rceil \right)$. We build W_{D} the same way. The TLindex has thus a size of nH_{k}(T) + nH_{0}(B_{A}) + a H_{0}(A′) + o(n log l) bits, assuming σ = O(l), and is built in $O\left(n+l+a\lceil \mathrm{log}l/\sqrt{\mathrm{log}a}\rceil \right)$ time. The TL_{BW}index has a size of nH_{k}(T) + d H_{0}(D′) + o(n log l) bits and is built in $O\left(n+l+d\lceil \mathrm{log}l/\sqrt{\mathrm{log}d}\rceil \right)$ time.
Shaping the WT for a label hierarchy
Labels may be further organized into a label hierarchy, given an additional set F = {F_{0}, …, F_{f − 1}} of label families (Fig. 5A). Both TL and TL_{BW}indexes can be adapted: The WT W (either W_{A} or W_{D}) will be shaped according to the hierarchy, and internal nodes q of W may have nonempty q.label values belonging to F. For example, in Fig. 5B, one can set on either index q_{1}.label = L_{1}, where L_{1} is the label family gathering L_{1.1}, L_{1.2} and L_{1.3}.
The findL() and findPL() queries naturally extend to label families. With the hierarchy depicted in Fig. 5, findL(L_{1}) has to find the positions that have a label L_{1.1}, L_{1.2} or L_{1.3}. Such a query does not need to be iterated on every label of the family L_{1}, but rather directly starts at the corresponding internal node (q_{1} on Fig. 5B).
Shaping W for a label hierarchy may increase the height w of the WT to O(l) in the worst case. To have a lower space consumption and a faster average query time, one can give a Huffman shape to W (Huffman, 1952). A leaf which corresponds to a frequently used label will be higher in the tree than a leaf of a rarely used label. Depending on the label hierarchy, the average height of the tree is H_{0}(A′) in the best case while in the worst case it could be O(l). If no label hierarchy is given, the average height w will be H_{0}(A′) (Mäkinen & Navarro, 2005).
Results and Discussion
HTindex: a baseline index
We compared the TL and TL_{BW}indexes with a baseline solution called HTindex, indexing the text T with a BWT. The labels are stored in a map linking each label to the list of its (start,end) positions. We also store the labels in the text order with the compressed bit vector B_{A} and, stored in plain form, A′. This enables the findL(L_{x}) query in O(y), where Y′ is the list of pairs (start,end) which represent the occurrences and y = Y′. Note that Y (in the TLindex) and Y′ represent the same information as the labels are stored in the text order in both indexes. The label(t_{i}) query runs in O(1) time. This solution is not spaceefficient with repeated labels: it needs nH_{k}(T) + nH_{0}(B_{A}) + a + l′ + o(n) bits, where a is the size of A′ and l′ the number of labeled factors. The query times are summarized in Table 1.
Requests  HTindex  TLindex  TL_{BW}index 

label(i)  O(1)  O(log l)  O(log^{1+ε} n + log l) 
findP(P)  O(P + occ_{P} × log^{1+ε} n)  
findL(L_{x})  O(y)  O(y × log l)  O(y × (log l + log^{1+ε} n)) 
countPL(P, L_{x})  O(P + occ_{P} × log^{1+ε} n)  O(P + occ_{p} × (log^{1+ε} n + log l))  O(P + Z × log l) 
findPL(P, L_{x})  O(P + Z × log l + y × log^{1+ε} n) 
Note:
Note that we have Z ≤ y ≤ occ_{p}. The label(i) and findL(L_{x}) queries are faster in the HTindex and the TLindex as the HTindex needs a sampling time. However, the countPL(P, L_{x}) and findPL(P, L_{x}) are faster in the HTindex.
Evaluation procedure
The three indexes were implemented in C++. We used the SDSLLite library (Gog et al., 2014) to build the bit vectors and the WT. We used the RopeBWT2 library (Li, 2014), which builds a BWT in O(n log n) time on small DNA sequences, as it is very efficient for storing and appending sequences corresponding to our projected application (Cox et al., 2012). As RopeBWT2 does not sample the suffix array, we iterate over the text until we find a $ symbol. To have results close to the usual FMindex sampling in O(log^{1+ε} n) steps, we use sequences of length 50, which is similar to the sampling distance usually chosen in practice. The real files have longer sequences, thus longer sampling distances. The queries relying on the BWT will be slower and therefore cannot be compared between real and simulated files. We build the B_{A} (or B_{D}) bit vectors, compress them using the rrr_vector class of SDSLLite, and finally build W_{A} (or W_{D}) using a shape we added in SDSLLite, which depends of the label hierarchy.
We evaluated the build time, the index size, as well as the run time of three of the queries detailed in “Queries”—findP(P) behaving similarly in all indexes and countPL(P, L_{x}) being very similar to findPL(P, L_{x}). The three indexes were tested on various datasets of labeled texts, each with 100 M characters (Table 2). Datasets and code are available at http://www.vidjil.org/data/#2018peerjcs.
Random  Fixed #1  Fixed #2  Genomic  

Sequence size  50  50  50  264  
Lab. (t/u)  3.7 M / 1,000  4 M / 100  4 M / 10  852 K / 355  
Lab. avg size  12.6  25  5  110.3  
Lab. letters (%)  47  100  20  92  
Variation (%)  100  5  50  ??  
a = …/d = …  0.06n/0.77n  0.06n/0.16n  0.08n/0.36n  0.016n/0.26n  
TL  TL_{BW}  HT  TL  TL_{BW}  HT  TL  TL_{BW}  HT  TL  TL_{BW}  HT  
Size (MB)  104  184  217  33  38  114  99  116  209  13  33  35 
Time (s)  18  80  15  13.2  77  11  14.6  73.8  13.2  10.2  65  10.2 
label(t_{i}) (μs)  3.84  21.54  0.73  4.50  20.02  0.55  1.22  21.2  0.67  2.98  81.1  0.37 
findL(L) (μs/l)  0.4  34  0.4  0.12  13.1  0.20  0.40  21.41  0.21  0.04  52.5  0.17 
findPL(P, L) (s)  34.7  5.13  29.7  26.0  3.19  20.9  30.64  3.00  28.91  131.1  3.83  127.8 
Notes:
All files have 100 M characters, and differ by the number of total and unique labels (“Lab (t/u)”) and their size (“Lab. avg size”), by the ratio of labeled letters (“Lab. letters”), and by the variation between sequences labeled by the same label (“Variation”). Queries use patterns P with three letters. Times were averaged on 1 M launches (label()) or at least five launches (other queries). Times for findL(L) are reported per letter. Best or closetothebest results are in bold.
Simulated files with random sequences and random labels. All sequences and labels are random (d ∼ 0.8n).
Simulated files, with random sequences but fixed labels. Here a given label is associated to the same pattern, possibly with some variation, and we alter the proportion of labeled letters (5–100%), the variation in the label’s pattern (0–50%, more variations giving random labels), the number of unique labels (10–1,000), the length of the labels (5–100 letters). The dataset has 546 files, two of those files are shown in Table 2.
Genomic sequences with immunologic labels. A person’s immune system can be described with a set of labeled DNA sequences with V(D)J recombinations. The dataset, detailed below, uses 838 K labels from 355 unique labels, with d ∼ 0.26n.
A dataset of DNA sequences with immunologic labels
The adaptive immunity is the mechanism thanks to which our body defends itself against infections. When B and Tcells, or lymphocytes, are under maturation, their DNA undergo a recombination which allows several billions possibilities from a register of a thousand genes (Tonegawa, 1983). For example, the V(D)J recombination V4*02 4/ACGT/0 J1*12 means that the DNA sequence is composed from the V4*02 gene without the four last letters, then the ACGT sequence, then the J1*12 gene.
A person’s immune system can thus be described with a set of labeled DNA sequences encoding V(D)J recombinations (Fig. 6). These sequences can be sampled by nextgeneration sequencing with bioinformatics analysis (Bystry et al., 2016, Duez et al., 2016).
The tested DNA sequences come from patients 09, 12, 14 and 63 from a public dataset on a study on acute lymphoblastic leukemia (Salson et al., 2017). They have 100 M letters and 838 K labels from 355 unique labels, making a 117 MB file. Each DNA sequence has between 100 and 350 letters and two or three labels, each label having a size between 5 and 200 letters (Fig. 7). For a given label, the labeled subsequences may vary up to 15% due to amplification and sequencing errors.
Results
Index sizes. The Table 1 shows the results. As expected, the size of U, B (either B_{A} or B_{D}) and W (either W_{A} or W_{D}) grows linearly with the number of indexed elements (data not shown). The TLindex is the smallest index, and the TL_{BW}index is generally slightly larger. The compression is directly related to a and d. The file with random labels (d = 0.77n) is hard to compress, whereas the files with a low d/n ratio give a 2× to 7× compression. Figure 8 further shows how these sizes vary: As expected, the indexes are larger when there are less consecutive identical labels in T or in U, thus more 1bits in B. Note that when there are more labeled letters, the text is more similar (as labels are placed on similar substrings), hence a decrease in the BWT size (A, C). W increases while the number of unique labels increases (D), the height of W increasing logarithmically with the number of unique labels.
Build time. Most of the build time of TL_{BW}index is devoted to build D′. For TLindex, the building of U takes most of the total time.
Queries. The label() query is faster on the HTindex. As expected, the TL_{BW}index needs more time for label() and findL(), as it needs to translate positions from the BWT back to the text. Note that locating the positions in the text takes about the same time as label(t_{i}) in the TLindex. However, for the complex findPL(P, L) query, the TL_{BW}index is the fastest solution because the position translation is only done on the letters which have both the label and the pattern. For the TLindex and HTindex, the actual time of the findPL(P, L) queries is more affected by the number of pattern occurrences than the number of final occurrences (between 0 and 100 K depending on the file).
On the genomic dataset, the sequences are longer: The TL_{BW}index suffers here even more from the missing suffix array sampling of the implementation for queries label() and findL(). However, on the findPL(P, L) query, the other indexes are penalized due to the sparsity of the sampling, bringing a more than 30× difference with TL_{BW}index.
Conclusion
The TLindex and TL_{BW}index store a labeled text and allow efficient queries. They can be constructed from files of some MB in a few seconds. Experiments confirm that the indexes stay small when the text is redundant (thus a smaller U), when each label describes a pattern with few variations (many 0bits in B, thus a smaller W), and when few letters are labeled (thus a small W). However, the TLindex and TL_{BW}index are robust even to nonredundant text with almost random labels everywhere. The TL_{BW}index needs more memory space than the TLindex but is more efficient in combined label/pattern queries. Those structures might be used on any labeled data, such as DNA sequences, but also on natural language texts or on music sheets with some semantic annotations.
Perspectives include improvement of the implementation, with label families queries or parameterizing the distance between samples in the FMindex to offer a spacetime trade off. Within SDSL we could use the sd_vector bit vector instead of the rrr_vector bit vector which should improve space consumption when the bit vectors are very sparse. However, this would only minimally improve the global space consumption of the index. We plan to use one of the indexes in a clone database for hematological malignancies: It will allow comparison of V(D)J recombinations between different samples of a patient or between several patients.