 Research
 Open Access
 Published:
Adaptive efficient compression of genomes
Algorithms for Molecular Biologyvolume 7, Article number: 30 (2012)
Abstract
Abstract
Modern highthroughput sequencing technologies are able to generate DNA sequences at an ever increasing rate. In parallel to the decreasing experimental time and cost necessary to produce DNA sequences, computational requirements for analysis and storage of the sequences are steeply increasing. Compression is a key technology to deal with this challenge. Recently, referential compression schemes, storing only the differences between a tobecompressed input and a known reference sequence, gained a lot of interest in this field. However, memory requirements of the current algorithms are high and run times often are slow. In this paper, we propose an adaptive, parallel and highly efficient referential sequence compression method which allows finetuning of the tradeoff between required memory and compression speed. When using 12 MB of memory, our method is for human genomes onpar with the best previous algorithms in terms of compression ratio (400:1) and compression speed. In contrast, it compresses a complete human genome in just 11 seconds when provided with 9 GB of main memory, which is almost three times faster than the best competitor while using less main memory.
Background
The development of novel highthroughput DNA sequencing techniques has led to an ever increasing flood of data. While it took roughly 12 years and an estimated amount of 3 billion USD to decipher the first human genome[1], current techniques, usually summarized under the term second generation sequencing (SGS), are able to produce roughly the same amount of data in about a week at a current cost of roughly 2000 USD. On top, third generation sequencing promises to deliver a further speedup, reducing the time and price for sequencing a human genome from weeks to days and from thousands to under a hundred USD, respectively[2].
Largescale projects are generating comprehensive surveys of the genomic landscape of various diseases by sequencing thousands of genomes[3]. Managing, storing and analyzing this quickly growing amount of data is challenging[4]. It requires large disk arrays for storage, and large compute clusters for analysis. A recent suggestion is to use cloud infrastructures for this purpose[5–7]. However, before being analyzed in a cloud, data first has to be shipped to the cloud, making bandwidth in file transfer one of the major bottlenecks in cloudbased DNA analysis[8]. Accordingly, sequence compression is a key technology to cope with the increasing flood of DNA sequences[9–11].
To store a complete genome of a human being, one needs roughly 3GB (uncompressed). Substitutional or statistic compression schemes can reduce the space requirements by up to 6:1 (one base is encoded with up to 1.3 Bit)[12, 13]. However, in many projects only genomes from one species are considered. This means that projects often deal with hundreds of highly similar genomes; for instance, two randomly selected human genomes are identical to an estimated 99.9%. This observation is exploited by socalled referential compression schemes, which only encode the differences of an input sequence with respect to a preselected reference sequence. Using spaceefficient encoding of differences and clever algorithms for finding long stretches of DNA without differences, the best current referential compression algorithm we are aware of reports a compression rates of up to 500:1 for human genomes[14].
However, all existing compression schemes have in common that they have very high demands on the underlying hardware (up to 25 GB, for instance[15, 16]). Furthermore, the time needed for compressing the amount of sequences corresponding to a human genome may be up to several hours, which easily outweights the time savings during file transfer. Furthermore, existing approaches cannot easily be parallelized to reduce compression time, because such a parallelization requires to share information about the generated compression model. This communication/synchronization overhead may easily mitigate the positive effects gained by parallelization.
In this paper, we present an adaptive, scalable and highly efficient algorithm for referential compression of DNA sequences genomes. It is able to gracefully tradeoff compression time and space requirements while consistently achieving very good compression rates. For instance, our algorithms requires only 12 MB memory to compress/decompress a human genome at the same speed as the fastest existing approach. Using 9 GB of memory usage, our method performs up to three times faster than the best competitor while still needing less main memory. Both variants achieve similar compression rates of approximately 400:1 for human DNA. For yeast sequences, the compression ratios are lower, since two yeast genomes can have substantial differences.
The remaining part of this paper is structured as follows. We first explain the main ideas behind our algorithm in Section “General Idea”. The concrete design choices and algorithms are described in detail in Section “Genome Compression”. We discuss related work in Section “Related Work”, before we provide an evaluation of our method in Section “Evaluation”. The paper is concluded with Section “Conclusions”.
General Idea
We denote strings with s,t. The length of a string s is denoted with s and the substring starting at position i with length n is denoted s(i,n). s(i) is an abbreviation for s(i,1). All positions in a string are zerobased, i.e. the first character is accessed by s(0). The concatenation of two strings s and t is denoted with s∘t. Although a genome can be encoded with four characters, i.e. A,C,G, and T, we allow arbitrary symbols. For instance, symbol N is often used to indicate an unknown base. Given two strings s and t, the longest prefixsuffix match of s in t, is the longest string t_{ m }, such that t = t_{1}∘t_{ m }∘t_{2} and s(0,t_{ m }) = t_{ m }.
We want to compress a given input genome with respect to a reference genome, by only encoding differences between the input and the reference. This yields lossless compression, i.e. based on the reference genome and the difference description it is possible to recover the input genome. The idea is to split up a given reference genome into blocks. In general, two genomes of the same species are very similar to each other, and blocks are chosen in a way that long matches of tobecompressed blocks can often be found in reference blocks by local search. Please note that splitting the reference into blocks of fixed length is not an optimal way, since different sizes of the blocks can provide better performance in terms of compression ratio. However, for the sake of compression speed and easier data handling, we only consider a fixed block length.
For each reference genome block, a suffix tree is computed. The suffix trees allow to find longest prefixsuffix matches of parts of the input genome and the reference genome efficiently. The compression process is informally summarized in Algorithm 1.
Algorithm 1: Sketch of our Compression Algorithm
1: while Input contains characters
2: Find matching reference block B for current input position
3: Perform referential compression with respect to B until we cannot find “long” matches anymore
4: end while
The compression algorithm generates a set of referential matches with respect to reference blocks. The output of our compression algorithm is a compressed file of entries, such that each entry is one of the following:

Blockchange entry BC(i): next entries are encoded with respect to reference block i.

Relative match entry RM(i,j): The input matches the reference block at position i for j characters.

Raw entry R(s): A string s is encoded raw (for instance if there is no good matching block).
An example compression is given in Figure1. An input sequence “TACGTAAT..” is compressed with respect to a reference sequence “ACGACGTA..”. The first character of the input is encoded raw, with entry R(T), then the first reference block is chosen, with entry BC(0) , in order to referentially encode the subsequence “ACGTA”, with entry RM(3,5).
Please note that block change entries can be actually avoided, since local positions in a block can be easily transformed into global positions in the reference. In the remaining part, we will still assume the usage of block change entries.
Genome Compression
Index Construction
In order to find matches of the input in reference blocks, we need an index structure over the reference blocks. Suffix trees[17] are tree data structures which allow for fast access to all suffixes of a given string. Each suffix is usually represented by a path from the root of the tree to a leaf. Please note that the longest prefixsuffix match problem of s in t can be be easily solved by depthfirst search given a suffix tree for the string t. The focus recently shifted towards socalled compressed suffix trees. One example of a compressed suffix tree implementation is CST++[18], which uses a new approach to compress the longest common prefixarray and achieves, depending on the sampling rates of the (inverse) suffix array, an usual space overhead of (4−6)∗n bits. In the following, the compressed suffix tree of a string s is denoted with CST(s).
In addition, we need a reference genome for our compression algorithm. The reference sequence is given as a set of compressed FASTAfiles, one for each chromosome. The index generation algorithm iterates over all chromosomes of the reference genome. Each chromosome is split up into blocks of a maximum size BS. For instance, given the textual representation s_{1} of Chromosome 1, we compute m substrings b_{1,1},…,b_{1,m}, such that

s_{1} = b_{1,1},…,b_{1,m},

∀i ≤ (m−1):b_{1,i} = BS, and

[b_{1,m}] ≤ BS.
We do the same for all other chromosomes of the reference genome. For each reference block a compressed suffix tree is computed and stored on hard disk together with the raw reference block. In addition, we keep track of the order of the blocks, which is induced by their position on the chromosomes. This meta information is used below for optimizing our compression algorithm.
The memory consumption during index creation is limited as follows: at each step of the index generation we have one raw reference block of size at most BS bytes in main memory plus (roughly) 4∗BS bytes for its compressed suffix tree. For example, given a BS of 4 MB, the main memory usage can be restricted to approximately 20 MB. The maximum value for BS is 300 MB, since the largest human chromosome (Chromosome 1) has about 247 million nucleotide base pairs, and each base is encoded as one byte.
Please note that we could have encoded bases of the reference genome with 3 bits (five entries: A,C,G,N,T), in order to further reduce memory usage. However, our tests indicate that the space saving yields a significant time overhead for accessing bases. Furthermore, the 3Bitencoding would not allow us to compress sequences against references with other symbols than these five.
Compression and Decompression
In the following, we present our compression algorithm for genome sequences in detail. Algorithms do not show range checks for the sake of readability. Algorithm 2 assumes the input genome in Input (as a string of bases). The input string is traversed from left to right, and depending on the current characters in the input and in the reference block, different subroutines are executed.
Algorithm 2: Compression Algorithm
1: P_{ in }←0
2: P_{ raw }←0
3: FINDMATCH
4: while P_{ in }≠ ∣Input∣do
5: if Input(P_{ in }) = B(P_{ raw }) then
6: ENCODEREF
7: else if Input(P_{ in })∉{A,C,G,T} then
8: ENCODERAW
9: else
10: FINDMATCH
11: end if
12: end while
In the beginning of the compression algorithm, a match for the current input position in the reference is searched, using function FINDMATCH, Algorithm 3, explained in detail below. If the current reference block B matches the input at the current position, then we try to generate a (as long as possible) reference match in function ENCODEREF, Algorithm 4. If the current input base does not equal the current reference block base, and the input is not a normal base, i.e. neither A, C, G, or T, then the base and all the following nonnormal bases are added as raw entries to the compressed genome file in function ENCODERAW, Algorithm 5. Finally, if neither of the conditions is satisfied, then the algorithm tries to find a new match (either in the current block or in another reference block) using function FINDMATCH.
Algorithm 3: FINDMATCH Function
1: Max←25
2: if not HASLOCALALTERNATIVE() then
3: for all B_{ j }∈Blocks do
4: if m(Input,P_{ in },B_{ j })≥Max then
5: Max←m(Input,P_{ in },B_{ j })
6: B←B_{ j }
7: end if
8: end for
9: if Max≤25then
10: Raws←Input(P_{ in },Max)
11: Add R(Raws) to output
12: P_{ in }←P_{ in } + Max
13: else
14: P_{ raw }← beginning of the longest match in B
15: Add BC(B) to output
16: end if
17: end if
Algorithm 4: ENCODEREF Function
1: M←0
2: while Input(P_{ in })=B(P_{ raw })do
3: M←M + 1
4: P_{ in }←P_{ in } + 1
5: P_{ raw }←P_{ raw } + 1
6: end while
7: Add RM(P_{ raw }M,M) to output
Algorithm 5: ENCODERAW Function
1: Raws←""
2: while Input(P_{ in })∉{A,C,G,T}do
3: Raws←Raws∘Input[P_{ in }];
4: P_{ in }←P_{ in } + 1;
5: end while
6: Add R(Raws) to output
In Algorithm 4 the number of matching characters between current input position and current reference position are determined and stored in variable M. A reference entry RM(P_{ raw }M,M) is added to the compressed output. The encoding of a raw sequence in Algorithm 5 is straightforward: The string Raws is filled with bases from the input until a normal base is found and R(Raws) is added to the output.
Algorithm 3 requires more thoughts. The function FINDMATCH is called, whenever there is a mismatch between the input and the reference. In this case, we need to find an alternative match for the current input position. First, we check whether there exists a match in the neighbourhood (mutations caused by few single nucleotide polymorphisms) of the current position of the reference block. The process is explained in detail later. If we cannot find a local match, then all the reference blocks are checked for a better match. The expression m(Input,P_{ in },B_{ j }) returns the position of the longest prefixsuffix match of the current input in B_{ j }.
In our implementation of Algorithm 3, we care about the order in which all reference blocks are traversed. This is necessary, since loading blocks from the disk is a time consuming effort and should be avoided. Reference blocks are traversed with the following heuristic:

1.
The block left and right of the current reference block

2.
All other blocks on the same chromosome as the current reference block

3.
All other blocks of the reference genome
If there is no long enough match in any reference block, then we just encode few raw bases. Matches in other blocks are required to be longer than 25 characters, in order to avoid random matches. Our experiments have shown that the mere length of the human DNA causes a lot of unrelated matches with less than 2025 characters.
Algorithm 3 uses the function HasLocalAlternative. The intuition is that we want to avoid searching all possible reference blocks all the time. Our experiments showed that one longest prefixsuffixmatch lookupoperation in a single compressed suffix tree can take up to few milliseconds depending on the size. This is basically caused by the high bitwise compression of the reference genome. Furthermore, potentially we have to load all the other reference block’s CTSs from the hard disk. Therefore, the naive strategy to search for a new reference block on each mismatch does not scale.
Instead we use a local search in the neighbourhood of the current input position and of the current reference position. This strategy has a biological foundation: Often two parts of a genome might only be different by few bases (base insertion, base removal, of base mutation). Whenever we find an appropriate match in the neighbourhood, we avoid checking other reference blocks, although they might contain better matches. In fact, if we searched for the best reference blocks each time, we could increase the compression rate slightly for the price of being orders of magnitude slower.
The algorithm for local neighbourhood search is shown in Algorithm 6. If a match of at least 20 characters can be found near the current input and reference positions, then the difference until the match is encoded raw, and the match is encoded referentially.
Decompression of the relative genome data is straightforward. Basically, all entries are processed according to their type. For decompression of genome data we do not need the compressed suffix trees any more, but only the raw blocks of the reference genome. The decompression of genome data is the least CPUintensive task. Our evaluation in the next section shows that index generation and compression are CPUbound, while the decompression phase is I/Obound.
Algorithm 6: HASLOCALALTERNATIVE Function
1: for i∈{0,1,…,6,7}do
2: for j∈{−7,−6,…,6,7}do
3: if Input(P_{ in } + i,20)=B(P_{ raw } + j,20)then
4: if i≥1then
5: Raws←Input(P_{ in },i)
6: Add R(Raws) to output
7: P_{ in }←P_{ in } + i
8: end if
9: P_{ raw }←P_{ raw } + j
10: ENCODEREF
11: RETURN
12: end if
13: end for
14: end for
We emphasize that our compression scheme is a heuristic, which mainly works when compressing a sequence with respect to a reference from the same species. The efficiency of this heuristic is evaluated in Section “Evaluation”. The worstcase time complexity of our compression algorithm is O(—Input—), since we need to traverse the whole input sequence one time from left to right and for each character we perform (in the worst case) one lookup in the reference. The complexity of the lookup depends on the length of the substring being looked up (we assume a fixed maximal match length) and is therefore constant. The worstcase space complexity is O(—Input—). Each character of the input is stored at most one time: either inside a raw block or as part of a referential block.
Related Work
In the following, we review existing work on biological data compression. In general, compression algorithms are either substitutional, statistical, or referential. Substitutional algorithms[19] replace long repeated substrings by references, e.g. LempelZivbased compression. Statistical algorithms[13, 20] derive a predictive model from (a subset of) the input, based on partial matches. If the model always indicates high probabilities for the next symbol, then high compression rates are possible. While referential algorithms replace long substrings as well, the source for these subsets is usually not part of the input (the sequence to be compressed). Referential compression algorithms have drawn a lot of attention recently, since they allow for very high compression rates.
In[16], RLZ, a selfindexing based approach is proposed as follows: Given a selfindex for a base sequence, compress other sequences with LZ77 encoding relative to the base sequence. In fact, a suffix array for the reference sequence is built and the reference entries are position references with length into the base sequence. Please note that the authors do not store raw sequences at any time, but only encode based on a list of references, no matter how long.
The authors of[21] proposed to store human genomes relatively to a reference chromosome. They used variable integers for storing absolute and relative positions of matches and represent often used kmers (sequences of length k) with Huffman encoding.
In[22], a LZ77style compression scheme is proposed. The main difference is that several reference sequences are taken into account. The matchfinding process is based on hashing. Compression is performed on input blocks with shared Huffman models, in order to support random access. In[23], another LZ77style compression scheme is proposed.
There exists further early work on nonreferential compression algorithms[24–27]. Furthermore, there is compression based on contextfree grammars[28], determination of Markovmodel based probabilities of sequence positions[29], and hybrid methods[30]. In[12], a splay treebased compression algorithm is proposed, which favors encoding of recently seen symbols. Finally, there exists previous work on relative compression of reads and alignment data, e.g.[31, 32].
Evaluation
In the following section, we evaluate our proposed compression scheme. All experiments have been run on a Acer Aspire 5950G with 16 GB RAM and Intel Core i72670QM, on Fedora 16 (64Bit, Linux kernel 3.1). The code was implemented in C++, using the BOOST library[33], CST[34], and libz. All size measures are in byte, e.g. 1 MB means 1,000,000 bytes. The source code is available for download^{1}.
First, we performed compression tests on human genomes. Our set of data genomes consists of 1000 genomes from the 1000 Genome project[35]. The 1000 Genome project group provides all sequenced genomes in Variant Call Format (VCF)[36] for download^{2}. The Variant Call Format describes differences of sets of genomes with respect to a reference sequence, based on SNPs and indels. We have extracted one consensus sequence each for in total 1000 genomes. Since the project contains slightly more than 1000 genomes, we have only extracted the first 1000 genomes (columns from left to right) named in these VCF files. The consensus sequences for each chromosome of each genome were stored GZipcompressed on a hard disk. These 1000 GZipcompressed genomes need in total 700 GB of storage.
Compression Ratio
First, we have compressed each chromosome referentially against the reference chromosome taken from HG19[37]. The results are shown in Figure2. The first chromosome of 1000 humans needs 236.6 GB of uncompressed storage, while the referentially compressed file needs only 0.57 GB, yielding a compression ratio of 415:1. The smallest human chromosome of our 1000 genomes, Chromosome 22, is compressed from 36.4 GB down to 0.1 GB, yielding a compression ratio of 364:1. The overall compression ratio obtained for the 1000 human genomes is 397:1.
For very similar sequences we achieve higher compression rates, than for less related sequences. This is inherent to all referential compression schemes: the more similar input and reference sequence are, longer referential matches can be found.
We have evaluated the impact of the block size on the compression ratio. The results (average over all human genomes and chromosomes ) are shown in Figure3. In general, a larger block size will yield better compression ratios. The intuition is that a larger block in main memory allows for finding longer matches. However, for a block size of 1 MB, our compression scheme can still achieve a compression ratio of 361:1, which is roughly competitive to existing relative compression schemes for human genome sequences. RLZ obtains a compression ration of 80:1 and RLZopt a compression ratio of 133:1 for human genomes. GDC achieves compression ratios of 200:1  500:1, depending on speedtradeoffs. There exists one variant GDCultra, which achieves a compression ratio of 1000:1, which switches the reference sequence during compression. Switching the reference naturally allows for higher compression ratios. However, since we only use one reference, it seems to be fair to only compare our results to the nonultra variant of GDC, obtaining a compression ratio of at most 500:1.
Please note that the compression ratio for 300 MB is actually smaller than for 50MB. One explanation could be that all these matches found with smaller blocks allow for a shorter encoding than the matches found in a longer block. At first sight this might sound counter intuitive. To the best of our knowledge, no research has been conducted in this area, since the compression gain mainly depends on the choice of referential entries, i.e. how to encode positions, length entries and raw base entries. Referential compression is an optimization problem, where the longest matches often, but not necessarily, yield the shortest compressed representation. For instance, sometimes more (shorter) matches can be encoded more efficiently than less (longer) matches. We think that these effects are important to be studied in Future Work.
The overall index size for the reference sequence (per chromosome) is in average 202 MB, while an average uncompressed input sequence is roughly 130 MB long. The size of the index structure is decreasing with decreasing block size, since the maximum length of paths in the suffix tree is limited.
Additional experiments have been conducted on yeast genomes. We have downloaded^{3} 39 yeast genomes, have chosen an arbitrary reference sequence (273614N) and referentially compressed the other 38 genomes (all chromosomes concatenated to each other) with respect to the reference. The average compression ratio is 61:1. The lower compression ratio compared to human sequences is not surprising, since it is well known that two yeast genomes can be less similar than even a human genome and a chimpanzee genome. In these cases (unoptimized) referential compression schemes do not obtain such nice compression ratios as with human genomes. RLZopt obtains compression ratios of 50:1, and GDC obtains compression ratios of 70:1100:1, depending on speedtradeoffs.
In average, the index size of an average yeast gnome (around 12 MB) was found to be around 17.5 MB.
Compression Times
For serial compression, the compression algorithm performs compression on one block at a time (in a single thread). The memory requirements during compression are roughly 6*BS (one reference raw block, one input genome block, and the compressed suffix tree of the input block). In addition we use a write buffer for output operations (size: 1 MB), which can be neglected. The results for serial compression of the 1000 human genomes are shown in Figure4. All the results were averaged over all chromosomes as input.
Clearly compression time decreases with increasing block index size, since less time is spent on traversing other blocks to find best matches. If the block index size is 1 MB (equals to roughly 6 MB main memory usage), the compression times are roughly 9 seconds per chromosome, i.e. around 200 seconds to compress a whole human genome. This yields a compression speed of around 15 MB/s. With a block index size of 300 MB (roughly 1.8 GB main memory usage), the compression time is down to 1.5 seconds per chromosome, i.e. around 35 seconds for compression a whole human genome. This yields a compression speed of 85 MB/s. The time necessary to create the compressed suffix tree for the reference is roughly 20 minutes. If one takes the position that indexing time should belong to the online search time, then the compression speed is reduced to 2.42 MB/s for a single genomes and to 82.87 MB/s for 1000 genomes. However, we are convinced that index construction should be taken as an offline task.
RLZopt obtains a compression speed of 1.34 MB/s, while GDC achieves compression speeds of 435 MB/s, depending on speedtradeoffs. Therefore, our adaptive algorithm seems to be competitive with existing relative compression schemes, while using less memory in a controllable (by changing the block size) way.
In addition, we have performed experiments with 39 yeast genomes as in the previous subsection. The compression time is around 70 seconds for one yeast genome, yielding a compression speed of 0.17 MB/s. As expected, our local search heuristic does not work as good as with human genomes. Most of the time is spent on consulting the compressed suffix tree which often only yields very short matches of length 1020. RLZopt has a compression speed of 1.6 MB/s and GDC achieves compression speeds of 233 MB/s, depending on speedtradeoffs. In this case, it seems to be advantageous to use a hash table as an index structure instead of a suffix tree, as proposed by the authors of GDC.
The parallelization on multi cores of our approach is straightforward. Blockprocessing can be easily distributed on several CPUs (or even machines). However, it has to be kept in mind that each compressor might work on different parts of the reference genome, which means that for n compressor threads the memory usage is increased by the factor n. The upper limit main memory usage for seven threads is 7 GB (maximum size of the complete index) + 7∗BS, if all index structures are loaded into main memory.
Further investigation showed that the compression time in the latter case is dominated by loading the index files into main memory. Basically, in the beginning each thread requests a set of index blocks, which (due to parallel access) might be loaded in a partial random access manner.
To counter act this effect, we have prebulk loaded all index structures at once after a reboot (taking roughly 80 seconds) and then performed compression. In this setting, a whole human genome can be compressed within 11 seconds with 7 threads.
We think that this bulkload scenario is actually quite realistic, as scientists often want to compress/decompress sets of genomes prior/after analysis.
Decompression Statistics
For all tests  independent of the index block size  we obtained an average decompression time of 22 seconds. Further parallelization did not improve these values, since the decompression algorithm is I/Obound. During the compression around 3.1 GB for the decompressed genome are written on the hard disk. Given that we measured the write speed of the hard disk with 150 MB/s, it seems hard to further optimize the decompression time. We have also tried to write only compressed FASTA files on the disk. Then, however, the decompression takes several minutes, because the (selfreferential) compression is CPUbound. The decompression times for competitors are similar, i.e. RLZopt achieves 130 MB/s and GDC up to 150 MB/s. We are convinced that for most decompressor implementations the actual limitation is the hard disk write speed.
Conclusions
We have proposed an adaptive referential compression schemes for genomes. The compression speed can be controlled by varying the amount of main memory for the compressor. Our variant with lowest main memory footprint (12 MB) achieves similar compression rates and compression speeds, while using almost two orders of magnitude less main memory. Our greedy variant using 9 GB of main memory is 23 times as fast as the best known variant. Compression speed can be further improved by massive parallelization on different machines.
Further work should be done on improving the compression ratio. For instance, it would be possible to find an efficient encoding for inverse complement matches or approximate matches.
Investigations on interspecies referential compression is challenging. So far, referential compression only works well, if input and reference belong to the same species. The development of a multispecies reference sequence would allow for multi purpose genome compression algorithms. Our initial experiments with human genome compression with respect to a mouse genome indicate that the matches are usually very short and the advantages of referential compression are mitigated.
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All authors have contributed equally to the main text. Sebastian Wandelt has implemented the presented algorithms on C++. All authors read and approved the final manuscript.
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Keywords
 Sequence compression
 Referential compression
 Heuristics
 Scalability