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In [[computer science]], a '''compressed suffix array'''<ref name="GV00">R. Grossi and J. S. Vitter, Compressed Suffix Arrays and Suffix Trees, with Applications to Text Indexing and String Matching, ''SIAM Journal on Computing,'' 35(2), 2005, 378-407.  An earlier version appeared in ''Proceedings of the 32nd ACM Symposium on Theory of Computing, May 2000, 397-406.</ref><ref name="opportunistic_2000">Paolo Ferragina and Giovanni Manzini (2000). "Opportunistic Data Structures with Applications". Proceedings of the 41st Annual Symposium on Foundations of Computer Science. p.390.</ref><ref name="GGV03">R. Grossi, A. Gupta, and J. S. Vitter, High-Order Entropy-Compressed Text Indexes, ''Proceedings of the 14th Annual SIAM/ACM Symposium on Discrete Algorithms,'' January 2003, 841-850.</ref> is a [[compressed data structure]] for [[pattern matching]].  Compressed suffix arrays are a general class of [[data structure]] that improve on the [[suffix array]].<ref name="GV00"/><ref name="opportunistic_2000"/> These data structures enable quick search for a arbitrary [[String (computer science)|string]] with a comparatively small index.

Given a text ''T'' of ''n'' characters from an alphabet Σ, a compressed suffix array supports searching for arbitrary patterns in ''T''.  For an input pattern ''P'' of ''m'' characters, the search time is typically O(''m'') or O(''m'' + ''log(n)''). The space used is typically <math>O(n H_k(T)) + o(n)</math>, where <math>H_k(T)</math> is the [[Entropy (information theory)|k-th order empirical entropy]] of the text ''T''. The time and space to construct a compressed suffix array are normally ''O(n)''.
 
The original instantiation of the compressed suffix array<ref name="GV00"/> solved a long-standing open problem by showing that fast pattern matching was possible using only a linear-space data structure, namely, one proportional to the size of the text ''T'', which takes <math>O(n \, {\log |\Sigma|})</math> bits.  The conventional suffix array and suffix tree use <math>\Omega(n \, {\log n})</math> bits, which is substantially larger.  The basis for the data structure is a recursive decomposition using the "neighbor function," which allows a suffix array to be represented by one of half its length.  The construction is repeated multiple times until the resulting suffix array uses a linear number of bits.  Following work showed that the actual storage space was related to the zeroth-order entropy and that the index supports self-indexing.<ref>K. Sadakane, Compressed Text Databases with Efficient Query Algorithms Based on the Compressed Suffix Arrays, ''Proceedings of the International Symposium on Algorithms and Computation'', Lecture Notes in Computer Science, vol. 1969, Springer, December 2000, 410-421.</ref>  The space bound was further improved achieving the ultimate goal of higher-order entropy; the compression is obtained by partitioning the neighbor function by high-order contexts, and compressing each partition with a [[Wavelet Tree|wavelet tree]].<ref name="GGV03"/>  The space usage is extremely competitive in practice with other state-of-the-art compressors,<ref>L. Foschini, R. Grossi, A. Gupta, and J. S. Vitter, Indexing Equals Compression: Experiments on Suffix Arrays and Trees'', ACM Transactions on Algorithms'', 2(4), 2006, 611-639.</ref> and it also supports fast pattern matching.
 
The memory accesses made by compressed suffix arrays and other compressed data structures for pattern matching are typically not localized, and thus these data structures have been notoriously hard to design efficiently for use in [[external memory]].  Recent progress using geometric duality takes advantage of the block access provided by disks to speed up the I/O time significantly<ref>W.-K. Hon, R. Shah, S. V. Thankachan, and J. S. Vitter, On Entropy-Compressed Text Indexing in External Memory, ''Proceedings of the Conference on String Processing and Information Retrieval'', August 2009.</ref> In addition, potentially practical search performance for a compressed suffix array in external-memory has been demonstrated.<ref>M. P. Ferguson, ''FEMTO: fast search of large sequence collections'', ''Proceedings of the 23rd Annual Conference on Combinatorial Pattern Matching'', July 2012</ref>

==Open Source Implementations==
There are several open source implementations of compressed suffix arrays available (see [[#External Links|External Links]] below). Bowtie and Bowtie2 are open-source compressed suffix array implementations of [[Sequence alignment|read alignment]] for use in [[bioinformatics]]. The Succinct Data Structure Library (SDSL) is a library containing a variety of compressed data structures including compressed suffix arrays. FEMTO is an implementation of compressed suffix arrays for external memory. In addition, a variety of implementations, including the original [[FM-index]] implementations, are available from the Pizza & Chili Website.

==See also==
[[FM-index]]

[[Suffix Array]]

==References==
<references/>

==External links==
Implementations:
* [http://bowtie-bio.sourceforge.net/index.shtml Bowtie] and [http://bowtie-bio.sourceforge.net/bowtie2/index.shtml Bowtie2]
* [https://github.com/simongog/sdsl-lite Succinct Data Structure Library (SDSL)]
* [https://github.com/femto-dev/femto FEMTO]
* [http://pizzachili.di.unipi.it/indexes.html Pizza&Chili website].

[[Category:String data structures]]
[[Category:Database index techniques]]
[[Category:Substring indices]]