BUNDLED SUFFIX TREES

Transcription

1 Motivation BUNDLED SUFFIX TREES Luca Bortolussi 1 Francesco Fabris 2 Alberto Policriti 1 1 Department of Mathematics and Computer Science University of Udine 2 Department of Mathematics and Computer Science University of Trieste BITS 2005, Milano, 18 th 19 th march 2005

2 Outline Motivation 1 Motivation Suffix Trees 2 Non-Transitive Relations Definition Size and Construction 3 4

3 Introduction Motivation Suffix Trees Since the discovery of DNA, biology gave birth to many thorough string problems. Important challenge: find repeated patterns in DNA that are biologically significant. Feature: patterns are repeated with errors. (Approximate pattern discovery is difficult) Other feature (more difficult): formalization of biologically significant.

4 Suffix Trees Motivation Suffix Trees bcabbabc A Suffix Tree is a data structure which exploits the internal structure of a string. Efficient for: Exact String Matching Problem Longest Exact Common Substring Problem Identifying Exactly Repeated Patterns

5 Suffix Trees Motivation Suffix Trees bcabbabc A Suffix Tree is a data structure which exploits the internal structure of a string. They are linear in size (w.r.t text length), and can be built in linear time. E. McCreight. A space-economical suffix tree construction algorithm, Journal of the ACM, 23(2), , E. Ukkonen. On-line construction of suffix-trees. Algorithmica, 14: , 1995.

6 Suffix Trees Motivation Suffix Trees bcabbabc Suffix Trees are not natural to deal with approximate string matching problems (positive Hamming or Edit distance) Landau G.M., Vishkin U., Efficient String Matching with k Mismatches, Theoretical Computer Science, 43, , Gusfield D., Algorithms on strings, trees and sequences, Cambridge University Press, 1997.

7 Suffix Trees Motivation Suffix Trees bcabbabc Suffix Trees are not natural to deal with approximate string matching problems (positive Hamming or Edit distance) The Longest Common Approximate Substring Problem or the extraction of approximate repeated patterns can t be solved as in the exact case.

8 Motivation Extending Suffix Trees Non-Transitive Relations Definition Size and Construction THE PROJECT Exploring the possibility of using different tree-based structures to tackle approximate string matching problems. SO FAR We developed, an extension of Suffix Trees such that: they incorporate information about errors ; they can be used for the Longest Common Approximate Substring Problem and for extracting approximate repeated patterns like Suffix Trees.

9 Motivation Extending Suffix Trees Non-Transitive Relations Definition Size and Construction THE PROJECT Exploring the possibility of using different tree-based structures to tackle approximate string matching problems. SO FAR We developed, an extension of Suffix Trees such that: they incorporate information about errors ; they can be used for the Longest Common Approximate Substring Problem and for extracting approximate repeated patterns like Suffix Trees.

10 Motivation Non-Transitive Relation Non-Transitive Relations Definition Size and Construction Character matching is a relation among letters (in fact, it is the equality relation) Approximate matching can also be modeled as a non-transitive relation among letters (bigger than equality!): two strings match if all their letters are in relation.

11 Motivation Non-Transitive Relation Non-Transitive Relations Definition Size and Construction Character matching is a relation among letters (in fact, it is the equality relation) Approximate matching can also be modeled as a non-transitive relation among letters (bigger than equality!): two strings match if all their letters are in relation.

12 Motivation Non-Transitive Relations Definition Size and Construction Non-Transitive Relation: An Example Modelling a relation based on Hamming Distance Start from a basic alphabet (e.g. binary: A = {0, 1}) Construct an alphabet composed of macrocharacters (e.g. A = {00, 01, 10, 11}) Impose that two letters x, y A are in relation if and only if d H (x, y) 1 (relation is non transitive) The Relation Graph

13 Motivation Non-Transitive Relations Definition Size and Construction Non-Transitive Relation: An Example Modelling a relation based on Hamming Distance Start from a basic alphabet (e.g. binary: A = {0, 1}) Construct an alphabet composed of macrocharacters (e.g. A = {00, 01, 10, 11}) Impose that two letters x, y A are in relation if and only if d H (x, y) 1 (relation is non transitive) The Relation Graph

14 Motivation Non-Transitive Relations Definition Size and Construction Bundled Suffix Tree: An Example bcabbabc; a b c We start from the suffix tree for the string bcabbabc. The alphabet is {a, b, c}, and the relation is a b c.

15 Motivation Non-Transitive Relations Definition Size and Construction Bundled Suffix Tree: An Example bcabbabc; a b c Let s compare suffix 3 (abbabc) and suffix 1 (bcabbabc) According to our relation, the maximal prefix of suffix 3, which is in relation with a prefix of suffix one, is abbab.

16 Motivation Non-Transitive Relations Definition Size and Construction Bundled Suffix Tree: An Example bcabbabc; a b c Therefore, after bcabb, we put in the tree a red node with label 3. Due to symmetry, there is also a red node with label 1 after abbab.

17 Motivation Non-Transitive Relations Definition Size and Construction Bundled Suffix Tree: An Example bcabbabc; a b c Therefore, after bcabb, we put in the tree a red node with label 3. Due to symmetry, there is also a red node with label 1 after abbab.

18 Motivation Non-Transitive Relations Definition Size and Construction Bundled Suffix Tree: An Example bcabbabc; a b c If we do this process for every couple of suffixes, we will build a Bundled Suffix Tree! Note that this data structure is in the middle between a suffix tree and a suffix trie.

19 Motivation Non-Transitive Relations Definition Size and Construction Bundled Suffix Tree: An Example bcabbabc; a b c If we do this process for every couple of suffixes, we will build a Bundled Suffix Tree! Note that this data structure is in the middle between a suffix tree and a suffix trie.

20 Motivation Non-Transitive Relations Definition Size and Construction Bundled Suffix Tree: An Example bcabbabc; a b c This tree can be use to solve the Longest Common Approximate Substring Problem with respect to a given relation. We just have to find the lowest red node! Similarly, we can also extract information about approximate repeated patterns.

21 How Big? Motivation Non-Transitive Relations Definition Size and Construction The number of red nodes inserted depends on: the relation the structure of the text. In the worst case, the number of red nodes is quadratic in the length of the text S. Example On average, the number of red nodes is limited by m 1+δ, δ = log 1/p + C. ( m is the length of the text, p + is the highest frequency of the most common letter in S and C depends on the relation) 1 + δ is slightly greater than one! Example

22 How Big? Motivation Non-Transitive Relations Definition Size and Construction The number of red nodes inserted depends on: the relation the structure of the text. In the worst case, the number of red nodes is quadratic in the length of the text S. Example On average, the number of red nodes is limited by m 1+δ, δ = log 1/p + C. ( m is the length of the text, p + is the highest frequency of the most common letter in S and C depends on the relation) 1 + δ is slightly greater than one! Example

23 How Fast? Motivation Non-Transitive Relations Definition Size and Construction Naive Algorithm The naive algorithm for building a BuST simply tries to match every suffix of the text along every branch of the suffix tree, until a mismatch is found. It can be quadratic in the worst case. Anyway, an analysis based on the average shape of a suffix tree, shows that its average complexity is bounded by m 1+δ (δ just slightly greater that δ). W. Szpankowski. A Generalized Suffix Tree and its (Un)expected Asymptotic Behaviors. SIAM J. Comput. 22(6): (1993) P. Jacquet, B. McVey, W. Szpankowski. Compact Suffix Trees Resemble PATRICIA Tries: Limiting Distribution of Depth, Journal of the Iranian Statistical Society, 3, , 2004.

24 Faster Motivation Non-Transitive Relations Definition Size and Construction Efficient Algorithm We found an McCreight-like algorithm that is linear in the size of the output. Intuitions It processes the suffixes backwards. It is based on the concept of inverse suffix links. Show Details It identifies the red nodes for suffix i by processing the red nodes for suffix i + 1. Show Details

25 The generall. concept Bortolussi, F. Fabris, of non-transitive A. Policriti BUNDLED relation SUFFIX TREES seems very Hunting TFBS Motivation We are using BuST to identify TFBS candidates in DNA sequences. The algorithm first constructs the BuST for the set of sequences under analysis, and then extracts and combines the information contained in it. The relation used is defined by an Hamming distance criterion. The algorithm is quite fast: for instance, we are able to solve the benchmark proposed by Pevzner et al. in few seconds. Show Benchmark s Details P.A. Pevzner and S.H. Sze. Combinatorial approaches to finding subtle signals in DNA sequences. Proc Int Conf Intell Syst Mol Biol. 2000;8:

26 Hunting TFBS Motivation We are using BuST to identify TFBS candidates in DNA sequences. The general concept of non-transitive relation seems very fruitful: it can be used to encode Hamming distance, but also to tackle edit distance or to encode other biologically-driven relations. G. Pavesi, G. Mauri and G. Pesole. In silico representation and discovery of transcription factor binding sites, Briefings in Bioinformatics. 5(3):1 20, 2004.

27 Conclusions Motivation We have introduced, a new data structure extending suffix trees. It can be used to extract approximate information from a string, and it is manipulated similarly to suffix trees. The structure is based on a very general concept of non-transitive relation among (macro)characters. Its size is slightly more than linear on average, and there s a fast (McCreight-like) algorithm to build it. It can be used to discover approximate patterns in a text. For instance, it can be used to identify candidates for TFBS.

28 Dimension of BuST Efficient Algorithm The Benchmark Tests We have implemented the naive algorithm for the construction of BuST. We have tested it with relations induced by hamming distance, defined over DNA-macrocharacters. With macrocharacters of size 4, such that two of them are in relation iff their Hamming distance is 1, the algorithm is quite fast, and can process texts of 100 Kb in few seconds. The number of red nodes grows with an exponent smaller than the predicted one. Show Details

29 Quadratic BuST Dimension of BuST Efficient Algorithm The Benchmark Quadratic BuST Delta Tests Let s consider the text a }. {{.. a } b }. {{.. b } c }. {{.. c }, m m 2m over {a, b, c, d}, with a b d c The number of nodes surrounded by the red box is quadratic in m! Return

30 Quadratic BuST Dimension of BuST Efficient Algorithm The Benchmark Quadratic BuST Delta Tests Let s consider the text a }. {{.. a } b }. {{.. b } c }. {{.. c }, m m 2m over {a, b, c, d}, with a b d c The number of nodes surrounded by the red box is quadratic in m! Return

31 Quadratic BuST Dimension of BuST Efficient Algorithm The Benchmark Quadratic BuST Delta Tests Let s consider the text a }. {{.. a } b }. {{.. b } c }. {{.. c }, m m 2m over {a, b, c, d}, with a b d c The number of nodes surrounded by the red box is quadratic in m! Return

32 The exponent δ Dimension of BuST Efficient Algorithm The Benchmark Quadratic BuST Delta Tests Return

33 The exponent δ Dimension of BuST Efficient Algorithm The Benchmark Quadratic BuST Delta Tests Return

34 Test Dimension of BuST Efficient Algorithm The Benchmark Quadratic BuST Delta Tests Number of macrocharacters of length 4 over DNA alphabet. Test strings are generated according to a uniform p.d. Return

35 Dimension of BuST Efficient Algorithm The Benchmark Inverse Suffix Links Ideas of the Algorithm Inverse Suffix Links A crucial role in the fast construction of suffix trees is played by suffix links. Suffix links are pointers from nodes with path label xα to nodes with path label α. Whenever there is a node with path label xα, there s also a node with path label α. Return

36 Dimension of BuST Efficient Algorithm The Benchmark Inverse Suffix Links Ideas of the Algorithm Inverse Suffix Links Inverse suffix links are pointers from nodes with path label α to positions in the tree labeled xα, for each x in the alphabet such that xα is a substring of S. They can point in the middle of an arc. If a ISL takes from α to xα, it is labeled with x. Return

37 Dimension of BuST Efficient Algorithm The Benchmark Inverse Suffix Links Ideas of the Algorithm The Algorithm Inverse suffix links con be used to identify the red nodes for suffix S[i]from the red nodes for suffix S[i + 1]. Suppose we know the location of a red node for suffix S[i + 1], and that it is just under a black node with path label α. Return

38 Dimension of BuST Efficient Algorithm The Benchmark Inverse Suffix Links Ideas of the Algorithm The Algorithm From this node, we can cross all inverse suffix links such that S(i) is in relation with the character labeling the ISL. With a skip and count trick, we can identify the positions of red nodes for S[i]. Return

39 Dimension of BuST Efficient Algorithm The Benchmark Inverse Suffix Links Ideas of the Algorithm The Algorithm From this node, we can cross all inverse suffix links such that S(i) is in relation with the character labeling the ISL. With a skip and count trick, we can identify the positions of red nodes for S[i]. Return

40 A First Application Dimension of BuST Efficient Algorithm The Benchmark The Benchmark There is a set of 20 strings of length 1000, generated according to a uniform distribution over the DNA alphabet. There is a pattern p of length 16, such that 20 of its occurrences are implanted in the strings, with 4 mutations occurring in random positions. The problem is to identify p (the signal), given the strings. P.A. Pevzner and S.H. Sze. Combinatorial approaches to finding subtle signals in DNA sequences. Proc Int Conf Intell Syst Mol Biol. 2000;8: Return

41 A First Application Dimension of BuST Efficient Algorithm The Benchmark A solution with BuST We used macrocharacters of length 4 (2 of them are in relation if their Hamming distance is 1). We built the generalized BuST for the strings (converted in macrocharacters in every possible way). For every substring of length 16 of the 20 strings, we looked at the set of substrings in relation with it, and we combined this information to find p. It s a naive use of BuST, but it works! Return