Combinatorial Pattern Matching

Transcription

1 Combinatorial Pattern Matching

2 Outline Exact Pattern Matching Keyword Trees Suffix Trees Approximate String Matching

3 Local alignment is to slow Quadratic local alignment is too slow while looking for similarities between long strings (e.g. the entire GenBank database) = ), ( ), ( ), ( 0 max 1 1, 1, 1,, i i i i i i w v s w s v s s δ δ δ

4 Local alignment is to slow Quadratic local alignment is too slow while looking for similarities between long strings (e.g. the entire GenBank database) = ), ( ), ( ), ( 0 max 1 1, 1, 1,, i i i i i i w v s w s v s s δ δ δ

5 Local alignment is to slow Quadratic local alignment is too slow while looking for similarities between long strings (e.g. the entire GenBank database) Guaranteed to find the optimal local alignment Sets the standard for sensitivity = ), ( ), ( ), ( 0 max 1 1, 1, 1,, i i i i i i w v s w s v s s δ δ δ

6 Local alignment is to slow Quadratic local alignment is too slow while looking for similarities between long strings (e.g. the entire GenBank database) Basic Local Alignment Search Tool Altschul, S., Gish, W., Miller, W., Myers, E. & Lipman, D.J. Journal of Mol. Biol., 1990 Search sequence databases for local alignments to a query s i, = 0 si max si si 1,, 1 1, 1 + δ ( v, ) + δ (, w i ) + δ ( v, w i )

7 Pattern Matching What if we want to find all sequences in a database that contain a given pattern? This leads us to a different problem, the Pattern Matching Problem

8 Pattern Matching Problem Goal: Find all occurrences of a pattern in a text Input: Pattern p = p 1 p n and text t = t 1 t m Output: All positions 1< i < (m n + 1) such that the n-letter substring of t starting at i matches p Motivation: Searching database for a known pattern

9 Exact Pattern Matching: A Brute-Force Algorithm PatternMatching(p,t) 1 n ß length of pattern p 2 m ß length of text t 3 for i ß 1 to (m n + 1) 4 if t i t i+n-1 = p 5 output i

10 Exact Pattern Matching: An Example PatternMatching algorithm for: Pattern GCAT Text CGCATC GCAT CGCATC GCAT CGCATC GCAT CGCATC GCAT CGCATC GCAT CGCATC

11 Exact Pattern Matching: Running Time PatternMatching runtime: O(nm) Probability-wise, it s more like O(m) Rarely will there be close to n comparisons in line 4 Better solution: suffix trees Can solve problem in O(m) time Conceptually related to keyword trees

12 Keyword Trees: Example Keyword tree: Apple

13 Keyword Trees: Example (cont d) Keyword tree: Apple Apropos

14 Keyword Trees: Example (cont d) Keyword tree: Apple Apropos Banana

15 Keyword Trees: Example (cont d) Keyword tree: Apple Apropos Banana Bandana

16 Keyword Trees: Example (cont d) Keyword tree: Apple Apropos Banana Bandana Orange

17 Keyword Trees: Properties Stores a set of keywords in a rooted labeled tree Each edge labeled with a letter from an alphabet Any two edges coming out of the same vertex have distinct labels Every keyword stored can be spelled on a path from root to some leaf

18 Keyword Trees: Threading (cont d) Thread appeal appeal

19 Keyword Trees: Threading (cont d) Thread appeal appeal

20 Keyword Trees: Threading (cont d) Thread appeal appeal

21 Keyword Trees: Threading (cont d) Thread appeal appeal

22 Keyword Trees: Threading (cont d) Thread apple apple

23 Keyword Trees: Threading (cont d) Thread apple apple

24 Keyword Trees: Threading (cont d) Thread apple apple

25 Keyword Trees: Threading (cont d) Thread apple apple

26 Keyword Trees: Threading (cont d) Thread apple apple

27 Multiple Pattern Matching Problem Goal: Given a set of patterns and a text, find all occurrences of any of patterns in text Input: k patterns p 1,,p k, and text t = t 1 t m Output: Positions 1 < i < m where substring of t starting at i matches p for 1 < < k Motivation: Searching database for known multiple patterns

28 Multiple Pattern Matching: Straightforward Approach Can solve as k Pattern Matching Problems Runtime: O(kmn) using the PatternMatching algorithm k times m - length of the text n - average length of the pattern

29 Suffix Trees=Collapsed Keyword Trees Similar to keyword trees, except edges that form paths are collapsed Each edge is labeled with a substring of a text All internal edges have at least two outgoing edges Leaves labeled by the index of the pattern.

30 Suffix Tree of a Text Suffix trees of a text is constructed for all its suffixes ATCATG TCATG CATG ATG TG G Keyword Tree Suffix Tree

31 Suffix Tree of a Text Suffix trees of a text is constructed for all its suffixes ATCATG TCATG CATG ATG TG G Keyword Tree Suffix Tree How much time does it take?

32 Suffix Tree of a Text Suffix trees of a text is constructed for all its suffixes ATCATG TCATG CATG ATG TG G quadratic Keyword Tree Suffix Tree Time is linear in the total size of all suffixes, i.e., it is quadratic in the length of the text

33 Suffix Trees: Advantages Suffix trees of a text is constructed for all its suffixes Suffix trees build faster than keyword trees ATCATG TCATG CATG ATG TG G quadratic Keyword Tree linear (Weiner suffix tree algorithm) Suffix Tree

34 Use of Suffix Trees Suffix trees hold all suffixes of a text i.e., ATCGC: ATCGC, TCGC, CGC, GC, C Builds in O(m) time for text of length m To find any pattern of length n in a text: Build suffix tree for text Thread the pattern through the suffix tree Can find pattern in text in O(n) time! O(n + m) time for Pattern Matching Problem Build suffix tree and lookup pattern

35 Pattern Matching with Suffix Trees SuffixTreePatternMatching(p,t) 1 Build suffix tree for text t 2 Thread pattern p through suffix tree 3 if threading is complete 4 output positions of all p-matching leaves in the tree 5 else 6 output Pattern does not appear in text

36 Suffix Trees: Example

37 Multiple Pattern Matching: Summary Keyword and suffix trees are used to find patterns in a text Keyword trees: Build keyword tree of patterns, and thread text through it Suffix trees: Build suffix tree of text, and thread patterns through it

38 Approximate vs. Exact Pattern Matching So far all we ve seen exact pattern matching algorithms Usually, because of mutations, it makes much more biological sense to find approximate pattern matches Biologists often use fast heuristic approaches (rather than local alignment) to find approximate matches

39 Approximate Pattern Matching Problem Goal: Find all approximate occurrences of a pattern in a text Input: A pattern p = p 1 p n, text t = t 1 t m, and k, the maximum number of mismatches Output: All positions 1 < i < (m n + 1) such that t i t i+n-1 and p 1 p n have at most k mismatches (i.e., Hamming distance between t i t i+n-1 and p < k)

40 Approximate Pattern Matching: A Brute-Force Algorithm ApproximatePatternMatching(p, t, k) 1 n ß length of pattern p 2 m ß length of text t 3 for i ß 1 to m n dist ß 0 5 for ß 1 to n 6 if t i+-1!= p 7 dist ß dist if dist < k 9 output i

41 Approximate Pattern Matching: Running Time That algorithm runs in O(nm). Landau-Vishkin algorithm: O(kn) We can generalize the Approximate Pattern Matching Problem into a Query Matching Problem : We want to match substrings in a query to substrings in a text with at most k mismatches Motivation: we want to see similarities to some gene, but we may not know which parts of the gene to look for