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Combinatorial Pattern Matching. CS 466 Saurabh Sinha. Genomic Repeats. Example of repeats: AT GGTC TA GGTC CTAGT GGTC Motivation to find them: Genomic rearrangements are often associated with repeats Trace evolutionary secrets Many tumors are characterized by an explosion of repeats.

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genomic repeats
Genomic Repeats
  • Example of repeats:
    • ATGGTCTAGGTCCTAGTGGTC
  • Motivation to find them:
    • Genomic rearrangements are often associated with repeats
    • Trace evolutionary secrets
    • Many tumors are characterized by an explosion of repeats
genomic repeats1
Genomic Repeats
  • The problem is often more difficult:
    • ATGGTCTAGGACCTAGTGTTC
  • Motivation to find them:
    • Genomic rearrangements are often associated with repeats
    • Trace evolutionary secrets
    • Many tumors are characterized by an explosion of repeats
l mer repeats
l -mer Repeats
  • Long repeats are difficult to find
  • Short repeats are easy to find (e.g., hashing)
  • Simple approach to finding long repeats:
    • Find exact repeats of short l-mers (l is usually 10 to 13)
    • Use l -mer repeats to potentially extend into longer, maximal repeats
l mer repeats cont d
l -mer Repeats (cont’d)
  • There are typically many locations where an l -mer is repeated:

GCTTACAGATTCAGTCTTACAGATGGT

  • The 4-mer TTAC starts at locations 3 and 17
extending l mer repeats
Extending l -mer Repeats

GCTTACAGATTCAGTCTTACAGATGGT

  • Extend these 4-mer matches:

GCTTACAGATTCAGTCTTACAGATGGT

  • Maximal repeat: CTTACAGAT
maximal repeats
Maximal Repeats
  • To find maximal repeats in this way, we need ALL start locations of all l -mers in the genome
  • Hashing lets us find repeats quickly in this manner
hashing maximal repeats
Hashing: Maximal Repeats
  • To find repeats in a genome:
    • For all l -mers in the genome, note the start position and the sequence
    • Generate a hash table index for each unique l -mer sequence
    • In each index of the hash table, store all genome start locations of the l -mer which generated that index
    • Extend l -mer repeats to maximal repeats
pattern matching
Pattern Matching
  • What if, instead of finding repeats in a genome, we want to find all sequences in a database that contain a given pattern?
  • This leads us to a different problem, the Pattern MatchingProblem
pattern matching problem
Pattern Matching Problem
  • Goal: Find all occurrences of a pattern in a text
  • Input: Pattern p = p1…pn and text t = t1…tm
  • 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
exact pattern matching running time
Exact Pattern Matching: Running Time
  • Naïve runtime: O(nm)
  • On average, it’s more like O(m)
    • Why?
  • Can solve problem in O(m) time ?
    • Yes, we’ll see how (later)
generalization of problem multiple pattern matching problem
Generalization of problem: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 p1,…,pk, and text t = t1…tm
  • Output: Positions 1 <i <m where substring of t starting at i matches pj for 1 <j<k
  • Motivation: Searching database for known multiple patterns
  • Solution: k “pattern matching problems”: O(kmn)
  • Solution: Using “Keyword trees” => O(kn+m) where n is maximum length of pi
keyword trees example
Keyword Trees: Example
  • Keyword tree:
    • Apple
keyword trees example cont d
Keyword Trees: Example (cont’d)
  • Keyword tree:
    • Apple
    • Apropos
keyword trees example cont d1
Keyword Trees: Example (cont’d)
  • Keyword tree:
    • Apple
    • Apropos
    • Banana
keyword trees example cont d2
Keyword Trees: Example (cont’d)
  • Keyword tree:
    • Apple
    • Apropos
    • Banana
    • Bandana
keyword trees example cont d3
Keyword Trees: Example (cont’d)
  • Keyword tree:
    • Apple
    • Apropos
    • Banana
    • Bandana
    • Orange
keyword trees properties
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
multiple pattern matching keyword tree approach
Multiple Pattern Matching: Keyword Tree Approach
  • Build keyword tree in O(kn) time; kn is total length of all patterns
  • Start “threading” at each position in text; at most n steps tell us if there is a match here to any pi
  • O(kn + nm)
  • Aho-Corasick algorithm: O(kn + m)
fail edges in keyword tree
“Fail” edges in keyword tree

Dashed edge out of internal node if matching edge not found

fail edges in keyword tree1
“Fail” edges in keyword tree
  • If currently at node q representing word L(q), find the longest proper suffix of L(q) that is a prefix of some pattern, and go to the node representing that prefix
  • Example: node q = 5 L(q) = she; longest proper suffix that is a prefix of some pattern: “he”. Dashed edge to node q’=2
automaton
Automaton
  • Transition among the different nodes by following edges depending on next character seen (“c”)
  • If outgoing edge with label “c”, follow it
  • If no such edge, and are at root, stay
  • If no such edge, and at non-root, follow dashes edge (“fail” transition); DO NOT CONSUME THE CHARACTER (“c”)

Example: search text “ushers” with the automaton

aho corasick algorithm1
Aho-Corasick algorithm
  • O(kn) to build the automaton
  • O(m) to search a text of length m
  • Key insight:
    • For every character “consumed”, we move at most one level deeper (away from root) in the tree. Therefore total number of such “away from root” moves is <= m
    • Each fail transition moves us at least one level closer to root. Therefore total number of such “towards root” moves is <= m (you cant climb up more than you climbed down)
suffix tree
Suffix tree
  • Build a tree from the text
  • Used if the text is expected to be the same during several pattern queries
  • Tree building is O(m) where m is the size of the text. This is preprocessing.
  • Given any pattern of length n, we can answer if it occurs in text in O(n) time
  • Suffix tree = “modified” keyword tree of all suffixes of text
suffix tree collapsed keyword trees
Suffix Tree=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.
suffix tree of a text
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

quadratic

Time is linear in the total size of all suffixes, i.e., it is quadratic in the length of the text

suffix trees advantages
Suffix Trees: Advantages
  • Suffix trees build faster than keyword trees

ATCATG TCATG CATG ATG TG G

Keyword Tree

Suffix Tree

quadratic

linear (Weiner suffix tree algorithm)

Time to find k patterns of length n: O(m + kn)

multiple pattern matching summary
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
approximate vs exact pattern matching
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 usefast heuristic approaches (rather than local alignment) to find approximate matches
heuristic similarity searches
Heuristic Similarity Searches
  • Genomes are huge: Smith-Waterman quadratic alignment algorithms are too slow
  • Alignment of two sequences usually has short identical or highly similar fragments
  • Many heuristic methods (i.e., FASTA) are based on the same idea of filtration
    • Find short exact matches, and use them as seeds for potential match extension
query matching problem
Query Matching Problem
  • Goal: Find all substrings of the query that approximately match the text
  • Input: Query q = q1…qw,

text t = t1…tm,

n (length of matching substrings),

k (maximum number of mismatches)

  • Output: All pairs of positions (i, j) such that the

n-letter substring of q starting at iapproximately matches the

n-letter substring of t starting at j,

with at most k mismatches

query matching main idea
Query Matching: Main Idea
  • Approximately matching strings share some perfectly matching substrings.
  • Instead of searching for approximately matching strings (difficult) search for perfectly matching substrings (easy).
filtration in query matching
Filtration in Query Matching
  • We want all n-matches between a query and a text with up to k mismatches
  • “Filter” out positions we know do not match between text and query
  • Potential match detection: find all matches of l -tuples in query and text for some small l
  • Potential match verification: Verify each potential match by extending it to the left and right, until (k + 1) mismatches are found
filtration match detection
Filtration: Match Detection
  • If x1…xn and y1…yn match with at most k mismatches, they must share an l -tuple that is perfectly matched, with l = n/(k + 1)
  • Break string of length n into k+1 parts, each each of length n/(k + 1)
    • k mismatches can affect at most k of these k+1 parts
    • At least one of these k+1 parts is perfectly matched
filtration match verification
Filtration: Match Verification
  • For each l -match we find, try to extend the match further to see if it is substantial

Extend perfect match of lengthl

until we find an approximate match of length n with k mismatches

query

text

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