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CSCI3070-Minimum Edit DistanceSolved
Minimum Edit Distance
Defini’on of Minimum Edit Distance How similar are two strings?
• Spell correc’on • Computa’onal Biology
• The user typed “graffe” • Align two sequences of nucleo’des
Which is closest? AGGCTATCACCTGACCTCCAGGCCGATGCCC
• graf TAGCTATCACGACCGCGGTCGATTTGCCCGAC
• graB • Resul’ng alignment:
• grail
• giraffe -AGGCTATCACCTGACCTCCAGGCCGA--TGCCC---
TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC
Also for Machine Transla’on, Informa’on Extrac’on, Speech Recogni’on
Edit Distance
• The minimum edit distance between two strings
• Is the minimum number of edi’ng opera’ons
• Inser’on
• Dele’on
• Subs’tu’on
• Needed to transform one into the other
Minimum Edit Distance
• Two strings and their alignment:
Minimum Edit Distance
• If each opera’on has cost of 1
• Distance between these is 5
• If subs’tu’ons cost 2 (Levenshtein)
• Distance between them is 8
Alignment in Computa9onal Biology
• Given a sequence of bases
AGGCTATCACCTGACCTCCAGGCCGATGCCC TAGCTATCACGACCGCGGTCGATTTGCCCGAC
• An alignment:
-AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC
• Given two sequences, align each leXer to a leXer or gap
Other uses of Edit Distance in NLP
• Evalua’ng Machine Transla’on and speech recogni’on
R Spokesman confirms senior government adviser was shot!
H Spokesman said the senior adviser was shot dead!
S I D I!
• Named En’ty Extrac’on and En’ty Coreference
• IBM Inc. announced today
• IBM profits
• Stanford President John Hennessy announced yesterday
• for Stanford University President John Hennessy
How to find the Min Edit Distance?
• Searching for a path (sequence of edits) from the start string to the final string:
• Ini9al state: the word we’re transforming
• Operators: insert, delete, subs’tute
• Goal state: the word we’re trying to get to
• Path cost: what we want to minimize: the number of edits
Minimum Edit as Search
• But the space of all edit sequences is huge!
• We can’t afford to navigate naïvely
• Lots of dis’nct paths wind up at the same state.
• We don’t have to keep track of all of them
• Just the shortest path to each of those revisted states.
Defining Min Edit Distance
• For two strings
• X of length n
• Y of length m
• We define D(i,j)
• the edit distance between X[1..i] and Y[1..j]
• i.e., the first i characters of X and the first j characters of Y
• The edit distance between X and Y is thus D(n,m)
Minimum Edit Distance
Defini’on of Minimum Edit Distance
Minimum Edit Distance
Compu’ng Minimum
Edit Distance Dynamic Programming for
Minimum Edit Distance
• Dynamic programming: A tabular computa’on of D(n,m)
• Solving problems by combining solu’ons to subproblems.
• BoXom-‐up
• We compute D(i,j) for small i,j
• And compute larger D(i,j) based on previously computed smaller values
• i.e., compute D(i,j) for all i (0 < i < n) and j (0 < j < m)
Defining Min Edit Distance (Levenshtein)
Ini’aliza’on
D(i,0) = i!
D(0,j) = j
Recurrence Rela’on:
For each i = 1…M!
! For each j = 1…N
D(i-1,j) + 1! D(i,j)= min D(i,j-1) + 1!
D(i-1,j-1) + 2; if X(i) ≠ Y(j) ! 0; if X(i) = Y(j)!
Termina’on:
D(N,M) is distance !
N 9
O 8
I 7
T 6
N 5
E 4
T 3
N 2
I 1
# 0 1 2 3 4 5 6 7 8 9
# E X E C U T I O N
The Edit Distance Table
N 9
O 8
I 7
T 6
N 5
E 4
T 3
N 2
I 1
# 0 1 2 3 4 5 6 7 8 9
# E X E C U T I O N
Edit Distance
The Edit Distance Table
N 9 8 9 10 11 12 11 10 9 8
O 8 7 8 9 10 11 10 9 8 9
I 7 6 7 8 9 10 9 8 9 10
T 6 5 6 7 8 9 8 9 10 11
N 5 4 5 6 7 8 9 10 11 10
E 4 3 4 5 6 7 8 9 10 9
T 3 4 5 6 7 8 7 8 9 8
N 2 3 4 5 6 7 8 7 8 7
I 1 2 3 4 5 6 7 6 7 8
# 0 1 2 3 4 5 6 7 8 9
# E X E C U T I O N
Minimum Edit Distance
Compu’ng Minimum
Edit Distance
Minimum Edit Distance
Backtrace for
Compu’ng Alignments Compu9ng alignments
• Edit distance isn’t sufficient
• We oBen need to align each character of the two strings to each other
• We do this by keeping a “backtrace”
• Every ’me we enter a cell, remember where we came from
• When we reach the end,
• Trace back the path from the upper right corner to read off the alignment
N 9
O 8
I 7
T 6
N 5
E 4
T 3
N 2
I 1
# 0 1 2 3 4 5 6 7 8 9
# E X E C U T I O N
Edit Distance
Adding Backtrace to Minimum Edit Distance
Base condi’ons: Termina’on:
D(i,0) = i D(0,j) = j D(N,M) is distance
Recurrence Rela’on:
For each i = 1…M!
! For each j = 1…N
D(i-1,j) + 1! dele’on
D(i,j)= min D(i,j-1) + 1! inser’on
subs’tu’on !
subs’tu’on
D(i-1,j-1) + 2; if X(i) ≠ Y(j) 0; if X(i) = Y(j)! LEFT! inser’on ptr(i,j)= DOWN! dele’on DIAG!
!
Result of Backtrace
• Two strings and their alignment:
Performance
• Time:
O(nm) • Space:
O(nm)
• Backtrace
O(n+m)
Minimum Edit Distance
Backtrace for
Compu’ng Alignments
Minimum Edit Distance
Weighted Minimum Edit Distance Weighted Edit Distance
• Why would we add weights to the computa’on?
• Spell Correc’on: some leXers are more likely to be mistyped than others
• Biology: certain kinds of dele’ons or inser’ons are more likely than others
Weighted Min Edit Distance
• Ini’aliza’on:
D(0,0) = 0!
D(i,0) = D(i-1,0) + del[x(i)]; 1 < i ≤ N!
D(0,j) = D(0,j-1) + ins[y(j)]; 1 < j ≤ M
• Recurrence Rela’on:
D(i-1,j) + del[x(i)]! D(i,j)= min D(i,j-1) + ins[y(j)]!
D(i-1,j-1) + sub[x(i),y(j)]!
• Termina’on:
D(N,M) is distance !
Where did the name, dynamic programming, come from?
…The 1950s were not good years for mathematical research. [the] Secretary of Defense …had a pathological fear and hatred of the word, research…
I decided therefore to use the word, “programming”.
I wanted to get across the idea that this was dynamic, this was multistage… I thought, let’s … take a word that has an absolutely precise meaning, namely dynamic… it’s impossible to use the word, dynamic, in a pejorative sense. Try thinking of some combination that will possibly give it a pejorative meaning. It’s impossible.
Thus, I thought dynamic programming was a good name. It was something not even a Congressman could object to.”
Richard Bellman, “Eye of the Hurricane: an autobiography” 1984.
Minimum Edit Distance
Weighted Minimum Edit Distance
Minimum Edit Distance
Minimum Edit Distance in Computa’onal Biology Sequence Alignment
AGGCTATCACCTGACCTCCAGGCCGATGCCC TAGCTATCACGACCGCGGTCGATTTGCCCGAC
-AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC
Why sequence alignment?
• Comparing genes or regions from different species
• to find important regions
• determine func’on
• uncover evolu’onary forces
• Assembling fragments to sequence DNA
• Compare individuals to looking for muta’ons
Alignments in two fields
• In Natural Language Processing
• We generally talk about distance (minimized)
• And weights
• In Computa’onal Biology
• We generally talk about similarity (maximized)
• And scores
The Needleman-‐Wunsch Algorithm
• Ini’aliza’on:
D(i,0) = -i * d! D(0,j) = -j * d
• Recurrence Rela’on:
D(i-1,j) - d! D(i,j)= min D(i,j-1) - d!
D(i-1,j-1) + s[x(i),y(j)]!
• Termina’on:
D(N,M) is distance !
!
A variant of the basic algorithm:
• Maybe it is OK to have an unlimited # of gaps in the beginning and end:
----------CTATCACCTGACCTCCAGGCCGATGCCCCTTCCGGC
GCGAGTTCATCTATCAC--GACCGC--GGTCG--------------
• If so, we don’t want to penalize gaps at the ends
Slide from Serafim Batzoglou
Example:
2 overlapping“reads” from a sequencing project
Example:
Search for a mouse gene within a human chromosome
Slide from Serafim Batzoglou
Changes:
1. Ini’aliza’on
2. Termina’on FOPT = max!
The Local Alignment Problem
Given two strings x = x1……xM,
y = y1……yN
Find substrings x’, y’ whose similarity
(op’mal global alignment value)
is maximum
x = aaaacccccggggXa
y = Xcccgggaaccaacc Slide from Serafim Batzoglou
The Smith-‐Waterman algorithm
Idea: Ignore badly aligning regions
Modifica’ons to Needleman-‐Wunsch:
F(0, j) = 0!
! !F(i, 0) = 0!
0 !!
Itera9on: F(i, j) = max F(i – 1, j) – d!
! ! ! F(i, j – 1) – d!
! ! ! F(i – 1, j – 1) + s(xi, yj) !
Slide from Serafim Batzoglou
The Smith-‐Waterman algorithm
Termina9on:
If we want the best local alignment…
FOPT = maxi,j F(i, j)
Find FOPT and trace back
If we want all local alignments scoring t
For all i, j find F(i, j) t, and trace back?
Complicated by overlapping local alignments Slide from Serafim Batzoglou
A! T! T! A! T! C!
0! 0! 0! 0! 0! 0! 0!
A! 0!
T! 0!
C! 0!
A! 0!
T! 0!
Local alignment example
X = ATCAT! Y = ATTATC!
m = 1 (1 point for match)
d = 1 (-‐1 point for del/ins/sub)
Local alignment example
A!T!T!A!T!C!
X = ATCAT! 0!0!0!0!0!0!0!
Y = ATTATC! A!0!1!0!0!1!0!0!
T!0!0!2!1!0!2!0!
C!0!0!1!1!0!1!3!
A!0!1!0!0!2!1!2!
Local alignment example
A!T!T!A!T!C!
X = ATCAT! 0!0!0!0!0!0!0!
Y = ATTATC! A!0!1!0!0!1!0!0!
T!0!0!2!1!0!2!0!
C!0!0!1!1!0!1!3!
A!0!1!0!0!2!1!2!
Local alignment example
A!T!T!A!T!C! X = ATCAT! 0!0!0!0!0!0!0!
Y = ATTATC! A!0!1!0!0!1!0!0!
T!0!0!2!1!0!2!0!
C!0!0!1!1!0!1!3!
A!0!1!0!0!2!1!2!
Minimum Edit Distance
Minimum Edit Distance in Computa’onal Biology
