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Bioinformatics Algorithms and Data Structures

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Bioinformatics Algorithms and Data Structures

Chapter 11: Core String Edits

Lecturer: Dr. Rose

Slides by: Dr. Rose

January 30 & February 1, 2007

- This chapter introduces inexact matching
- Inexact matching is used to compute similarity.
- Sequences similarity is a key concept.
- Sequence similarity implies
- Structural similarity
- Functional similarity

- We will consider a dynamic programming approach to inexact matching.

- One measure of similarity between two strings is their edit distance.
- This is a measure of the number of operations required to transform the first string into the other.
- Single character operations:
- Deletion of a character in the first string
- Insertion of a character in the first string
- Substitution of a character from the second character into the second string
- Match a character in the first string with a character of the second.

Example from textbook: transform vintner to writers

vintner replace v with w wintner

wintner insert r after w wrintner

wrintner match i wrintner

wrintner delete n writner

writnermatch t writner

writnerdelete n writer

writermatch e writer

writermatch r writer

writerinsert s writers

Let S = {I, D, R, M} be the edit alphabet

Defn. An edit transcript of two strings is a string over S describing a transformation of one string into another.

Defn. The edit distance between two strings is defined as the minimum number of edit operations needed to transform the first into the second. Matches are not included in the count.

Edit distance is also called Levenshtein distance.

Defn. An optimal transcript is an edit transcript with the minimal number of edit operations for transforming one string into another.

Note: optimal transcripts may not be unique.

Defn. The edit distance problem entails computing the edit distance between two strings along with an optimal transcript.

Defn. A global alignment of strings S1 and S2 is obtained by:

- Inserting dashes/spaces into or at the ends of S1 and S2.
- Aligning the two strings s.t. each character/space in either string is opposite a unique character/space in the other string.
Example 1: S1 = qacdbd S2 = qawxb

q a c - d b d

q a w x - b -

Example 2: S1 = vintner S2 = writers

v - i n t n e r -

w r i - t - e r s

- Mathematically, string alignment and edit transcripts are equivalent.
- From a modeling perspective they are not equivalent.
- Edit transcripts express the idea of mutational changes.

- Observation 1: There are many possible ways to transform one string into another.
- Observation 2: This is like the knapsack problem
- Recall: dynamic programming is used to solve knapsack-like problems.
- Defn. Let D(i,j) denote the edit distance of S1[1..i] and S2[1..j].
- That is, D(i,j) is the minimum number of edit ops needed to transform the first i characters of S1 into the first j characters of S2.

- Notice that we can solve D(i,j) for all combination of lengths of prefixes of S1 and S2.
- Examples: D(0,0),.., D(0,j), D(1,0),..,D(1,j), … D(i,j)
- Dynamic programming is a divide and conquer method.
- The three parts to dynamic programming are:
- The recurrence relation
- Tabular computation
- Traceback

- The recurrence relation expresses the recursive relation between a problem and smaller instances of the problem.
- For any recursive relation, the base condition(s) must be specified.
- Base conditions for D(i,j) are:
- D(i,0) = i
Q: Why is this true? What does it mean in terms of edit ops?

- D(0,j) = j
Q: Why is this true? What does it mean in terms of edit ops?

- D(i,0) = i

- The general recurrence is given by:
D(i,j) = min[D(i - 1, j) + 1, D(i, j - 1) + 1, D(i - 1, j - 1) + t (i,j) ]

Here t (i,j) = 1 if S1(i) S2(j), o/w t (i,j) = 0.

- Proof of correctness on Pages 218-219
- Basic argument: D(i,j) must be one of :
- D(i - 1, j) + 1
- D(i, j - 1) + 1
- D(i - 1, j - 1) + t (i,j)
There are NO other ways of creating S2[1..j] from S1[1..i].

Q: How do we use the recurrence relation to efficiently compute D(i,j) ?

Wrong Answer: simply use recursion.

Q: Why is this the wrong answer?

A: recursion results in inefficient duplication of computations for subproblems.

Q: How much duplication?

A: Exponential duplication!

Example: Fibonacci numbers

Example: Fibonacci numbers

f(n) = f(n - 1) + f(n - 2)

Base conditions: f(0) = 0, f(1) = 1

- Note: In calculating D(n,m), there are only (n + 1) (m + 1) unique combinations of i and j.
- Clearly an exponential number of computations is NOT required.
- Soln: instead of going top-down with recursion, go bottom-up. Compute each combination only once.
- Decide on a data structure to hold intermediate results.
- Start from base conditions. These are the smallest D(i,j) values and are already defined.
- Compute D(i,j) for larger values of i and j.

- Example: Fibonacci numbers

- Decide on a data structure: simple array

- Start from base conditions: f(0) = 0, f(1) = 1

- Compute f(i) for larger values of i. From bottom up.

- Each f(i) is computed only once!

- Q: What kind of data structure should we use for edit distance?
- Has to be a random access data structure.
- Has to support the dimensionality of the problem.

- D(i,j) is two-dimensional: S1 and S2.
- We will use a two-dimensional array, i.e., a table.

Example: edit distance from vintner to writers.

Fill in the base condition values.

- Q: How do we fill in the other values?
- A: use the recurrence:
D(i,j) = min[D(i - 1, j) + 1, D(i, j - 1) + 1, D(i - 1, j - 1) + t (i,j) ]

where t (i,j) = 1 if S1(i) S2(j), o/w t (i,j) = 0.

- We can first compute D(1,1) because we have D(0,0), D(0,1), and D(1,0)
- D(1,1) = min[ 1+1, 1+1, 0+1] = 1

- Then we have all the values needed to compute in turn D(1,2), D(1,3),..,D(1,m)

First compute D(1,1) because we have D(0,0), D(0,1), and D(1,0)

Then compute in turn D(1,2), D(1,3),..,D(1,m)

Fill in subsequent values, row by row, from left to right.

Alternatively, first compute D(1,1) from D(0,0), D(0,1), and D(1,0)

Then compute in turn D(2,1), D(3,1),..,D(n,1)

Fill in subsequent values, column by column, from top to bottom.

- Filling each cell entails a constant number of operations.
- Cell (i,j) depends only on characters S1(i) and S2(j) and cells (i - 1, j - 1), (i, j - 1), and (i - 1, j).

- There are O(nm) cells in the table
- Consequently, we can compute the edit distance D(n, m) in O(nm) time by computing the table in O(nm).

- Having computed the table we know the value of the optimal edit transcript.
- Q: How do we extract the optimal edit transcript from the table?
- A: One way would be to establish pointers from each cell, to predecessor cell(s) from which its value was derived, i.e,
- If D(i,j) = D(i - 1, j) + 1 add a pointer from (i,j) to (i - 1, j)
- If D(i,j) = D(i, j - 1) + 1add a pointer from (i,j) to (i, j - 1)
- If D(i,j) = D(i - 1, j - 1) + t(i,j)add a pointer from (i,j) to (i - 1, j - 1)

- We can recover an optimal edit sequence simply by following any path from (n,m) to (0,0)
- The interpretation of the path links are:
- A horizontal link , (i,j) (i,j-1), corresponds to an insertion of character S2(j) into S1.
- A vertical link, (i,j) (i-1,j), corresponds to a deletion of S1(i) from S1.
- A diagonal link, (i,j) (i-1,j-1), corresponds to a match S1(i) = S2(j) and a substitution if S1(i) S2(j)

An optimal edit path.

What edit transcript does

this path correspond to?

S,S,S,M,D,M,M,I

Another optimal edit path.

What edit transcript does

this path correspond to?

I,S,M,D,M,D,M,M,I

The third possible optimal edit

path. What edit transcript

does this path correspond to?

S,I,M,D,M,D,M,M,I

- Alternatively we can interpret any path from (n,m) to (0,0) as an alignment of S1 and S2.
- The interpretation of the path links are:
- A horizontal link , (i,j) (i,j-1), corresponds to an insertion of a space/dash into S1.
- A vertical link, (i,j) (i-1,j), corresponds to an insertion of a space/dash into S2.
- A diagonal link, (i,j) (i-1,j-1), corresponds to a match if S1(i) = S2(j) or a mismatch if S1(i) S2(j)

Possible optimal path.

What alignment does this

optimal path correspond to?

w r i t - e r s

v i n t n e r -

A second possible optimal path.

What alignment does this

optimal path correspond to?

w r i - t - e r s

v - i n t n e r -

A third possible optimal path.

What alignment does this

optimal path correspond to?

w r i - t - e r s

- v i n t n e r -

- Any path from (n,m) to (0,0) corresponds to an optimal edit sequence and an optimal alignment
- We can recover all optimal edit sequences and alignments simply by extracting all paths from (n,m) to (0,0)
- The correspondence between paths and edit sequences is one-to-one.
- The correspondence between paths and alignments is one-to-one.