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Background

“Dynamic programming” originates with Richard Bellman (1940s) in multistage decision process problems.

While at RAND Corp, he wanted his work to appear more practical (“real work”) as opposed to theoretical. To shield himself from scrutiny, Bellman chose the word “programming,” which implies fruitful, deliberate effort and embellished it with “dynamic.” As he puts it “it’s impossible to use dynamic in a pejorative sense.”

Applications:

String alignments / problems

Pattern recognition:

Image matching / image recognition (2D & 3D)

Speech recognition (Viterbi algorithm)

Manufacturing – find fastest way through factory

Order of matrices in matrix multiplication to minimize cost

Build optimal binary search tree – minimize number of nodes visited during search

Language translator – most common words near root of tree

Used to solve problems exhibiting:

- Overlapping Subproblems: “they occur as a subproblem of different problems”
- Optimal Substructure:“An optimal solution to the problem contains within it optimal solutions to subproblems.”
- Subproblem Independence:“the solution to one subproblem does not affect the solution to another subproblem, i.e., they do not share resources”

Tops Down and Bottoms Up

- Top-down: problem is broken down to subproblems then solved using memoization to remember the solutions to subproblems already solved.
Topdown:

function fib(n)

if n = 0 return 0

if n = 1 return 1

else return fib(n − 1) + fib(n − 2)

Top down with memoization (not memorization)

var m := map(0 → 1, 1 → 1)

function fib(n)

if map m does not contain key n

m[n] := fib(n − 1) + fib(n − 2)

return m[n]

- Bottom-up: all subproblems must be solved in advance to build solutions to larger problems
function fib(n)

var previousFib := 0, currentFib := 1

repeat n − 1 times

var newFib := previousFib + currentFib

previousFib := currentFib

currentFib := newFib

return currentFib

Biological Sequence Matching Problems 1

- DNA
- Two strands
- Four letter alphabet (four bases)
- Base pairing rules
- Strands are directional and, within a gene, only one strand is translated

- RNA
- Functional or intermediate step of protein manufacturing
- Four letter alphabet

- Proteins
- 20 letter alphabet

Biological Sequence Matching Problems 2

- Applications
- Identify strains of viruses, bacteria
- Identify genes (hair, skin, eye color, height) and genetic basis for diseases (lethal or susceptibility to cancer, etc.)
- Identify evolutionary relationships

- Dynamic programming is the basis of BLAST (Basic Local Alignment Search Tool) – in top 3of most cited papers in recent bioscience history (was #1 in 1990s)

Sequence Alignment Algorithm 1

AGGCGGATC

TAGCATCTAC

-AGGCGGATC---

TAG-C--ATCTAC

Given two strings x = x1x2...xM, y = y1y2…yN,

Find the alignment with maximum score

F = (# matches) m - (# mismatches) s – (#gaps) d

Sequence Alignment Algorithm 2

AGTGCCCTGGAACCCTGACGGTGGGTCACAAAACTTCTGGA

There are > 2N possible alignments.

AGTGACCTGGGAAGACCCTGACCCTGGGTCACAAAACTC

Sequence Alignment Algorithm 3

Note:

The score of aligning x1……xM

y1……yN

is additive

Say that x1…xixi+1…xM

aligns to y1…yjyj+1…yN

Add the two scores:

F(x1…xM, y1…yN) = F(x1…xi, y1...yj) + F(xi+1…xm, yj+1…yN)

Sequence Alignment Algorithm 4

- Original problem
- Align x1…xM to y1…yN

- Divide into a finite number of subproblems (non-overlapping for efficiency)
- Align x1…xi to y1…yj

- Subdivide the subproblem and construct the solution from smaller subproblems
- Classic problem type for dynamic programming
Let F(i, j) = optimal score of aligning

x1……xi

y1……yj

- F is the “matrix” or “table” or “program.”Hence the term “dynamic programming.”

Sequence Alignment Algorithm 5

F = (# matches) m - (# mismatches) s – (# gaps) d

F(i, j) calculated with scoring function s(xi, yj) or gap function g

Scoring function s(xi, yj)

Three cases:

- xi aligns to yj
x1……xi-1 xi

y1……yj-1 yj

2. xi aligns to a gap

x1……xi-1 xi

y1……yj -

- yj aligns to a gap
x1……xi -

y1……yj-1 yj

diagonal move

m, if xi = yj

F(i, j) = F(i – 1, j – 1) +

-s, if not

horizontal move

F(i, j) = F(i – 1, j) – d

Gap function

vertical move

F(i, j) = F(i, j – 1) – d

Sequence Alignment Algorithm 6

How do we choose the case for each matrix position?

Assume that the subproblems are solved:

F(i, j – 1), F(i – 1, j), F(i – 1, j – 1) are optimal

Therefore,

F(i – 1, j – 1) + s(xi, yj)

F(i, j) = max F(i – 1, j) – d

F(i, j – 1) – d

Where s(xi, yj) = m, if xi = yj; -s, if not

Sequence Alignment Algorithm 7

Set d = 1, m = 1, s = -0.5

F(i – 1, j – 1) + s(xi, yj)

F(i, j) = max F(i – 1, j) – 1

F(i, j – 1) – 1

Where s(xi, yj) = 1, if xi = yj

-0.5, if not

Needleman-Wunsch Algorithm 1:Finds Global Optimal Alignment

- Initialization
- F(0, 0) = 0
- F(0, j) = - j d
- F(i, 0) = - i d

- Main IterationFilling-in partial alignments
- For each i = 1……M
For each j = 1……N

F(i – 1,j – 1) + s(xi, yj) [case 1]

F(i, j) = max F(i – 1, j) – d [case 2]

F(i, j – 1) – d [case 3]

% if [case 1]

Ptr(i, j) = ! if [case 2]

# if [case 3]

- For each i = 1……M
- TerminationF(M, N) is the optimal score, and from Ptr(M, N) can trace back optimal alignment

Needleman-Wunsch Algorithm 2

Initialization

F(0, 0) = 0

F(0, j) = - j d

F(i, 0) = - i d

(1) F(i – 1,j – 1) + s(xi, yj)

F(i, j) = max (2) F(i – 1, j) – d

(3) F(i, j – 1) – d

% (1)

Ptr(i, j) = ! (2)

# (3)

Smith-Waterman Algorithm 1:Finds local optimal alignment(s)

Ignore poorly aligned regions

- Initialization
- F(0, 0) = 0
- F(0, j) = 0
- F(i, 0) = 0

- Main IterationFilling-in partial alignments
- For each i = 1……M
For each j = 1……N

0

F(i – 1,j – 1) + s(xi, yj) [case 1]

F(i, j) = max F(i – 1, j) – d [case 2]

F(i, j – 1) – d [case 3]

% if [case 1]

Ptr(i, j) = ! if [case 2]

# if [case 3]

- For each i = 1……M
- Termination
F(M, N) is the optimal score, and from Ptr(M, N) can trace back optimal alignment

Smith-Waterman Algorithm 2

Initialization

F(0, 0) = 0

F(0, j) = 0

F(i, 0) = 0

(1) F(i – 1,j – 1) + s(xi, yj)

F(i, j) = max (2) F(i – 1, j) – d

(3) F(i, j – 1) – d

% (1)

Ptr(i, j) = ! (2)

# (3)

Overlap Detection 1

- When searching for matches of a short string in database of long strings, we don’t want to penalize overhangs

x

x

y

y

x1 …………………… xM

x1 …………………… xM

y1 ………… yN

y1 ………………… yN

Overlap Detection 2

F(i – 1, 0)

F(i, 0) = maxF(i – 1, m) – T

F(i – 1,j – 1) + s(xi, yj)

F(i, j) = max F(i – 1, j) – d

F(i, j – 1) – d

Overlap Detection 3

0

F(i – 1,j – 1) + s(xi, yj)

F(i, j) = max F(i – 1, j) – d

F(i, j – 1) – d

F(i – 1, 0)

F(i, 0) = maxF(i – 1, m) – T

Needleman-WunschwithOverlap Detection

Smith-WatermanwithOverlap Detection

Bounded Dynamic Programming

Initialization:

F(i,0), F(0,j) undefined for i, j > k

Iteration:

For i = 1…M

For j = max(1, i – k)…min(N, i+k)

F(i – 1, j – 1)+ s(xi, yj)

F(i, j) = max F(i, j – 1) – d, if j > i – k(N)

F(i – 1, j) – d, if j < i + k(N)

Termination: same

x1 ………………………… xM

y1 ……………………… yN

k(N)

Largest Common Subsequence 1

- Initialization
- F(0, 0) = 0
- F(0, j) = 0
- F(i, 0) = 0

- Main Iteration
- For each i = 1……M
For each j = 1……N

F(i – 1,j – 1) + 1, if xi = yj [case 1]

F(i, j) = max F(i – 1, j), if not(xi = yj) [case 2]

F(i, j – 1), if not(xi = yj) [case 3]

% if [case 1]

Ptr(i, j) = ! if [case 2]

#if [case 3]

- For each i = 1……M
- TerminationF(M, N) is the optimal score, and from Ptr(M, N) can trace back optimal alignment

Largest Common Subsequence 2

Initialization

F(0, 0) = 0

F(0, j) = 0

F(i, 0) = 0

(1) F(i – 1,j – 1) + 1, if xi = yj

F(i, j) = max (2) F(i – 1, j), if not(xi = yj)

(3) F(i, j – 1), if not(xi = yj)

% (1)

Ptr(i, j) = ! (2)

# (3)

Largest Common Subsequence 3

Cormen: error on page 353

Corrected (to obtain figure 15.6)

m = length[X]

n = length[Y]

for i = 1 to m

do c[i,0] = 0

for j = 0 to n

do c[0,j] = 0

for i = 1 to m

for j = 1 to n

if xi = yj then

c[i,j] = c[i-1, j-1] + 1]

b[i,j] = “%”

else if c[i-1, j] > c[i, j-1] then

c[i,j] = c[i-1, j]

b[i,j] = “!”

else

c[i,j] = c[i, j-1]

b[i,j] = “#”

return c and b

Performance

- Running Time: O(mn) + O(m+n) for output
- Storage: O(mn)
- Possible to eliminate backpointer matrix for some problems

- Improvements
- Overlap detection
- Partitioning: Find local alignments to seed global alignment
- Bounded DP
- Gap opening vs. gap extension
- Biochemically significant scoring function

Sources

Altschul, S.F., et al. Basic Local Alignment Search Tool.J. Molec. Biol. 215(3): 403-10, 1990.

Bellman, Richard. Dynamic Programming. Princeton University Press, Princeton: 1957.

Cormen et al. Introduction to Algorithms. MIT Press, Cambridge: 2001.

Dreyfus, Stuart. 2002. Richard Bellman on the birth of dynamic programming. Operations Research 50: 48-51.

Durbin et al. Biological Sequence Analysis: Probabilistic models of proteins and nucleic acids. Cambridge University Press, New York: 1998.

Gotoh, O. 1982. An improved algorithm for matching biological sequences. Journal of Molecular Biology 162: 705-708.

Gusfield, Dan. Algorithms on Strings, Trees, and Sequences, Cambridge University Press, New York: 1997.

Needleman, S.B. and Wunsch, C.D. 1970. A general method applicable to the search for similarities in the amino acid sequence of two proteins. Journal of Molecular Biology 48: 443-453.

Preiss. B.R. Data Structures and Algorithms with Object-Oriented Design Patterns in C#.

Smith, T. F. and Waterman, M.S. 1981. Identification of common molecular subsequences. Journal of Molecular Biology 147: 195-197.

Wikipedia

Sequence Alignment Algorithm X

AGGCTATCACCTGACCTCCAGGCCGATGCCC

TAGCTATCACGACCGCGGTCGATTTGCCCGAC

-AGGCTATCACCTGACCTCCAGGCCGA--TGCCC---

TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC

Given two strings x = x1x2...xM, y = y1y2…yN,

Find the alignment with maximum score

F = (# matches) m - (# mismatches) s – (#gaps) d

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