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Lecture 2 Dynamic Programming

- ??????

Content

- What is Dynamic Programming?
- Matrix Chain-Products
- Sequence Alignments
- Knapsack Problem
- All-Pairs Shortest Path Problem
- Traveling Salesman Problem
- Conclusion

Lecture 2 Dynamic Programming

- What is Dynamic Programming?

What is Dynamic Programming?

- Dynamic Programming (DP) tends to break the

original problem to sub-problems, i.e., in a

smaller size - The optimal solution in the bigger sub-problems

is found through a retroactive formula which

connects the optimal solutions of sub-problems. - Used when the solution to a problem may be viewed

as the result of a sequence of decisions.

Properties for Problems Solved by DP

- Simple Subproblems
- The original problem can be broken into smaller

subproblems with the same structure - Optimal Substructure of the problems
- The solution to the problem must be a composition

of subproblem solutions (the principle of

optimality) - Subproblem Overlap
- Optimal subproblems to unrelated problems can

contain subproblems in common

The Principle of Optimality

- The basic principle of dynamic programming
- Developed by Richard Bellman
- An optimal path has the property that whatever

the initial conditions and control variables

(choices) over some initial period, the control

(or decision variables) chosen over the remaining

period must be optimal for the remaining problem,

with the state resulting from the early decisions

taken to be the initial condition.

Example Shortest Path Problem

Goal

Start

Example Shortest Path Problem

Start

Goal

Example Shortest Path Problem

25

10

28

5

Start

Goal

40

3

Recall ? Greedy Method forShortest Paths on a

Multi-stage Graph

Is the greedy solution optimal?

- Problem
- Find a shortest path from v0 to v3

Recall ? Greedy Method forShortest Paths on a

Multi-stage Graph

?

Is the greedy solution optimal?

- Problem
- Find a shortest path from v0 to v3

The optimal path

Example ? Dynamic Programming

Lecture 2 Dynamic Programming

- Matrix Chain-Products

Matrix Multiplication

- C A B
- A is d e and B is e f
- O(def )

Matrix Chain-Products

- Given a sequence of matrices, A1, A2, , An, find

the most efficient way to multiply them together. - Facts
- A(BC) (AB)C
- Different parenthesizing may need different

numbers of operation. - Example A10 30, B 30 5, C 5 60
- (AB)C (10305) (10560) 1500 3000

4500 ops - A(BC) (30560) (103060) 9000 18000

27000 ops

Matrix Chain-Products

- Given a sequence of matrices, A1, A2, , An, find

the most efficient way to multiply them together. - A Brute-force Approach
- Try all possible ways to parenthesize

AA1?A2??An - Calculate number of operations for each one
- Pick the best one
- Time Complexity
- paranethesizations binary trees of n nodes
- O(4n)

A Greedy Approach

- Idea 1
- repeatedly select the product that uses the most

operations. - Counter-example
- A 10 ? 5, B 5 ? 10, C 10 ? 5, and D 5 ? 10
- Greedy idea 1 gives (AB)(CD), which takes

5001000500 2000 ops - A((BC)D) takes 500250250 1000 ops

Another Greedy Approach

- Idea 2
- repeatedly select the product that uses the least

operations. - Counter-example
- A 101 ? 11, B 11 ? 9, C 9 ? 100, and D 100 ?

999 - Greedy idea 2 gives A((BC)D), which takes

1099899900108900228789 ops - (AB)(CD) takes 99998999189100189090 ops

DP ? Define Subproblem

Subproblem (Pij, i? j)

Original Problem

(P1n)

Suppose operations for the optimal solution of

Pij is Nij

operations for the optimal solution of the

original problem P1n is N1n

DP ? Define Subproblem

Subproblem (Pij, i? j)

Original Problem

(P1n)

Suppose operations for the optimal solution of

Pij is Nij

operations for the optimal solution of the

original problem P1n is N1n

DP ? Define Subproblem

What is the relation btw Nij (Pij) and N1n

(P1n)?

Subproblem (Pij, i? j)

Original Problem

(P1n)

Suppose operations for the optimal solution of

Pij is Nij

operations for the optimal solution of the

original problem P1n is N1n

DP ? Principle of Optimality

dk?dj1

di?dk1

Nk1,n

Nik

DP ? Implementation

Nij

DP ? Implementation

Nij

DP ? Implementation

Nij

?

DP ? Implementation

Nij

1

2

j

n

1

2

?

i

n

DP ? Implementation

Nij

1

2

j

n

1

2

?

i

n

DP ? Implementation

Nij

1

2

j

n

1

2

i

n

DP ? Implementation

Nij

1

2

j

n

1

?

2

i

n

DP ? Implementation

Nij

1

2

j

n

1

2

?

i

n

DP ? Implementation

Nij

1

2

j

n

1

2

i

n

DP ? Implementation

Nij

1

2

j

n

1

?

2

i

n

DP ? Implementation

Nij

1

2

j

n

1

2

i

?

n

DP ? Implementation

Nij

1

2

j

n

1

2

i

n

DP for Matrix Chain-Products

Algorithm matrixChain(S) Input sequence S of n

matrices to be multiplied Output number of

operations in an optimal parenthesization of

S for i ?1 to n // main diagonal terms are all

zero Ni,i ? 0 for d ? 2 to n // each diagonal

do following for i ?1 to n?d1 // do from top to

bottom for each diagonal j ? id?1 Ni,j ?

infinity for k ? i to j?1 // counting

minimum Ni,j ? min(Ni,j, Ni,k Nk1,j di

dk1 dj1)

Time Complexity

Algorithm matrixChain(S) Input sequence S of n

matrices to be multiplied Output number of

operations in an optimal parenthesization of

S for i ?1 to n // main diagonal terms are all

zero Ni,i ? 0 for d ? 2 to n // each diagonal

do following for i ?1 to n?d1 // do from top to

bottom for each diagonal j ? id?1 Ni,j ?

infinity for k ? i to j?1 // counting

minimum Ni,j ? min(Ni,j, Ni,k Nk1,j di

dk1 dj1)

O(n3)

Exercises

- The matrixChain algorithm only computes

operations of an optimal parenthesization. But,

it doesnt report the optimal parenthesization

scheme. Please modify the algorithm so that it

can do so. - Given an example with 5 matrices to illustrate

your idea using a table.

Lecture 2 Dynamic Programming

- Sequence Alignment

Question

- Given two strings
- are they similar?
- what is their distance?

and

Example

applicable

X

Y

plausibly

How similar they are?

Can you give them a score?

Example

applica---ble

X

Match

Match

Match

Match

Match

Mismatch

Indel

Indel

Indel

Indel

Indel

Indel

Indel

-p-l--ausibly

Y

Matches Mismatches Insertions deletions (indel)

Three cases

Example

applica---ble

X

Match

Match

Match

Match

Match

Mismatch

Indel

Indel

Indel

Indel

Indel

Indel

Indel

-p-l--ausibly

Y

Matches Mismatches Insertions deletions (indel)

(1)

(?1)

Three cases

(?1)

Example

applica---ble

X

Score 5?(1) 1?(?1) 7 ?(?1) ?3

Match

Match

Match

Match

Match

Mismatch

Indel

Indel

Indel

Indel

Indel

Indel

Indel

-p-l--ausibly

Y

Matches Mismatches Insertions deletions (indel)

(1)

(?1)

Three cases

(?1)

Example

applica---ble

X

Is the alignment optimal?

Score 5?(1) 1?(?1) 7 ?(?1) ?3

Match

Match

Match

Match

Match

Mismatch

Indel

Indel

Indel

Indel

Indel

Indel

Indel

-p-l--ausibly

Y

Matches Mismatches Insertions deletions (indel)

(1)

(?1)

Three cases

(?1)

Sequence Alignment

- In bioinformatics, a sequence alignment is a way

of arranging the primary sequences of DNA, RNA,

or protein to identify regions of similarity that

may be a consequence of functional, structural,

or evolutionary relationships between the

sequences.

Global and Local Alignments

L G P S S K Q T G K G S - S R I W D N

Global alignment L N -

I T K S A G K G A I M R L G D A - - - - - - - T

G K G - - - - - - - -

Local alignment - - - - - - - A G K

G - - - - - - - -

Global and Local Alignments

Global and Local Alignments

- Global Alignment
- attempts to align the entire sequence
- most useful when the sequences in the query set

are similar and of roughly equal size. - NeedlemanWunsch algorithm (1971).
- Local Alignment
- Attempts to align partial regions of sequences

with high level of similarity. - Smith-Waterman algorithm (1981)

NeedlemanWunsch Algorithm

- Find the best global alignment of any two

sequences under a given substitution matrix. - Maximize a similarity score, to give maximum

match - Maximum match largest number of residues of one

sequence that can be matched with another

allowing for all possible gaps - Based on dynamic programming
- Involves an iterative matrix method of

calculation

Substitution Matrix

- In bioinformatics, a substitution matrix

estimates the rate at which each possible residue

in a sequence changes to each other residue over

time. - Substitution matrices are usually seen in the

context of amino acid or DNA sequence alignment,

where the similarity between sequences depends on

the mutation rates as represented in the matrix.

Substitution Matrix (DNA) w/o Gap Cost

Substitution Matrix (DNA) w/ Gap Cost

Substitution Matrix (3D-BLAST)

DP ? Define Subproblem

- Consider two strings, s of length n and t of

length m. Let S be the substitution matrix. - Subproblem Let Pij is defined to be the optimal

aligning for the two substrings t1..i and

s1..j, - and let Mij be the matching score.
- Original Problem Pmn (matching score Mmn)

DP ? Principle of Optimality

?

?

?

?

Example

- Step 1. Create a scoring matrix
- Step 2. Make an empty table for Mij
- Step 3. Initialize base conditions
- Step 4. Fill table by
- Step 5. Trace back

Example

- Step 1. Create a scoring matrix
- Step 2. Make an empty table for Mij
- Step 3. Initialize base conditions
- Step 4. Fill table by
- Step 5. Trace back

0

?2

?4

?6

?8

?10

?12

?14

?2

?4

?6

?8

?10

?12

Example

- Step 1. Create a scoring matrix
- Step 2. Make an empty table for Mij
- Step 3. Initialize base conditions
- Step 4. Fill table by
- Step 5. Trace back

0

?2

?4

?6

?8

?10

?12

?14

?2

?1

0

?2

?4

?6

?8

?10

?4

?3

1

?1

?3

?3

?5

?7

?6

?5

?1

0

?2

?1

?3

?5

?8

?4

?3

0

1

?1

1

?1

?10

?6

?5

?2

1

0

1

2

?12

?8

?7

?3

0

0

1

3

Example

s t

G G

A A

T -

G A

G T

C C

A C

- Step 1. Create a scoring matrix
- Step 2. Make an empty table for Mij
- Step 3. Initialize base conditions
- Step 4. Fill table by
- Step 5. Trace back

0

?2

?4

?6

?8

?10

?12

?14

0

?2

?1

0

?2

?4

?6

?8

?10

?1

?4

?3

1

?1

?3

?3

?5

?7

1

?6

?5

?1

0

?2

?1

?3

?5

0

?8

?4

?3

0

1

?1

1

?1

?1

1

?10

?6

?5

?2

1

0

1

2

1

?12

?8

?7

?3

0

0

1

3

3

NeedlemanWunsch Algorithm

- Step 1. Create a scoring matrix
- Step 2. Make an empty table for Mij
- Step 3. Initialize base conditions
- Step 4. Fill table by
- Step 5. Trace back

NeedlemanWunsch Algorithm

s ? , t ? while i lt 1 and j lt 1 do

if s ?sj s t ?ti t else

if s ?sj s t ?gap t

else s ?gap s t ?ti t while

i gt 1 do t ?gap t while j gt 1 do s ?gap s

- Step 1. Create a scoring matrix
- Step 2. Make an empty table for Mij
- Step 3. Initialize base conditions
- Step 4. Fill table by
- Step 5. Trace back

Local Alignment Problem

- Given two strings s s1sn,
- t t1.tm
- Find substrings s, t whose similarity
- (optimal global alignment value) is maximum.

Example Local Alignment

GTAGT CATCAT ATG TGACTGAC G TC CATDOGCAT CC

TGACTGAC A

Best aligned subsequeces

Recursive Formulation

- Global Alignment (NeedlemanWunsch Algorithm)
- Local Alignment (Smith-Waterman Algorithm)

Exercises

- Find the best local aligned substrings for the

following two DNA strings - GAATTCAGTTA
- GGATCGA
- You have to give the detail.
- Hint start from the left table.

Exercises

- What is longest common sequence (LCS) problem?

How to solve LCS using dynamic programming

technique?

Lecture 2 Dynamic Programming

- Knapsack Problem

Knapsack Problems

- Given some items, pack the knapsack to get the

maximum total value. Each item has some weight

and some benefit. Total weight that we can carry

is no more than some fixed capacity. - Fractional knapsack problem
- Items are divisible you can take any fraction of

an item. - Solved with a greedy algorithm.
- 0-1 knapsack problem
- Items are indivisible you either take an item or

not. - Solved with dynamic programming.

0-1 Knapsack Problem

- Given a knapsack with maximum capacity W, and a

set S consisting of n items - Each item i has some weight wi and benefit value

bi (all wi and W are integer values) - Problem How to pack the knapsack to achieve

maximum total value of packed items?

Why it is called a 0-1 Knapsack Problem?

Example 0-1 Knapsack Problem

Which boxes should be chosen to maximize the

amount of money while still keeping the overall

weight under 15 kg ?

Example 0-1 Knapsack Problem

- Objective Function
- Unknowns or Variables
- Constraints

Formulation 0-1 Knapsack Problem

0-1 Knapsack Problem Brute-Force Approach

- Since there are n items, there are 2n possible

combinations of items. - We go through all combinations and find the one

with maximum value and with total weight less or

equal to W - Running time will be O(2n)

DP ? Define Subproblem

- Suppose that items are labeled 1,..., n.
- Define a subproblem, say, Pk as to finding an

optimal solution for items in Sk 1, 2,..., k. - ? original problem is Pn.
- Is such a scheme workable?
- Is the principle of optimality held?

A Counterexample

1. 2kgs, 3

P1

P2

P3

P4

P5

2. 3kgs, 4

3. 4kgs, 5

4. 5kgs, 8

20 kgs

5. 9kgs, 10

A Counterexample

1. 2kgs, 3

2. 3kgs, 4

3. 4kgs, 5

4. 5kgs, 8

20 kgs

5. 9kgs, 10

A Counterexample

Solution for P4 is not part of the solution for

P5 !!!

1. 2kgs, 3

2. 3kgs, 4

3. 4kgs, 5

4. 5kgs, 8

20 kgs

5. 9kgs, 10

DP ? Define Subproblem

- Suppose that items are labeled 1,..., n.
- Define a subproblem, say, Pk as to finding an

optimal solution for items in Sk 1, 2,..., k. - ? original problem is Pn.
- Is such a scheme workable?
- Is the principle of optimality held?

?

?

DP ? Define Subproblem

New version

- Suppose that items are labeled 1,..., n.
- Define a subproblem, say, Pk,w as to finding an

optimal solution for items in Sk 1, 2,..., k

and with total weight no more than w. - ? original problem is Pn,W.
- Is such a scheme workable?
- Is the principle of optimality held?

DP ? Principle of Optimality

Denote the benefit for the optimal solution of

Pk,w as Bk,w.

DP ? Principle of Optimality

In this case, it is impossible to include the kth

object.

Denote the benefit for the optimal solution of

Pk,w as Bk,w.

include the kth object

Not include the kth object

There are two possible choices.

Example

1. 2kgs, 3

2. 3kgs, 4

3. 4kgs, 5

4. 5kgs, 6

5 kgs

Example

Step 1. Setup table and initialize base

conditions.

1. 2kgs, 3

2. 3kgs, 4

3. 4kgs, 5

4. 5kgs, 6

5 kgs

Example

Step 2. Fill all table entries progressively.

1. 2kgs, 3

2. 3kgs, 4

0

3

3. 4kgs, 5

3

4. 5kgs, 6

5 kgs

3

3

Example

Step 2. Fill all table entries progressively.

1. 2kgs, 3

2. 3kgs, 4

0

0

3

3

3. 4kgs, 5

3

4

4. 5kgs, 6

5 kgs

3

4

3

7

Example

Step 2. Fill all table entries progressively.

1. 2kgs, 3

2. 3kgs, 4

0

0

0

3

3

3

3. 4kgs, 5

3

4

4

4. 5kgs, 6

5 kgs

3

4

5

3

7

7

Example

Step 2. Fill all table entries progressively.

1. 2kgs, 3

2. 3kgs, 4

0

0

0

0

3

3

3

3

3. 4kgs, 5

3

4

4

4

4. 5kgs, 6

5 kgs

3

4

5

5

3

7

7

7

Example

Step 3. Trace back

1. 2kgs, 3

2. 3kgs, 4

0

0

0

0

3

3

3

3

3. 4kgs, 5

3

4

4

4

4. 5kgs, 6

5 kgs

3

4

5

5

3

7

7

7

Pseudo-Polynomial Time Algorithm

- The time complexity for 0-1 knapsack using DP is

O(Wn). - Not a polynomial-time algorithm if W is large.
- This is a pseudo-polynomial time algorithm.

Lecture 2 Dynamic Programming

- All-Pairs
- Shortest Path Problem

All-Pairs Shortest Path Problem

- Given weighted graph G(V,E), we want to determine

the cost dij of the shortest path between each

pair of nodes in V.

Floyd's Algorithm

- Let be the minimum cost of a path from node i

to node j, using only nodes in Vkv1,,vk.

k

The all-pairs shortest path problem is to find

all paths with costs

i

j

Floyd's Algorithm

Input Parameter D Output Parameter D,

next all_paths(D, next) n D.NumberOfRows //

initialize next if no intermediate // vertices

are allowed nextij j for i 1 to n for j

1 to n nextij j for k 1 to n //

compute D(k) for i 1 to n for j 1 to

n if (Dik Dkj lt Dij)

Dij Dik Dkj nextij

nextik

O(n3)

Floyd's Algorithm

Input Parameters next, i, j Output Parameters

None print_path(next, i, j) // if no

intermediate vertices, just // print i and j

and return if (j nextij) print(i

j) return // output i and

then the path from the vertex // after i

(nextij) to j print(i )

print_path(next,nextij, j)

Lecture 2 Dynamic Programming

- Traveling Salesman Problem

Traveling Salesman Problem (TSP)

Traveling Salesman Problem (TSP)

How many feasible paths?

n cities

Example (TSP)

(1234) 18

(1243) 19

(1324) 23

(1342) 19

(1423) 23

(1432) 18

Subproblem Formulation for TSP

length of the shortest path from i to 1 visiting

each city in S exactly once.

g(i, S)

1

g(1, V ? 1)

length of the optimal TSP tour.

i

Subproblem Formulation for TSP

Goal g(1, V ? 1)

length of the shortest path from i to 1 visiting

each city in S exactly once.

g(i, S)

1

j

i

Example

Goal g(1, V ? 1)

Example

Goal g(1, V ? 1)

18

2

6

4

16

13

14

7

6

7

5

6

5

9

8

13

11

10

9

5

5

6

6

7

7

4

6

4

2

6

2

DP ? TSP Algorithm

Goal g(1, V ? 1)

Input Parameter D Output Parameter P //

path TSP(D) n Dim(D) for i 1 to n gi, ?

Di, 1 for k 1 to n?2 // compute g for

subproblems for all S ? V?1 with Sk

for all i ? S ? 1 gi, S minj?SDi,

j, gj, S ? j Pi, S arg minj?SDi,

j, gj, S ? j // compute the TSP tour g1,

V?1 minj?V?1D1, j, gj, V ? 1,

j P1, V?1 arg minj?V?1D1, j, gj,

V ? 1, j

DP ? TSP Algorithm

Goal g(1, V ? 1)

Input Parameter D Output Parameter P //

path TSP(D) n Dim(D) for i 1 to n gi, ?

Di, 1 for k 1 to n?2 // compute g for

subproblems for all S ? V?1 with Sk

for all i ? S ? 1 gi, S minj?SDi,

j, gj, S ? j Pi, S arg minj?SDi,

j, gj, S ? j // compute the TSP tour g1,

V?1 minj?V?1D1, j, gj, V ? 1,

j P1, V?1 arg minj?V?1D1, j, gj,

V ? 1, j

O(2n)

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