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Algorithms. Dynamic Programming. Dynamic Programming. General approach – combine solutions to subproblems to get the solution to an problem Unlike Divide and Conquer in that subproblems are dependent rather than independent bottom-up approach

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algorithms

Algorithms

Dynamic Programming

dynamic programming
Dynamic Programming
  • General approach – combine solutions to subproblems to get the solution to an problem
  • Unlike Divide and Conquer in that
    • subproblems are dependent rather than independent
    • bottom-up approach
    • save values of subproblems in a table and use more than one time
usual approach
Usual Approach
  • Start with the smallest, simplest subproblems
  • Combine “appropriate” subproblem solutions to get a solution to the bigger problem
  • If you have a D&C algorithm that does a lot of duplicate computation, you can often find a DP solution that is more efficient
a simple example
A Simple Example
  • Calculating binomial coefficients
  • n choose k is the number of different combinations of n things taken k at a time
  • These are also the coefficients of the binomial expansion (x+y)n
algorithm for recursive definition
Algorithm for Recursive Definition

function C(n,k)

if k = 0 or k = n then

return 1

else

return C(n-1, k-1) + C(n-1, k)

slide7

C(5,3)

C(4,2)

C(4,3)

C(3,1)

C(3,2)

C(3,2)

C(3,3)

complexity of d c algorithm
Complexity of D & C Algorithm
  • Time complexity is (n!)
  • But we did a lot of duplicate computation
  • Dynamic programming approach
    • Store solutions to sub-problems in a table
    • Bottom-up approach
slide9

k

n k

0 1 2 3

0

1

2

3

4

5

n

analysis of dp version
Analysis of DP Version
  • Time complexity

O(n k)

  • Storage requirements
    • full table O(n k)
    • better solution
typical problem
Typical Problem
  • Dynamic Programming is often used for optimization problems that satisfy the principle of optimality
  • Principle of optimality
    • In an optimal sequence of decisions or choices, each subsequence must be optimal
optimization problems
Optimization Problems
  • Problem has many possible solutions
  • Each solution has a value
  • Goal is to find the solution with the optimal value
steps in dp solutions to optimization problems
Steps in DP Solutions to Optimization Problems

1 Characterize the structure of an optimal solution

2 Recursively define the value of an optimal solution

3 Compute the value of an optimal solution in a bottom-up manner

4 Construct an optimal solution from computed information

matrix chain multiplication
Matrix Chain Multiplication

A1A2A3An

  • Matrix multiplication is associative
  • All ways that a sequence can be parenthesized give the same answer
  • But some are much less expensive to compute
matrix chain multiplication problem
Matrix Chain Multiplication Problem
  • Given a chain < A1,A2, . . .,An> of n matrices, where i = 1, 2, . . ., n and matrix Ai has dimension pi-1pi, fully parenthesize the product A1A2An in a way that minimizes the number of scalar multiplications
example
Example

Matrix Dimensions

A 13 x 5

B 5 X 89

C 89 X 3

D 3 X 34

M = A B C D 13 x 34

slide17
Parenthesization Scalar multiplications

1 ((A B) C) D 10,582

2 (A B) (C D) 54,201

3 (A (B C)) D 2, 856

4 A ((B C) D) 4, 055

5 A (B (C D)) 26,418

1 13 x 5 x 89 to get (A B) 13 x 89 result

13 x 89 x 3 to get ((AB)C) 13 x 3 result

13 x 3 x 34 to get (((AB)C)D) 13 x 34

slide19

T(n) ways to parenthesize

n 1 2 3 4 5 10 15

T(n) 1 1 2 5 14 4,862 2,674,440

slide20

Steps in DP Solutions to Optimization Problems

1 Characterize the structure of an optimal solution

2 Recursively define the value of an optimal solution

3 Compute the value of an optimal solution in a bottom-up manner

4 Construct an optimal solution from computed information

step 1
Step 1
  • Show that the principle of optimality applies
    • An optimal solution to the problem contains within it optimal solutions to sub-problems
    • Let Ai..j be the optimal way to parenthesize AiAi+1. . .Aj
    • Suppose the optimal solution has the first split at position k

A1..k Ak+1..n

    • Each of these sub-problems must be optimally parenthesized
step 2
Step 2
  • Define value of the optimal solution recursively in terms of optimal solutions to sub-problems.
  • Consider the problem Ai..j
  • Let m[i,j] be the minimum number of scalar multiplications needed to compute matrix Ai..j
  • The cost of the cheapest way to compute A1..n is m[1,n]
step 2 continued
Step 2 continued
  • Define m[i,j]

If i = j the chain has one matrix and the cost m[i,i] = 0 for all i  n

If i < j

Assume the optimal split is at position k where i k < j

m[i,j] = m[i,k] + m[k+1,j] + pi-1 pkpj

to compute Ai..k and Ak+1..j

step 2 continued24
Step 2 continued
  • Problem: we don’t know what the value for k is, but there are only j-i possibilities
step 3
Step 3
  • Compute optimal cost by using a bottom-up approach
  • Example: 4 matrix chain

Matrix Dimensions

A1 100 x 1

A2 1 x 100

A3 100 x 1

A4 1 x 100

  • po p1 p2 p3 p4

100 1 100 1 100

slide26

j

j

1 2 3 4

1 2 3 4

i

i

1

2

3

4

1

2

3

4

s

m

po p1 p2 p3 p4

100 1 100 1 100

slide27

MATRIX-CHAIN-ORDER(p)

1 n length[p] - 1

2 for i 1 to n

3 do m[i,i] 0

4 for l 2 to n

5 do for i 1 to n - l + 1

6 do ji + l - 1

7 m[i,j] 

8 for k i to j - 1

9 do q m[i,k]+m[k+1,j]+ pi-1 pkpj

10 if q < m[i,j]

11 then m[i,j] = q

12 s[i,j] = k

13 return m and s

time and space complexity
Time and Space Complexity
  • Time complexity
    • Triply nested loops
    • O(n3)
  • Space complexity
    • n x n two dimensional table
    • O(n2)
step 4 constructing the optimal solution
Step 4: Constructing the Optimal Solution
  • Matrix-Chain-Order determines the optimal number of scalar multiplications
  • Does not directly compute the product
  • Step 4 of the dynamic programming paradigm is to construct an optimal solution from computed information.
  • The s matrix contains the optimal split for every level
slide30

MATRIX-CHAIN-MULTIPLY(A,s,i,j)

1 if j > i

2 then X MATRIX-CHAIN-MULTIPLY(A,s,i,s[i,j])

3 Y MATRIX-CHAIN-MULTIPLY(A,s,s[i,j]+1,j)

4 return MATRIX-MULTIPLY(X,Y)

5 else return Ai

Initial call

MATRIX-CHAIN-MULTIPLY(A,s,1,n) where

A = < A1,A2, . . .,An>

algorithms31

Algorithms

Dynamic Programming Continued

comments on dynamic programming
Comments on Dynamic Programming
  • When the problem can be reduced to several “overlapping” sub-problems.
  • All possible sub-problems are computed
  • Computation is done by maintaining a large matrix
  • Usually has large space requirement
  • Running times are usually at least quadratic
memoization
Memoization
  • Variation on dynamic programming
  • Idea–memoize the natural but inefficient recursive algorithm
  • As each sub-problem is solved, store values in table
  • Initialize table with values that indicate if the value has been computed
slide34

MEMOIZED-MATRIX-CHAIN(p)

1 n  length(p) - 1

2 for i 1 to n

3 do for j i to n

4 do m[i,j] 

5 return LOOKUP-CHAIN(p,1,n)

slide35

LOOKUP-CHAIN(p,i,j)

1 if m[i,j] 

2 then return m[i,j]

3 if i = j

4 then m[i,j] 

5 else for k i to j - 1

6 do q LOOKUP-CHAIN(p,i,k) +

LOOKUP-CHAIN(p,k+1,j) +

pi-1pkpj

7 if q < m[i,j]

8 then m[i,j] q

9 return m[i,j]

space and time requirements of memoized version
Space and Time Requirementsof Memoized Version
  • Running time (n3)
  • Storage (n2)
longest common subsequence
Longest Common Subsequence
  • Definition 1: Subsequence

Given a sequence

X = < x1, x2, . . . , xm>

then another sequence

Z = < z1, z2, . . . , zk>

is a subsequence of X if there exists a strictly increasing sequence <i1, i2, . . . , ik> of indices of x such that for all j = 1,2,...k we have xij= zj

example38
Example

X = <A,B,D,F,M,Q>

Z = <B, F, M>

Z is a subsequence of X with index sequence <2,4,5>

more definitions
More Definitions
  • Definition 2: Common subsequence
    • Given 2 sequences X and Y, we say Z is a common subsequence of X and Y if Z is a subsequence of X and a subsequence of Y
  • Definition 3: Longest common subsequence problem
    • Given X = < x1, x2, . . . , xm> and Y = < y1, y2, . . . , yn> find a maximum length common subsequence of X and Y
example40
Example

X = <A,B,C,B,D,A,B>

Y = <B,D,C,A,B,A>

brute force algorithm
Brute Force Algorithm

1 for every subsequence of X

2 Is there subsequence in Y?

3 If yes, is it longer than the longest subsequence found so far?

Complexity?

yet more definitions
Yet More Definitions
  • Definition 4: Prefix of a subsequence

If X = < x1, x2, . . . , xm> , the ith prefix of X for i = 0,1,...,m is Xi = < x1, x2, . . . , xi>

  • Example
    • if X = <A,B,C,D,E,F,H,I,J,L> then

X4 = <A,B,C,D> and X0 = <>

optimal substructure
Optimal Substructure
  • Theorem 16.1 Optimal Substructure of LCS

Let X = < x1, x2, . . . , xm> and Y = < y1, y2, . . . , yn> be sequences and let Z = < z1, z2, . . . ,zk> be any LCS of X and Y

1. if xm = yn then zk = xm = yn and zk-1 is an LCS of xm-1 and yn-1

2. if xm  yn and zk  x m Z is an LCS of Xm-1 and Y

3. if xm  yn and zk  yn Z is an LCS of Xm and Yn-1

sub problem structure
Sub-problem structure
  • Case 1
    • if xm = yn then there is one sub-problem to solve

find a LCS of Xm-1 and Yn-1 and append xm

  • Case 2
    • if xm  yn then there are two sub-problems
      • find an LCS of Xmand Yn-1
      • find an LCS of Xm-1 and Yn
      • pick the longer of the two
cost of optimal solution
Cost of Optimal Solution
  • Cost is length of the common subsequence
  • We want to pick the longest one
  • Let c[i,j] be the length of an LCS of the sequences Xi and Yj
  • Base case is an empty subsequence--then c[i,j] = 0 because there is no LCS
dynamic programming solution
Dynamic Programming Solution
  • Table c[0..m, 0..n] stores the length of an LCS of Xi and Yj c[i,j]
  • Table b[1..m, 1..n] stores pointers to optimal sub-problem solutions
slide48

Y

C B R F T S Q

0 1 2 3 4 5 6 7

0

A 1

B 2

D 3

F 4

M 5

Q 6

X

Matrix C

slide49

Y

j

C B R F T S Q

1 2 3 4 5 6 7

i

A 1

B 2

D 3

F 4

M 5

Q 6

X

Matrix B

slide50

LCS-LENGTH(X,Y)

1 m length[X]

2 n length[Y]

3 for i  to m

4 do c[i,0] 

5 for j  to n

6 do c[0,j] 

7 for i  to m

8 do for j  to n

9 do if xi = yj

then c[i,j] c[i-1,j-1] + 1

b[i,j] ””

12 else if c[i-1,j] c[i,j-1]

13 then c[i,j] c[i-1,j]

14 b[i,j] ””

15 else c[i,j] c[i,j-1]

16 b[i,j] ””

17 return c and b

slide51

PRINT-LCS (b,X,i,j)

1if i = 0 or j = 0

2 then return

3 if b[i,j] = ””

4 then PRINT-LCS(b,X,i-1,j-1)

5 print xi

6 else if b[i,j] ””

7 then PRINT-LCS(b,X,i-1,j)

8 else PRINT-LCS(b,X,i,j-1)

code complexity
Code Complexity
  • Time complexity
  • Space complexity
  • Improved space complexity?
optimal polygon triangulation
Optimal Polygon Triangulation
  • Turns out to be very similar to matrix chain multiplication
  • Mapping one problem to another kind of problem for which you already have a good solution is a useful concept
lots of definitions
Lots of definitions
  • A polygon is a piecewise-linear, closed curve in the plane.
  • The “pieces” are called sides.
  • A point joining two consecutive sides is called a vertex.
  • The set of points in the plane enclosed by a simple polygon forms the interior of the polygon.
  • the set of point ons the polygon forms its boundary
  • the set of points surrounding the polygon form its exterior
convex polygons
Convex Polygons
  • A simple polygon is convex if given any two points on its boundary or in its interior, all points on the line segment drawn between them are contained in the polygon’s boundary or interior.
labeling a convex polygon
Labeling a convex polygon

v5

v0= vn

v4

v1

v3

P = <v0, v1, . . . vn-1>

v2

slide57

Chords

v5

v0

v5

v0

v4

v4

v1

v1

v3

v3

v2

v2

triangulation
Triangulation
  • A triangulation of a polygon is a set T of chords of the polygon that divide the polygon into disjoint triangles.
    • No chords intersect
    • The set of chords T is maximal; every chord not in T intersects some chord in T
  • Every triangulation of an n-vertex convex polygon
    • has n-3 chords
    • divides polygon into n-2 triangles
optimal polygon triangulation problem
Optimal (polygon) triangulation problem
  • Given a convex polygonP = <v0, v1, . . . vn-1> and a weight function w defined on triangles formed by sides and chords of P.
    • Find a triangulation that minimizes the sum of the weights of the triangles in the triangulation
    • One possible weight function is to minimize the sum of the lengths of the sides
correspondance of parenthesization
Correspondance of Parenthesization
  • We will show how to view both parenthesization and triangulation as parse trees and show the correspondence between the two views.
  • The result will be that a slight modification of the matrix chain problem can be used to solve the triangulation problem.
slide61

((A1(A2A3)(A4(A5A6)))

A1(A2A3)

A4(A5A6)

A1

A2A3

A4

A5A6

A2

A3

A5

A6

slide62

v0

v6

v1

v5

v2

v4

v3

slide63

((A1(A2A3)(A4(A5A6)))

v0

v6

v1

v5

v2

v4

v3

slide64

A1 A2 A3 A4 A5 A6

6 matrices

7 dimensions

p0 p1 p2 p3 p4 p5 p6

v0

v6

v1

v5

v2

v4

6 sides

7 vertices

v3

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