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1
Dynamic
Programming
ALG00S, [email protected]^ LECT-16, S-^2 Javed I. Khan@ DESIGN & ANALYSIS OF ALGORITHM
Dynamic Programming
- Dynamic Programming, like the divide-and-conquer method, solves problems by combining the solutions to sub-problems.
- Pure divide-and-conquer:
- divides problems into independent sub-problems,
- solves the sub-problem recursively, and then,
- combines their solutions to solve the original sub-problem.
- Dynamic programming in contrast is used when the sub-problems are not independent, that is sub- problems share sub-problems.
- It is typically applied to optimization problems.
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Example:
Matrix Chain
Multiplication
ALG00S, [email protected]^ LECT-16, S-^4 Javed I. Khan@ DESIGN & ANALYSIS OF ALGORITHM
Matrix Multiplication
- Cost of multiplying A[p][q] x B[q][r] is p.q.r
- What is the cost of multiplying three matrices A, B, and C of sizes 10x100, 100x5, and 5x50?
- How to find the best way of multiplying?
LECT-16, S- 7 ALG00S, [email protected] Javed I. Khan@ DESIGN & ANALYSIS OF ALGORITHM
Recursive Solution
- Running time is exponential O(2n)! ALG00S, [email protected]^ LECT-16, S-^8 Javed I. Khan@ DESIGN & ANALYSIS OF ALGORITHM
Observation- 2
- Existence of Overlapping Sub-problem: the same sub-sequence is part of many super sequences.
- For a string of limited size, the actual number of subproblems are quite small. O(n^2 ) only!
LECT-16, S- 9 ALG00S, [email protected] Javed I. Khan@ DESIGN & ANALYSIS OF ALGORITHM
Dynamic Programming Solution:
A Bottom up approach
- Compute the optimum cost for multiplying all matrix chains of size 2.
- Store them in a matrix m[i,j], when i-j spans two matrices.
- Use the above values to compute optimum cost for multiplying all matrix chains of size 3.
- Then size 4 .. Up to size n. ALG00S, [email protected]^ LECT-16, S-^10 Javed I. Khan@ DESIGN & ANALYSIS OF ALGORITHM
Algorithm
LECT-16, S- 13 ALG00S, [email protected] Javed I. Khan@ DESIGN & ANALYSIS OF ALGORITHM
Constructing the Optimal Solution
ALG00S, [email protected]^ LECT-16, S-^14 Javed I. Khan@ DESIGN & ANALYSIS OF ALGORITHM
Complexity of Algorithm
15
Example:
Optimal Polygon
Triangulation
ALG00S, [email protected]^ LECT-16, S-^16 Javed I. Khan@ DESIGN & ANALYSIS OF ALGORITHM
Polygon Triangulation
- We are given a convex polygon P=<v 0 ,v 1 ,…v (^) n- 1 > and a weight function w defined on triangles formed by sides and chords of P. The problem is to find a triangulation that minimizes the sum of the weights of the triangles in the triangulation. v v1 (^) v v v v v v1 (^) v v v v v w (∆ v (^) i vjvk )= vivj + vjvk + vkv i v
LECT-16, S- 19 ALG00S, [email protected] Javed I. Khan@ DESIGN & ANALYSIS OF ALGORITHM
Final Exam
- 4 Questions Total:
- 1 True-False
- 1 Overall Concept
- 1 Tree & Graph
- 1 String Matching & Dynamic Programming
- Open Book 60 min.
- Project Due/Demo on “Exam Day”
- Quiz Today or extra 5% goes to project?
- End Term (20%)
- Last Assignment (~5%)
