1. Catalan Numbers

2. Multiplying n Numbers n multiplication order 2 (x1 · x2) 3
(x1 · (x2 · x3))
((x1 · x2) · x3) 4
(x1 · (x2 · (x3 · x4)))
(x1 · ((x2 · x3) · x4))
((x1 · x2) · (x3 · x4))
((x1 · (x2 · x3)) · x4)
(((x1 · x2) · x3) · x4)
Objective:
Find C(n), the number of ways to compute product x1 . x2 …. xn.

3. Multiplying n Numbers – small n
C(n) 1 2 3 4 5 14 6 42 7
132

4. Chain-Matrix Multiplication
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

5. Problem Statement: Chain Matrix Multiplication
Statement: The chain-matrix multiplication problem can be stated as below:
Given a chain of [A1, A2, , An] of n matrices where for i = 1, 2, , n, matrix Ai has dimension pi-1 x pi, find the order of multiplication which minimizes the number of scalar multiplications.
Note:
Order of A1 is p0 x p1,
Order of A2 is p1 x p2,
Order of A3 is p2 x p3, etc.
Order of A1 x A2 x A3 is p0 x p3,
Order of A1 x A2 x x An is p0 x pn

6. Chain Matrix Multiplication
(Brute Force Approach)

7. Brute Force Chain Matrix Multiplication
Example
Given a sequence [A1, A2, A3, A4]
Order of A1 = 10 x 100
Order of A2 = 100 x 5
Order of A3 = 5x 50
Order of A4 = 50x 20
Compute the order of the product A1 . A2 . A3 . A4 in such a way that minimizes the total number of scalar multiplications.

8. Brute Force Chain Matrix Multiplication
There are five ways to parenthesize this product
Cost of computing the matrix product may vary, depending on order of parenthesis.
All possible ways of parenthesizing
(A1 · (A2 . (A3 . A4)))
(A1 · ((A2 . A3). A4))
((A1 · A2). (A3 . A4))
((A1 · (A2 . A3)). A4)
(((A1 · A2). A3). A4)

9. Kinds of problems solved by algorithms

10. Second Chain : (A1 · ((A2 . A3). A4))

11. Third Chain : ((A1 · A2). (A3 . A4))

12. Fourth Chain : ((A1 · (A2 . A3)). A4)
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

13. Fifth Chain : (((A1 · A2). A3). A4)

14. Chain Matrix Cost ((A1 · A2). (A3 . A4)) First Chain 35,000
Second Chain
145,000
Third Chain
11,000
Fourth Chain
85,000
Fifth Chain
17,500
((A1 · A2). (A3 . A4))