Number of Ways to Multiply a Chain of M Matrices understanding the Problem: When multiplying a chain of matrices, the order in which we perform the multiplications can significantly impact the number of scalar multiplications required. This is because matrix multiplication is not commutative. The Catalan Number Connection: The number of ways to parenthesize a chain of M matrices is given by the Catalan number C(M-1) Catalan Numbers: Catalan numbers are a sequence of natural numbers that occur in many counting problems. The nth Catalan number, denoted as C_n, can be calculated using the following formula: C_n (1/(n+1)) (2n choose n) where (2n choose n) is a binomial coefficient. Why Catalan Numbers? Each way to parenthesize a chain of matrices corresponds to a unique way to divide a convex polygon with M+1 sides into triangles using non-intersecting diagonals. This is a classic combinatorial problem solved by Catalan numbers. Example: Consider a chain of 4 matrices: A, B, C, and D. We can parenthesize them in the following ways: ((ABC)D (ABC))D ABC)D) A(BICD)) (ABCD) There are 5 ways to parenthesize this chain, which corresponds to C 3 - 5. In Summary: The number of ways to multiply a chain of M matrices is given by the (M-1)th Catalan number, C(M-1). This number grows rapidly with M, highlighting the importance of finding the optimal parenthesization to minimize the number of scalar multiplications.
Number of Ways to Multiply a Chain of M Matrices understanding the Problem: When multiplying a chain of matrices, the order in which we perform the multiplications can significantly impact the number of scalar multiplications required. This is because matrix multiplication is not commutative. The Catalan Number Connection: The number of ways to parenthesize a chain of M matrices is given by the Catalan number C(M-1) Catalan Numbers: Catalan numbers are a sequence of natural numbers that occur in many counting problems. The nth Catalan number, denoted as C_n, can be calculated using the following formula: C_n (1/(n+1)) (2n choose n) where (2n choose n) is a binomial coefficient. Why Catalan Numbers? Each way to parenthesize a chain of matrices corresponds to a unique way to divide a convex polygon with M+1 sides into triangles using non-intersecting diagonals. This is a classic combinatorial problem solved by Catalan numbers. Example: Consider a chain of 4 matrices: A, B, C, and D. We can parenthesize them in the following ways: ((ABC)D (ABC))D ABC)D) A(BICD)) (ABCD) There are 5 ways to parenthesize this chain, which corresponds to C 3 - 5. In Summary: The number of ways to multiply a chain of M matrices is given by the (M-1)th Catalan number, C(M-1). This number grows rapidly with M, highlighting the importance of finding the optimal parenthesization to minimize the number of scalar multiplications.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Transcribed Image Text:Number of Ways to Multiply a Chain of M Matrices
understanding the Problem:
When multiplying a chain of matrices, the order in which we perform the multiplications can
significantly impact the number of scalar multiplications required. This is because matrix
multiplication is not commutative.
The Catalan Number Connection:
The number of ways to parenthesize a chain of M matrices is given by the Catalan number
C(M-1)
Catalan Numbers:
Catalan numbers are a sequence of natural numbers that occur in many counting problems. The
nth Catalan number, denoted as C_n, can be calculated using the following formula:
C_n (1/(n+1)) (2n choose n)
where (2n choose n) is a binomial coefficient.
Why Catalan Numbers?
Each way to parenthesize a chain of matrices corresponds to a unique way to divide a convex
polygon with M+1 sides into triangles using non-intersecting diagonals. This is a classic combinatorial
problem solved by Catalan numbers.
Example:
Consider a chain of 4 matrices: A, B, C, and D. We can parenthesize them in the following ways:
((ABC)D
(ABC))D
ABC)D)
A(BICD))
(ABCD)
There are 5 ways to parenthesize this chain, which corresponds to C 3 - 5.
In Summary:
The number of ways to multiply a chain of M matrices is given by the (M-1)th Catalan number,
C(M-1). This number grows rapidly with M, highlighting the importance of finding the optimal
parenthesization to minimize the number of scalar multiplications.
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