Fast Multiplication The Divide and Conquer Method Omar

Objectives Define Divide and Conquer n Explain the Karatsuba Multiplication Method (Fast Multiplication) n

Divide et impera (Divide and Conquer) A Divide and Conquer algorithm works by recursively

About the Algorithm n n n Discovered by Anatolii Karatsuba in 1960. Published in

How It Works Assume two numbers A, B in base 10 n. A= n.

How It Works (cont. ) 1 2 3 4 Multiplying the four subproducts separately

How It Works (cont. ) Let n What? How? n

How It Works (cont. ) n n n Let Let 1 2 3 Now

New Way 1962 * 1960 = (19 * 19) * 10000 + (91 *

New Way 1962 * 1960 = (19361 * 19) * 10000 + (91 *

New Way 3720 1962 * 1960 = (19361 * 19) * 10000 + (91

Observations n 13 Multiplications in the example n Time Complexity = O( n )

Sources n n http: //en. wikipedia. org/wiki/Karatsuba_algorithm http: //en. wikipedia. org/wiki/Divide_and_conquer_algorithm

Questions 1. 2. Who discovered the Karatsuba Multiplication Method? What is the significance of

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Fast Multiplication The Divide and Conquer Method Omar Hemmali

Objectives Define Divide and Conquer n Explain the Karatsuba Multiplication Method (Fast Multiplication) n Work an example using the Karatsuba Method n

Divide et impera (Divide and Conquer) A Divide and Conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly. n The solutions to the sub-problems are then combined to give a solution to the original problem. n [taken from wikipedia]

About the Algorithm n n n Discovered by Anatolii Karatsuba in 1960. Published in a joint paper with Yu. Ofman in 1962. O( ) Disproved Andrey Kolmogorov's 1956 conjecture that the fastest multiplication algorithm was O( ) First Divide and Conquer Algorithm

How It Works Assume two numbers A, B in base 10 n. A= n. B= n m = half the number of digits in A, B n

How It Works (cont. ) 1 2 3 4 Multiplying the four subproducts separately gives O( ) n Example: 25 * 16 n We can do better n

How It Works (cont. ) Let n What? How? n

How It Works (cont. ) n n n Let Let 1 2 3 Now only 3 multiplications n So n

The “Old” Way Multiplying

New Way 1962 * 1960 = (19 * 19) * 10000 + (91 * 79 – X 0 Y 0 Z 0) * 100 + (62 * 60) = Z 0

New Way 1962 * 1960 = (19 * 19) * 10000 + (91 * 79 – X 0 Y 0 Z 0) * 100 + (62 * 60) = Z 0 11 – 81 19 * 19 = (1 1 * 1) * 100 + (10 * 10 – X Z 1 ) * 10 + (981 * 9) = X 1 Y 1 Z 1

New Way 1962 * 1960 = (19 * 19) * 10000 + (91 * 79 – X 0 Y 0 Z 0) * 100 + (62 * 60) = Z 0 11 – 81 19 * 19 = (1 1 * 1) * 100 + (10 * 10 – X Z 1 ) * 10 + (981 * 9) = X 1 Y 1 Z 1 * 1) * 100 + (1 1* 1 – 0 X 12 – Z 02 ) * 10 + (0 0* 0) = 100 10 * 10 = (1 1 Z 2 X 2 Y 2

New Way 1962 * 1960 = (19 * 19) * 10000 + (91 * 79 – X 0 Y 0 Z 0) * 100 + (62 * 60) = Z 0 11 – 81 19 * 19 = (1 1 * 1) * 100 + (10100 * 10 18 –X Z 1 ) * 10 + (981 * 9) = X 1 Y 1 Z 1 * 1) * 100 + (1 1* 1 – 0 X 12 – Z 02 ) * 10 + (0 0* 0) = 100 10 * 10 = (1 1 Z 2 X 2 Y 2

New Way 1962 * 1960 = (19 * 19) * 10000 + (91 * 79 – X 0 Y 0 Z 0) * 100 + (62 * 60) = Z 0 11 – 81 19 * 19 = (1 1 * 1) * 100 + (10100 * 10 18 –X Z 1 ) * 10 + (981 * 9) = 361 X 1 Y 1 Z 1

New Way 1962 * 1960 = (19361 * 19) * 10000 + (91 * 79 – 361 X 0 – X 0 Y 0 Z 0) * 100 + (62 * 60) = Z 0 11 – 81 19 * 19 = (1 1 * 1) * 100 + (10100 * 10 18 –X Z 1 ) * 10 + (981 * 9) = 361 X 1 Y 1 Z 1

New Way 1962 * 1960 = (19361 * 19) * 10000 + (91 * 79 – 361 X 0 – X 0 Y 0 Z 0) * 100 + (62 * 60) = Z 0 36 62 * 60 = (636 * 6) * 100 + (848 * 6 – 12 X 1 – Z 01 ) * 10 + (2 0* 0) = 3720 X 1 Y 1 Z 1

New Way 3720 1962 * 1960 = (19361 * 19) * 10000 + (91 * 79 – 361 X 0 – 3720) Z 0) * 100 + (62 * 60) = Z 0 X 0 Y 0 62 * 60 = (636 * 6) * 100 + (8 * 6 – 12 X 1 – Z 1 ) * 10 + (0 0* 2) = 3720 X 1 Y 1 Z 1

New Way 3720 1962 * 1960 = (19361 * 19) * 10000 + (91 * 79 – 361 X 0 – 3720 Z 0) * 100 + (62 * 60) = 3720) Z 0 X 0 Y 0 91 * 79 = (963 * 7) * 100 + (10 * 16 – 63 X 1 – Z 91 ) * 10 + (1 9 * 9) = X 1 Y 1 Z 1

New Way 3720 1962 * 1960 = (19361 * 19) * 10000 + (91 * 79 – 361 X 0 – 3720) Z 0) * 100 + (62 * 60) = Z 0 X 0 Y 0 91 * 79 = (963 * 7) * 100 + (10 * 16 – 63 X 1 – Z 91 ) * 10 + (1 9 * 9) = X 1 Y 1 Z 1 10 * 16 = (1 1* 1) * 100 + (1 7* 7 – 6 X 12 – Z 02 ) * 10 + (0 0* 6) = 160 Z 2 X 2 Y 2

New Way 3720 1962 * 1960 = (19361 * 19) * 10000 + (91 * 79 – 361 X 0 – 3720) Z 0) * 100 + (62 * 60) = Z 0 X 0 Y 0 91 * 79 = (963 * 7) * 100 + (10160 * 16 – 63 X 1 – Z 91 ) * 10 + (1 9 * 9) = X 1 Y 1 Z 1 10 * 16 = (1 1* 1) * 100 + (1 7* 7 – 6 X 12 – Z 02 ) * 10 + (0 0* 6) = 160 Z 2 X 2 Y 2

New Way 3720 1962 * 1960 = (19361 * 19) * 10000 + (91 * 79 – 361 X 0 – 3720) Z 0) * 100 + (62 * 60) = Z 0 X 0 Y 0 91 * 79 = (963 * 7) * 100 + (10160 * 16 88 – 63 X 1 – Z 91 ) * 10 + (1 9 * 9) = 7189 X 1 Y 1 Z 1

New Way 3720 1962 * 1960 = (19361 * 19) * 10000 + (91 * 79 – 361 X 0 – 3720) Z 0) * 100 + (62 * 60) = 7189 Z 0 X 0 Y 0 91 * 79 = (963 * 7) * 100 + (10160 * 16 88 – 63 X 1 – Z 91 ) * 10 + (1 9 * 9) = 7189 X 1 Y 1 Z 1

New Way 3720 1962 * 1960 = (19361 * 19) * 10000 + (91 * 79 – 3108 X 0 – 3720) Z 0) * 100 + (62 * 60) = 361 7189 Z 0 X 0 Y 0 3845520

Observations n 13 Multiplications in the example n Time Complexity = O( n )

Sources n n http: //en. wikipedia. org/wiki/Karatsuba_algorithm http: //en. wikipedia. org/wiki/Divide_and_conquer_algorithm

Questions 1. 2. Who discovered the Karatsuba Multiplication Method? What is the significance of the numbers in the example?