1. Greatest Common Divisor Jordi Cortadella Department of Computer Science

2. Simplifying fractions greatest common divisor Introduction to Programming© Dept. CS, UPC2

3. The largest square tile What is the largest square tile that can exactly cover a w  h rectangle? w = 60 h=24 44444444 90 tiles of side-length 4 Introduction to Programming© Dept. CS, UPC3

4. The largest square tile What is the largest square tile that can exactly cover a w  h rectangle? w = 60 h=24 66666666 90 tiles of side-length 4 40 tiles of side-length 6 Introduction to Programming© Dept. CS, UPC4

5. The largest square tile What is the largest square tile that can exactly cover a w  h rectangle? w = 60 h=24 90 tiles of side-length 4 40 tiles of side-length 6 10 tiles of side-length 12 12  12 Introduction to Programming© Dept. CS, UPC5

6. The largest square tile What is the largest square tile that can exactly cover a w  h rectangle? w = 60 h=24 12  12 60  24 = (5  12)  (2  12) = (5  2)  (12  12) gcd(60, 24) largest tile number of tiles Introduction to Programming© Dept. CS, UPC6

7. Euclid (300 B.C.) A fragment of Euclid’s Elements http://www.math.ubc.ca/~cass/Euclid/papyrus/tha.jpg Introduction to Programming© Dept. CS, UPC7

8. Euclidean algorithm for gcd Introduction to Programming© Dept. CS, UPC8

9. Euclidean algorithm for gcd Introduction to Programming© Dept. CS, UPC9 1071 462 147 21

10. Euclidean algorithm for gcd Properties: gcd(a, a) = a; gcd(a, 0) = a If a  b, then gcd(a, b) = gcd(a  b, b) Example: ab 11442 7242 3042 3012 1812 6 66 gcd (114, 42) Introduction to Programming© Dept. CS, UPC10

11. Euclidean algorithm for gcd // Pre: a > 0, b > 0 // Returns the greatest common divisor of a and b int gcd(int a, int b) { while (a != b) { if (a > b) a = a – b; else b = b – a; } return a; } Introduction to Programming© Dept. CS, UPC11

12. Euclidean algorithm for gcd iterationab 010,000,001154 19,999,847154 29,999,693154 39,999,539154 …… 64,934165154 64,93511154 64,93611143 64,93711132 64,93811121 64,93911110 …11… 64,9471122 64,94811 10000001 154 11 64935 Introduction to Programming© Dept. CS, UPC12

13. Faster Euclidean algorithm for gcd Properties: gcd(a, 0) = a If b  0 then gcd(a, b) = gcd(a%b, b) Example ab 10,000,001154 11154 110 Introduction to Programming© Dept. CS, UPC13

14. // Pre: a  0, b  0 // Returns the greatest common divisor of a and b int gcd(int a, int b) { while (a != 0 and b != 0) { if (a > b) a = a%b; else b = b%a; } return a + b; } Faster Euclidean algorithm for gcd Termination: a == 0 or b == 0 not Introduction to Programming© Dept. CS, UPC14

15. ab 42114 42 30 12 6 60 Faster Euclidean algorithm for gcd // Pre: a  0, b  0 // Returns the greatest common divisor of a and b int gcd(int a, int b) { while (b > 0) { int r = a%b; a = b; b = r; } return a; } yyxx yyx%yx%y > a b Introduction to Programming© Dept. CS, UPC15

16. Comparing algorithms for gcd int gcd(int a, int b) { while (b > 0) { int r = a%b; a = b; b = r; } return a; } int gcd(int a, int b) { while (a > 0 and b > 0) { if (a > b) a = a%b; else b = b%a; } return a + b; } Every iteration: 3 comparisons 1 mod operation Every iteration: 1 comparison 1 mod operation Introduction to Programming© Dept. CS, UPC16

17. Efficiency of the Euclidean algorithm How many iterations will Euclid’s algorithm need to calculate gcd(a,b) in the worst case (assume a > b)? – Subtraction version: a iterations (consider gcd(1000000, 1)) – Modulo version:  5  d(b) iterations, where d(b) is the number of digits of b (Gabriel Lamé, 1844) Introduction to Programming© Dept. CS, UPC17

18. Binary Euclidean algorithm Computers can perform  2 and /2 operations efficiently (217) 10 = (11011001) 2 (2172) 10 = (434) 10 = (110110010) 2 (217/2) 10 = (108) 10 = (1101100) 2 (217%2) 10 = 1 (least significant bit) Introduction to Programming© Dept. CS, UPC18

19. Binary Euclidean algorithm Assume a  b: abgcd(a,b) a0a even 2gcd(a/2, b/2) evenoddgcd(a/2, b) oddevengcd(a,b/2) odd gcd((a-b)/2, b) ab 13284 6642 3321 6 3 39 33 30 22 22 33 12 Introduction to Programming© Dept. CS, UPC19

20. // Pre: a  0, b  0 // Returns the greatest common divisor of a and b int gcd(int a, int b) { int r = 1; // Accumulates common powers of two while (a != 0 and b != 0) { if (a%2 == 0 and b%2 == 0) { a = a/2; b = b/2; r = r2; } else if (a%2 == 0) a = a/2; else if (b%2 == 0) b = b/2; else if (a > b) a = (a - b)/2; else b = (b – a)/2; } return (a + b)r; } Binary Euclidean algorithm Introduction to Programming© Dept. CS, UPC20

21. // Pre: a  0, b  0 // Returns the greatest common divisor of a and b int gcd(int a, int b) { if (a == 0 or b == 0) return a + b; int r = 1; // Accumulates common powers of two while (a%2 == 0 and b%2 == 0) { a = a/2; b = b/2; r = r2; } while (a != b) { if (a%2 == 0) a = a/2; else if (b%2 == 0) b = b/2; else if (a > b) a = (a - b)/2; else b = (b – a)/2; } return ar; } Binary Euclidean algorithm Introduction to Programming© Dept. CS, UPC21

22. Summary Euclid’s algorithm is very simple. It is widely used in some applications related to cryptography (e.g., electronic commerce). Euclid’s algorithm efficiently computes the gcd of very large numbers. Question: how would you compute the least common multiple of two numbers efficiently? Introduction to Programming© Dept. CS, UPC22