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Greatest Common Divisor
Jordi CortadellaDepartment of Computer Science
Simplifying fractions
(114, 42) = 6
(19, 7) = 1
greatestcommondivisor
Introduction to Programming © Dept. CS, UPC 2
The largest square tile
What is the largest square tile that can exactly cover a wh rectangle?
w = 60
h=24
44
90 tiles of side-length 4
Introduction to Programming © Dept. CS, UPC 3
The largest square tile
What is the largest square tile that can exactly cover a wh rectangle?
w = 60
h=24
66
90 tiles of side-length 440 tiles of side-length 6
Introduction to Programming © Dept. CS, UPC 4
The largest square tile
What is the largest square tile that can exactly cover a wh rectangle?
w = 60
h=24
90 tiles of side-length 440 tiles of side-length 610 tiles of side-length 12
1212
Introduction to Programming © Dept. CS, UPC 5
The largest square tile
What is the largest square tile that can exactly cover a wh rectangle?
w = 60
h=24
1212
60 24 = (5 12) (2 12) = (5 2) (12 12)
gcd(60, 24)largest
tilenumberof tiles
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Euclid (300 B.C.)
A fragment of Euclid’s Elementshttp://www.math.ubc.ca/~cass/Euclid/papyrus/tha.jpg
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Euclidean algorithm for gcdObservation: if d divides 114 and 42, it also divides 114 42
Therefore, all common divisors of 114 and 42are also divisors of 72.
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Euclidean algorithm for gcd
Introduction to Programming © Dept. CS, UPC 9
1071
462
147
21
Euclidean algorithm for gcd• Properties:
gcd(a, a) = a; gcd(a, 0) = aIf a b, then gcd(a, b) = gcd(a b, b)
• Example: a b114 42
72 4230 4230 1218 12
6 126 6
gcd (114, 42)
Introduction to Programming © Dept. CS, UPC 10
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, UPC 11
Euclidean algorithm for gcditeration a b
0 10,000,001 154
1 9,999,847 154
2 9,999,693 154
3 9,999,539 154
… … 154
64,934 165 154
64,935 11 154
64,936 11 143
64,937 11 132
64,938 11 121
64,939 11 110
… 11 …
64,947 11 22
64,948 11 11
10000001 154
11 64935
Introduction to Programming © Dept. CS, UPC 12
Faster Euclidean algorithm for gcd• Properties:
gcd(a, 0) = a If b 0 then gcd(a, b) = gcd(a%b, b)
• Example a b
10,000,001 15411 15411 0
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// 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, UPC 14
a b42 114
114 4242 3030 1212 6
6 0
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;}
yx
y x%y>
a b
Introduction to Programming © Dept. CS, UPC 15
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
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Efficiency of the Euclidean algorithmHow 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: 5d(b) iterations,where d(b) is the number of digits of b(Gabriel Lamé, 1844)
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Binary Euclidean algorithmComputers can perform 2 and /2 operations efficiently
(217)10 = (11011001)2
(2172)10 = (434)10 = (110110010)2
(217/2)10 = (108)10 = (1101100)2
(217%2)10 = 1 (least significant bit)
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Binary Euclidean algorithmAssume a b:
a b gcd(a,b)
a 0 a
even even 2gcd(a/2, b/2)even odd gcd(a/2, b)
odd even gcd(a,b/2)
odd odd gcd((a-b)/2, b)
a b132 84
66 4233 21
6 213 213 93 33 0
2
2
3
12
Introduction to Programming © Dept. CS, UPC 19
// 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 = r2; } 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, UPC 20
// 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 = r2; }
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 ar;}
Binary Euclidean algorithm
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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 thegcd of very large numbers.
• Question: how would you compute the least common multiple of two numbers efficiently?
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