This document summarizes a seminar on algorithm verification and efficiency. It discusses verifying that algorithms are correct through mathematical proofs and analyzing computational efficiency by calculating the number of operations. Basic arithmetic algorithms like multiplication and finding the greatest common divisor are analyzed. The most efficient multiplication algorithm runs in O(log y) time while the naive approach is O(y). Euclid's algorithm for finding the greatest common divisor runs in O(log n) time.

1. POSCAT Seminar 2 :
Algorithm Verification & Efficiency
yougatup @ POSCAT
1

2. Topic
 Topic today
− Problem Solving Procedure
• Algorithm Verification
• Computational Efficiency
− Basic arithmetic
• Multiplication
• Power
• Greatest Common Divisor
− Primarity Testing
• Naïve approach
• Eratosthenes’ sieve
POSCAT Seminar 1-2
9 July 2014
yougatup
− Stable Matching
− Implementation
• Basic implementation
• Covering today’s topic

3. Problem Solving Procedure
 문제를 푸는 과정
− 알고리즘을 개발한 후에 이 알고리즘이 정확하다는 것을 증명
− 이 알고리즘이 충분히 빠른 알고리즘인지를 생각
− 괜찮은 알고리즘이면 위의 알고리즘을 오류 없이 구현
 정확성의 증명과 효율성
− 알고리즘의 정확성에 대한 수학적 증명
− 알고리즘의 시간이 얼마나 걸릴지 계산하는 과정
POSCAT Seminar 1-3
9 July 2014
yougatup

4. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥
𝑥 ∙ 𝑦 − 1 + 𝑥
𝑖𝑓 𝑦 = 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Verification : Is this algorithm true ?
Efficiency : How long does it takes ?
POSCAT Seminar 1-4
9 July 2014
yougatup

5. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥
𝑥 ∙ 𝑦 − 1 + 𝑥
𝑖𝑓 𝑦 = 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Verification : Is this algorithm true ?
POSCAT Seminar 1-5
9 July 2014
yougatup

6. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥
𝑥 ∙ 𝑦 − 1 + 𝑥
𝑖𝑓 𝑦 = 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Verification : Is this algorithm true ?
POSCAT Seminar 1-6
9 July 2014
yougatup
Proof : use Mathematical Induction
𝑥 ∙ 𝑦 = 𝑥 + 𝑥 + ⋯ + 𝑥

7. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥
𝑥 ∙ 𝑦 − 1 + 𝑥
𝑖𝑓 𝑦 = 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Efficiency : How long does it takes ?
POSCAT Seminar 1-7
9 July 2014
yougatup

8. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥
𝑥 ∙ 𝑦 − 1 + 𝑥
𝑖𝑓 𝑦 = 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Efficiency : How long does it takes ?
POSCAT Seminar 1-8
9 July 2014
yougatup
What is the unit ?

9. Computational Efficiency
 알고리즘의 대략적인 연산횟수를 계산
− 연산을 오래 하는 알고리즘이면 시간도 더 느릴 것이다
− 사칙연산, 비교연산 등 연산에 관계없이 모두 1번이라 생각
− 데이터 크기에 따라 연산 횟수가 달라짐  n에 대한 함수 필요
− 정확하게는 세기 어려움
몇 번의 연산이 필요한가 ?
POSCAT Seminar 1-9
9 July 2014
yougatup

10. Computational Efficiency
 알고리즘의 대략적인 연산횟수를 계산
− 연산을 오래 하는 알고리즘이면 시간도 더 느릴 것이다
− 사칙연산, 비교연산 등 연산에 관계없이 모두 1번이라 생각
− 데이터 크기에 따라 연산 횟수가 달라짐  n에 대한 함수 필요
− 정확하게는 세기 어려움
몇 번의 연산이 필요한가 ? 약 n번
POSCAT Seminar 1-10
9 July 2014
yougatup

11. Computational Efficiency
 알고리즘의 대략적인 연산횟수를 계산
− 연산을 오래 하는 알고리즘이면 시간도 더 느릴 것이다
− 사칙연산, 비교연산 등 연산에 관계없이 모두 1번이라 생각
− 데이터 크기에 따라 연산 횟수가 달라짐  n에 대한 함수 필요
− 정확하게는 세기 어려움
몇 번의 연산이 필요한가 ? O(𝒏)
POSCAT Seminar 1-11
9 July 2014
yougatup

12. Computational Efficiency
 Example
POSCAT Seminar 1-12
9 July 2014
yougatup

13. Computational Efficiency
 Example
POSCAT Seminar 1-13
9 July 2014
yougatup
O(𝑛2
+ 𝑛)

14. Computational Efficiency
 Example
POSCAT Seminar 1-14
9 July 2014
yougatup
O(𝒏 𝟐
)

15. Computational Efficiency
 Example
POSCAT Seminar 1-15
9 July 2014
yougatup

16. Computational Efficiency
 Example
POSCAT Seminar 1-16
9 July 2014
yougatup
O(3 ∙ { 𝑛 − 1 + 𝑛 − 2 + ⋯ + 1})
swap loop

17. Computational Efficiency
 Example
POSCAT Seminar 1-17
9 July 2014
yougatup
O(3 ∙ {
𝑛 ∙ (𝑛−1)
2
})

18. Computational Efficiency
 Example
POSCAT Seminar 1-18
9 July 2014
yougatup
O(3 ∙ {
𝑛2
2
})

19. Computational Efficiency
 Example
POSCAT Seminar 1-19
9 July 2014
yougatup
O(3𝑛2)

20. Computational Efficiency
 Example
POSCAT Seminar 1-20
9 July 2014
yougatup
O(𝒏 𝟐)

21. Computational Efficiency
 Example
POSCAT Seminar 1-21
9 July 2014
yougatup
O(𝒏 𝟐)
사실 대부분의 경우에는 최고차 항만 고려하면 됨

22. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥
𝑥 ∙ 𝑦 − 1 + 𝑥
𝑖𝑓 𝑦 = 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Efficiency : How long does it takes ?
POSCAT Seminar 1-22
9 July 2014
yougatup
What is the unit ?

23. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥
𝑥 ∙ 𝑦 − 1 + 𝑥
𝑖𝑓 𝑦 = 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Efficiency : How long does it takes ?
POSCAT Seminar 1-23
9 July 2014
yougatup
What is the unit ?
 The number of operations !

24. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥
𝑥 ∙ 𝑦 − 1 + 𝑥
𝑖𝑓 𝑦 = 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Efficiency : How long does it takes ?
POSCAT Seminar 1-24
9 July 2014
yougatup
So, what is the efficiency ?

25. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥
𝑥 ∙ 𝑦 − 1 + 𝑥
𝑖𝑓 𝑦 = 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Efficiency : How long does it takes ?
POSCAT Seminar 1-25
9 July 2014
yougatup
So, what is the efficiency ?

26. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥
𝑥 ∙ 𝑦 − 1 + 𝑥
𝑖𝑓 𝑦 = 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Efficiency : How long does it takes ?
POSCAT Seminar 1-26
9 July 2014
yougatup
So, what is the efficiency ?
 We need pseudo-code

27. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥
𝑥 ∙ 𝑦 − 1 + 𝑥
𝑖𝑓 𝑦 = 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Efficiency : How long does it takes ?
POSCAT Seminar 1-27
9 July 2014
yougatup

28. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥
𝑥 ∙ 𝑦 − 1 + 𝑥
𝑖𝑓 𝑦 = 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Efficiency : How long does it takes ?
POSCAT Seminar 1-28
9 July 2014
yougatup
O(y)

29. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥
𝑥 ∙ 𝑦 − 1 + 𝑥
𝑖𝑓 𝑦 = 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Efficiency : How long does it takes ?
POSCAT Seminar 1-29
9 July 2014
yougatup
O(y)
Done!

30. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥
𝑥 ∙ 𝑦 − 1 + 𝑥
𝑖𝑓 𝑦 = 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Efficiency : How long does it takes ?
POSCAT Seminar 1-30
9 July 2014
yougatup
O(y)
Done!
Is there another algorithm faster ?

31. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥 + (2 ∙ 𝑥 ∙
𝑦
2
)
2 ∙ 𝑥 ∙
𝑦
2
𝑖𝑓 𝑦 𝑖𝑠 𝑜𝑑𝑑
𝑖𝑓 𝑦 𝑖𝑠 𝑒𝑣𝑒𝑛
Verification : Is this algorithm true ?
Efficiency : How long does it takes ?
POSCAT Seminar 1-31
9 July 2014
yougatup

32. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥 + (2 ∙ 𝑥 ∙
𝑦
2
)
2 ∙ 𝑥 ∙
𝑦
2
𝑖𝑓 𝑦 𝑖𝑠 𝑜𝑑𝑑
𝑖𝑓 𝑦 𝑖𝑠 𝑒𝑣𝑒𝑛
Verification : Is this algorithm true ? Mathematical Induc.
Efficiency : How long does it takes ? O(log 𝑦)
POSCAT Seminar 1-32
9 July 2014
yougatup

33. Basic Arithmetic
 Multiplication
𝑥 ∙ 𝑦 =
𝑥 + (2 ∙ 𝑥 ∙
𝑦
2
)
2 ∙ 𝑥 ∙
𝑦
2
𝑖𝑓 𝑦 𝑖𝑠 𝑜𝑑𝑑
𝑖𝑓 𝑦 𝑖𝑠 𝑒𝑣𝑒𝑛
Verification : Is this algorithm true ? Mathematical Induc.
Efficiency : How long does it takes ? O(log 𝑦)
POSCAT Seminar 1-33
9 July 2014
yougatup
그냥 x*y 하면 될걸 무슨 이런 헛짓을…

34. Basic Arithmetic
 Power
Derive the efficient algorithm to calculate 𝑥 𝑦
Find correct algorithm which takes O(log 𝑦)
Also, give me the pseudo-code
POSCAT Seminar 1-34
9 July 2014
yougatup

35. Basic Arithmetic
 Euclid’s Rule
gcd 𝑥, 𝑦 = gcd(𝑥 𝑚𝑜𝑑 𝑦, 𝑦)
Claim. G = gcd 𝑥, 𝑦 = gcd 𝑥 − 𝑦, 𝑦 𝑖𝑓 𝑥 ≥ 𝑦
Proof.
POSCAT Seminar 1-35
9 July 2014
yougatup

36. Basic Arithmetic
 Euclid’s Rule
gcd 𝑥, 𝑦 = gcd(𝑥 𝑚𝑜𝑑 𝑦, 𝑦)
Claim. G = gcd 𝑥, 𝑦 = gcd 𝑥 − 𝑦, 𝑦 𝑖𝑓 𝑥 ≥ 𝑦
Proof. Any divisor of both x and y also divides x-y
Therefore, gcd 𝑥, 𝑦 is the divisor or gcd(𝑥 − 𝑦, 𝑦)
∴ gcd 𝑥, 𝑦 ≤ gcd(𝑥 − 𝑦, 𝑦)
POSCAT Seminar 1-36
9 July 2014
yougatup

37. Basic Arithmetic
 Euclid’s Rule
gcd 𝑥, 𝑦 = gcd(𝑥 𝑚𝑜𝑑 𝑦, 𝑦)
Claim. G = gcd 𝑥, 𝑦 = gcd 𝑥 − 𝑦, 𝑦 𝑖𝑓 𝑥 ≥ 𝑦
Proof. Also, any divisor of both 𝑥 − 𝑦 and 𝑦 also divides x
Therefore, gcd 𝑥, 𝑦 ≥ gcd(𝑥 − 𝑦, 𝑦)
∴ gcd 𝑥, 𝑦 = gcd(𝑥 − 𝑦, 𝑦)
POSCAT Seminar 1-37
9 July 2014
yougatup

38. Basic Arithmetic
 Euclid’s Rule
gcd 𝑥, 𝑦 = gcd(𝑥 𝑚𝑜𝑑 𝑦, 𝑦)
Then G = gcd 𝑥, 𝑦 = gcd(𝑥 𝑚𝑜𝑑 𝑦 , 𝑦) 𝑖𝑓 𝑥 ≥ 𝑦
POSCAT Seminar 1-38
9 July 2014
yougatup

39. Basic Arithmetic
 Euclid’s Rule
gcd 𝑥, 𝑦 = gcd(𝑥 𝑚𝑜𝑑 𝑦, 𝑦)
gcd 248, 32
POSCAT Seminar 1-39
9 July 2014
yougatup

40. Basic Arithmetic
 Euclid’s Rule
gcd 𝑥, 𝑦 = gcd(𝑥 𝑚𝑜𝑑 𝑦, 𝑦)
gcd 248, 32
gcd(24, 32)
POSCAT Seminar 1-40
9 July 2014
yougatup

41. Basic Arithmetic
 Euclid’s Rule
gcd 𝑥, 𝑦 = gcd(𝑥 𝑚𝑜𝑑 𝑦, 𝑦)
gcd 248, 32
gcd 24, 32
gcd(32, 24)
POSCAT Seminar 1-41
9 July 2014
yougatup

42. Basic Arithmetic
 Euclid’s Rule
gcd 𝑥, 𝑦 = gcd(𝑥 𝑚𝑜𝑑 𝑦, 𝑦)
gcd 248, 32
gcd 24, 32
gcd 32, 24
gcd(8, 24)
POSCAT Seminar 1-42
9 July 2014
yougatup

43. Basic Arithmetic
 Euclid’s Rule
gcd 𝑥, 𝑦 = gcd(𝑥 𝑚𝑜𝑑 𝑦, 𝑦)
gcd 248, 32
gcd 24, 32
gcd 32, 24
gcd 8, 24
gcd(24, 8)
POSCAT Seminar 1-43
9 July 2014
yougatup

44. Basic Arithmetic
 Euclid’s Rule
gcd 𝑥, 𝑦 = gcd(𝑥 𝑚𝑜𝑑 𝑦, 𝑦)
gcd 248, 32
gcd 24, 32
gcd 32, 24
gcd 8, 24
gcd(24, 8)
POSCAT Seminar 1-44
9 July 2014
yougatup
8 divides 24 ! Therefore, gcd 24, 8 = 8

45. Basic Arithmetic
 Euclid’s Rule
gcd 𝑥, 𝑦 = gcd(𝑥 𝑚𝑜𝑑 𝑦, 𝑦)
How long does it takes ?
POSCAT Seminar 1-45
9 July 2014
yougatup

46. Basic Arithmetic
 Euclid’s Rule
gcd 𝑥, 𝑦 = gcd(𝑥 𝑚𝑜𝑑 𝑦, 𝑦)
How long does it takes ?
Claim. If 𝑎 ≥ 𝑏 then 𝑎 𝑚𝑜𝑑 𝑏 < 𝑎/2
Consider the cases 𝑏 ≤ 𝑎/2 and 𝑏 > 𝑎/2
POSCAT Seminar 1-46
9 July 2014
yougatup

47. Basic Arithmetic
 Euclid’s Rule
gcd 𝑥, 𝑦 = gcd(𝑥 𝑚𝑜𝑑 𝑦, 𝑦)
How long does it takes ?
Claim. If 𝑎 ≥ 𝑏 then 𝑎 𝑚𝑜𝑑 𝑏 ≤ 𝑎/2
Consider the cases 𝑏 ≤ 𝑎/2 and 𝑏 > 𝑎/2
If 𝑏 ≤ 𝑎/2 then 𝑎 𝑚𝑜𝑑 𝑏 < 𝑏 ≤ 𝑎/2
𝑏 > 𝑎/2 then 𝑎 𝑚𝑜𝑑 𝑏 = 𝑎 − 𝑏 < 𝑎/2
POSCAT Seminar 1-47
9 July 2014
yougatup

48. Basic Arithmetic
 Euclid’s Rule
gcd 𝑥, 𝑦 = gcd(𝑥 𝑚𝑜𝑑 𝑦, 𝑦)
How long does it takes ?
After two consecutive round, a and b are at least half
 O(𝐥𝐨𝐠 𝒏)
POSCAT Seminar 1-48
9 July 2014
yougatup

49. Primarity Testing
 Problem Definition
Given p, determine whether p is prime number or not
Given a interval [s, e], show all of the prime numbers in the interval
 Approach
− Naïve approach
− Eratosthenes’ seive
POSCAT Seminar 1-49
9 July 2014
yougatup

50. Primarity Testing
Given p, determine whether p is prime number or not
 Naïve approach
For all 2 ≤ 𝑖 ≤ 𝑝 − 1, test whether 𝑖 divides p
If there is such 𝑖, then p is NOT prime number
Otherwise, p is prime number
Verification : Is this algorithm true ?
Efficiency : How long does it takes ?
POSCAT Seminar 1-50
9 July 2014
yougatup

51. Primarity Testing
Given p, determine whether p is prime number or not
 Naïve approach
For all 2 ≤ 𝑖 ≤ 𝑝 − 1, test whether 𝑖 divides p
If there is such 𝑖, then p is NOT prime number
Otherwise, p is prime number
Verification : Is this algorithm true ? By definition of prime number
Efficiency : How long does it takes ? O(𝑝)
POSCAT Seminar 1-51
9 July 2014
yougatup

52. Primarity Testing
Given p, determine whether p is prime number or not
 Naïve approach
For all 2 ≤ 𝑖 ≤ 𝑝 − 1, test whether 𝑖 divides p
If there is such 𝑖, then p is NOT prime number
Otherwise, p is prime number
Verification : Is this algorithm true ? By definition of prime number
Efficiency : How long does it takes ? O(𝑝)
POSCAT Seminar 1-52
9 July 2014
yougatup
Is there another algorithm faster ?

53. Primarity Testing
Given p, determine whether p is prime number or not
 We don’t have to consider all such 𝑖 !
It is sufficient to consider 2 ≤ 𝑖 ≤ √𝑝, Why ?
Verification : Is this algorithm true ? By definition of prime number
Efficiency : How long does it takes ? O(𝑝)
POSCAT Seminar 1-53
9 July 2014
yougatup

54. Primarity Testing
Given p, determine whether p is prime number or not
 We don’t have to consider all such 𝑖 !
It is sufficient to consider 2 ≤ 𝑖 ≤ √𝑝, Why ?
If p is tested as prime number when 𝑖 > √𝑝, It must be tested before !
Verification : Is this algorithm true ? By definition of prime number
Efficiency : How long does it takes ? O(𝑝)
POSCAT Seminar 1-54
9 July 2014
yougatup

55. Primarity Testing
Given p, determine whether p is prime number or not
 We don’t have to consider all such 𝑖 !
It is sufficient to consider 2 ≤ 𝑖 ≤ √𝑝, Why ?
If p is tested as prime number when 𝑖 > √𝑝, It must be tested before !
Verification : Is this algorithm true ? By definition of prime number
Efficiency : How long does it takes ? O(√𝒑)
POSCAT Seminar 1-55
9 July 2014
yougatup

56. Given a interval [s, e], show all of the prime numbers in the interval
 Naïve approach
For all numbers 𝑥 in the interval, test it
Verification : Is this algorithm true ? Trivial
Efficiency : How long does it takes ? O(𝒏√𝒏)
POSCAT Seminar 1-56
9 July 2014
yougatup
Primarity Testing

57. Given a interval [s, e], show all of the prime numbers in the interval
 Naïve approach
For all numbers 𝑥 in the interval, test it
Verification : Is this algorithm true ? Trivial
Efficiency : How long does it takes ? O(𝒏√𝒏)
POSCAT Seminar 1-57
9 July 2014
yougatup
Is there another algorithm faster ?
Primarity Testing

58. Eratosthenes’ seive
 Simple Algorithm
1. ‘1’ is not a prime number by definition
2. Pick the smallest value among we have, which is 2
3. Erase all the numbers divided by 2 ( except 2 )
4. Pick the smallest value among we have, which is 3
5. Erase all the numbers divided by 3 ( except 3 )
6. Repeat this procedure
POSCAT Seminar 1-58
9 July 2014
yougatup

59. Eratosthenes’ seive
 Simple Algorithm
POSCAT Seminar 1-59
9 July 2014
yougatup
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60. Eratosthenes’ seive
 Simple Algorithm
‘1’ is not a prime number by definition
POSCAT Seminar 1-60
9 July 2014
yougatup
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61. Eratosthenes’ seive
 Simple Algorithm
Pick the smallest value among we have, which is 2
POSCAT Seminar 1-61
9 July 2014
yougatup
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

62. Eratosthenes’ seive
 Simple Algorithm
Erase all the numbers divided by 2 ( except 2 )
POSCAT Seminar 1-62
9 July 2014
yougatup
2 3 5 7 9 11 13 15 17

63. Eratosthenes’ seive
 Simple Algorithm
Pick the smallest value among we have, which is 3
POSCAT Seminar 1-63
9 July 2014
yougatup
2 3 5 7 9 11 13 15 17

64. Eratosthenes’ seive
 Simple Algorithm
Erase all the numbers divided by 3 ( except 3 )
POSCAT Seminar 1-64
9 July 2014
yougatup
2 3 5 7 11 13 17

65. Eratosthenes’ seive
 Simple Algorithm
Pick the smallest value among we have, which is 5
POSCAT Seminar 1-65
9 July 2014
yougatup
2 3 5 7 11 13 17

66. Eratosthenes’ seive
 Simple Algorithm
Erase all the numbers divided by 5 ( except 5 )
POSCAT Seminar 1-66
9 July 2014
yougatup
2 3 5 7 11 13 17

67. Eratosthenes’ seive
 Simple Algorithm
Repeat this procedure
POSCAT Seminar 1-67
9 July 2014
yougatup
2 3 5 7 11 13 17

68. Eratosthenes’ seive
 Analysis
Verification : Is this algorithm true ? It looks like …
Efficiency : How long does it takes ?
POSCAT Seminar 1-68
9 July 2014
yougatup

69. Eratosthenes’ seive
 Analysis
Verification : Is this algorithm true ? It looks like …
Efficiency : How long does it takes ?
Suppose that we want to find all the prime numbers in [1, n]
Let’s focus on a integer 𝑥 in the interval
Then 𝑥 is “clicked” as many as the number of divisor of 𝑥
It must be bounded by O(log 𝑥) for each 𝑥
Therefore, It takes O(𝒏 𝐥𝐨𝐠 𝒏)
POSCAT Seminar 1-69
9 July 2014
yougatup
Much better than O(𝑛√𝑛) !

70. Bipartite Matching
 Problem Definition (weak)
Maximize the number of matched pair on bipartite graph
POSCAT Seminar 1-70
9 July 2014
yougatup
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3
4
1
2
3
4

71. Bipartite Matching
 Problem Definition (weak)
Maximize the number of matched pair on bipartite graph
POSCAT Seminar 1-71
9 July 2014
yougatup
1
2
3
4
1
2
3
4

72. Stable Matching
 Problem Definition (weak)
N명의 남자와 여자가 미팅을 하기 위해서 만남
남자와 여자 모두 원하는 상대방의 우선순위를 정함
서로 짝이 아닌 두 남녀가 자신의 짝보다 상대방을 더 선호하면 ?
POSCAT Seminar 1-72
9 July 2014
yougatup
1
2
3
1
2
3
난 1번보다 3번이 더 좋은데…난 1번보다 3번이 더 좋은데…
男 女

73. Stable Matching
 Problem Definition (weak)
N명의 남자와 여자가 미팅을 하기 위해서 만남
남자와 여자 모두 원하는 상대방의 우선순위를 정함
서로 짝이 아닌 두 남녀가 자신의 짝보다 상대방을 더 선호하면 ?
POSCAT Seminar 1-73
9 July 2014
yougatup
1
2
3
1
2
3
男 女
도망ㅋ
헐
헐
??

74. Stable Matching
 Problem Definition (weak)
N명의 남자와 여자가 미팅을 하기 위해서 만남
남자와 여자 모두 원하는 상대방의 우선순위를 정함
서로 짝이 아닌 두 남녀가 자신의 짝보다 상대방을 더 선호하면 ?
이런 불상사가 나지 않도록 짝을 짓는 것이 가능한가?
POSCAT Seminar 1-74
9 July 2014
yougatup

75. Stable Matching
 Problem Definition (weak)
N명의 남자와 여자가 미팅을 하기 위해서 만남
남자와 여자 모두 원하는 상대방의 우선순위를 정함
서로 짝이 아닌 두 남녀가 자신의 짝보다 상대방을 더 선호하면 ?
이런 불상사가 나지 않도록 짝을 짓는 것이 가능한가?
 Yes ! 가능하다는 것을 해를 구하는 알고리즘으로 증명
2012년 노벨 경제학상을 받게 한 알고리즘 (Gale-Shapely Algorithm)
POSCAT Seminar 1-75
9 July 2014
yougatup

76. Stable Matching
 Problem Definition (weak)
1. 처음에 남성이 모두 가장 선호하는 여성에게 프로포즈를 함
2. 여성이 그 중 가장 마음에 드는 남성을 고르고 나머지는 퇴짜
3. 퇴짜맞은 남성은 그 다음으로 선호하는 여성이 파트너가 있던
말던 프로포즈를 함
4. 여성은 현재 자신의 파트너보다 프로포즈 한 남성이 더 마음에
든다면 자신의 파트너에게 퇴짜를 놓음
5. 이 과정을 계속해서 반복 !
POSCAT Seminar 1-76
9 July 2014
yougatup

77. Stable Matching
 Correctness
− Is it terminate ? // 언젠간 종료하는가 ?
− Is there any lonely man or woman ? // 짝 없는 사람이 존재하는가 ?
− Is it stable ? // 불상사가 없는가 ?
POSCAT Seminar 1-77
9 July 2014
yougatup

78. Stable Matching
 Correctness
− Is it terminate ? // 언젠간 종료하는가 ?
POSCAT Seminar 1-78
9 July 2014
yougatup

79. Stable Matching
 Correctness
− Is it terminate ? // 언젠간 종료하는가 ?
각 남성은 많아야 n명에게 프로포즈를 하고, 같은 사람에게 두 번 하지 않는다.
따라서 이 과정은 언젠간 종료된다.
POSCAT Seminar 1-79
9 July 2014
yougatup

80. Stable Matching
 Correctness
− Is there any lonely man or woman ? // 짝 없는 사람이 존재하는가 ?
POSCAT Seminar 1-80
9 July 2014
yougatup

81. Stable Matching
 Correctness
− Is there any lonely man or woman ? // 짝 없는 사람이 존재하는가 ?
여성이 짝이 없다고 가정하자. 그렇다면 한 명도 프로포즈한 남성이 없다.
그러면 모든 남성이 n-1명의 여성과 짝을 이룬다는 것이므로 모순.
남성이 짝이 없다고 가정하자. 그렇다면 모두 퇴짜를 맞았다는 것이다.
그러면 모든 여성이 n-1명의 남성과 짝을 이룬다는 것이므로 모순.
POSCAT Seminar 1-81
9 July 2014
yougatup

82. Stable Matching
 Correctness
− Is it stable ? // 불상사가 없는가 ?
POSCAT Seminar 1-82
9 July 2014
yougatup

83. Stable Matching
 Correctness
− Is it stable ? // 불상사가 없는가 ?
불상사가 생긴 남성과 여성이 존재한다고 가정하자.
편의상 현재 파트너를 X, 불상사가 생기는 파트너를 Y 이라고 하자.
남성은 Y를 X보다 더 좋아한다. 그러면 X보다 Y에게 더 먼저 프로포즈 했을 것.
여성은 현재 이 남성과 파트너가 아니므로 퇴짜를 놓았다는 뜻이다.
하지만 여성 역시 현재 자신의 파트너보다 이 남성이 더 좋은 상황이다.
더 좋은 사람을 퇴짜 놓은 상황이므로 모순.
POSCAT Seminar 1-83
9 July 2014
yougatup

84. Question ?
 To-do List
− 코딩 합시다 
POSCAT Seminar 1-84
9 July 2014
yougatup