1. Computational Processes II
Mathematics for Computer Science MIT 6.042J/18.062J
Computational Processes II
Copyright © Albert Meyer, 2002.
Prof. Albert Meyer & Dr. Radhika Nagpal

2. Computational Processes
Last lecture:
Invariants & Partial Correctness.

3. Computational Processes
Today:
Termination, Derived Variables,
Getting Married.

4. A derived variable, v, is a function giving a value to each state.
Derived variables
A derived variable, v, is a function
giving a value to each state.

5. A derived variable, v, is a function giving a value to each state.
Derived variables
A derived variable, v, is a function
giving a value to each state.
Robot example:
Sum of coordinates, v ::= x + y.

6. A derived variable, v, is a function giving a value to each state.
Derived variables
A derived variable, v, is a function
giving a value to each state.
Robot example:
Sum of coordinates, v ::= x + y.
Parity of x + y, w ::= v mod 2.

7. Derived variables
For GCD, already have
variables x, y.

8. Proof of GCD termination: variable y is strictly decreasing and
Derived variables
Proof of GCD termination:
variable y is strictly decreasing and
natural number valued.

9. Termination follows by least number principle:
Derived variables
Termination follows by least number
principle:
y must take a least value, and thus
the algorithm is stuck.

10. Strictly Decreasing Variable16 12 8 4
State

11. Strictly Decreasing Variable16 12 8 4
Goes down at
every step
State

12. Weakly Decreasing Variable16 12 8 4
Down or constant
after each step
State

13. Weakly Decreasing Variable
(We used to call weakly
decreasing variables
“nonincreasing” variables.)

14. Weakly Decreasing Variable
OK terminology but remember:
nonincreasing is
NOT SAME as “not increasing.”

15. In-class Problem
Problem 1

16. Stable Marriage
A Marriage Problem
Boys
Girls
A B C D E

17. Preferences: Boys Girls 1: CBEAD A : 35214 2 : ABECD B : 52143
Stable Marriage
Preferences:
Boys Girls
1: CBEAD A : 35214
2 : ABECD B : 52143
3 : DCBAE C : 43512
4 : ACDBE D : 12345
5 : ABDEC E :

18. Try “greedy” strategy for boys Preferences 1: CBEAD 2 : ABECD
Stable Marriage
1: CBEAD
2 : ABECD
3 : DCBAE
4 : ACDBE
5 : ABDEC
Preferences
Try “greedy”
strategy for boys

19. Marry Boy 1 with Girl C Preferences (his 1st choice) 1: CBEAD
Stable Marriage
Marry Boy 1 with Girl C
(his 1st choice)
Preferences
1: CBEAD
2 : ABECD
3 : DCBAE
4 : ACDBE
5 : ABDEC 1 C

20. Marry Boy 1 with Girl C Preferences (his 1st choice) 1: CBEAD
Stable Marriage
Marry Boy 1 with Girl C
(his 1st choice)
Preferences
1: CBEAD
2 : ABECD
3 : DCBAE
4 : ACDBE
5 : ABDEC 1 C

21. Marry Boy 1 with Girl C Preferences (his 1st choice) 2 : ABE D
Stable Marriage
Marry Boy 1 with Girl C
(his 1st choice)
Preferences
2 : ABE D
3 : D BAE
4 : A DBE
5 : ABDE 1 C

22. Marry Boy 1 with Girl C Preferences (his 1st choice) 2 : ABED 3 : DBAE
Stable Marriage
Marry Boy 1 with Girl C
(his 1st choice)
Preferences
2 : ABED
3 : DBAE
4 : ADBE
5 : ABDE 1 C

23. (best remaining choice) Preferences
Stable Marriage
Next:
Marry Boy 2 with Girl A:
(best remaining choice)
Preferences
2 : ABED
3 : DBAE
4 : ADBE
5 : ABDE 1 C 2 A

24. (best remaining choice) Preferences
Stable Marriage
Next:
Marry Boy 2 with Girl A:
(best remaining choice)
Preferences
2 : ABED
3 : DBAE
4 : ADBE
5 : ABDE 1 C 2 A

25. Final “boy greedy” marriages
Stable Marriage
Final “boy greedy” marriages
1 C
3 D
2 A
4 B
5 E

26. Stable Marriage
Trouble! 1 C 4 B

27. Boy 4 likes Girl C better than his wife.
Stable Marriage
Boy 4 likes Girl C better than his wife. 1 C 4 B

28. Stable Marriage
and vice-versa 1 C 4 B

29. Stable Marriage
Rogue Couple 1 C 4 B

30. Stable Marriage Problem: Marry everyone without any rogue couples!

31. Stable Marriage
Do it!
PROBLEM 2

32. Stable Marriage
More than a puzzle:

33. More than a puzzle: College Admissions Stable Marriage
(original Gale & Shapley paper, 1962)

34. Matching Hospitals & Residents.
Stable Marriage
More than a puzzle:
College Admissions
Matching Hospitals & Residents.
(original Gale & Shapley paper, 1962)

35. Matching Hospitals & Residents. Matching Dance Partners.
Stable Marriage
More than a puzzle:
College Admissions
Matching Hospitals & Residents.
Matching Dance Partners.
(original Gale & Shapley paper, 1962)

36. Stable Marriage

37. Stable Marriage
The Mating Algorithm

38. Stable Marriage
The Mating Algorithm:
day by day

39. Morning: boy serenades favorite girl
Stable Marriage
Morning: boy serenades favorite girl
Ted
Alice
Bob

40. Morning: boy serenades favorite girl
Stable Marriage
Morning: boy serenades favorite girl
Afternoon: girl rejects all but favorite
Ted
Alice
Bob

41. … … Morning: boy serenades favorite girl
Stable Marriage
Morning: boy serenades favorite girl
Afternoon: girl rejects all but favorite
Evening: rejected boy writes off girl …
Alice …
Ted

42. … … Morning: boy serenades favorite girl
Stable Marriage
Morning: boy serenades favorite girl
Afternoon: girl rejects all but favorite
Evening: rejected boy writes off girl …
Alice …
Ted

43. Stop when no rejects. favorite suitor. Girl marries her
Stable Marriage
Stop when no rejects.
Girl marries her
favorite suitor.

44. In-class Problem
Problem 3

45. Stable Marriage
Partial Correctness:

46. Stable Marriage
Partial Correctness:
Everyone is married.

47. Partial Correctness: Everyone is married. Marriages are stable.
Stable Marriage
Partial Correctness:
Everyone is married.
Marriages are stable.

48. Partial Correctness: Termination: Everyone is married.
Stable Marriage
Partial Correctness:
Everyone is married.
Marriages are stable.
Termination:
there exists a Wedding Day.

49. State q: Each boy’s set of eligible girls
Stable Marriage
Model as State Machine
State q: Each boy’s set of eligible girls
q(Bob)={Carole, Alice, …}

50. Derived Variable: serenading
Stable Marriage
Derived Variable: serenading
Bob’s favorite eligible girl.

51. Derived Variable: suitors
Stable Marriage
Derived Variable: suitors
all boys serenading Alice.

52. Derived Variable: favorite
Stable Marriage
Derived Variable: favorite
Carole’s preferred suitor.

53. Different girls have different favorites,
Stable Marriage
Different girls have different favorites,
because boys serenade one girl at
a time.

54. Lemma: A girl’s favorite tomorrow
Stable Marriage
Lemma: A girl’s favorite tomorrow
will be at least as desirable as today’s.

55. Lemma: A girl’s favorite tomorrow
Stable Marriage
Lemma: A girl’s favorite tomorrow
will be at least as desirable as today’s.
favorite(G) is weakly increasing
for each G.

56. Lemma: A boy’s love tomorrow will be no more desirable than today’s.
Stable Marriage
Lemma: A boy’s love tomorrow
will be no more desirable than today’s.
serenading(B) is weakly decreasing
for each B.

57. Derived Variable: total-girls-names
Stable Marriage
Derived Variable: total-girls-names
::= total number of names not crossed
off boy’s lists

58. Derived Variable: total-girls-names
Stable Marriage
Derived Variable: total-girls-names
::= total number of names not crossed
off boy’s lists
Strictly decreasing & N-valued.

59. So  Wedding Day. Derived Variable: total-girls-names
Stable Marriage
Derived Variable: total-girls-names
::= total number of names not crossed
off boy’s lists
Strictly decreasing & N-valued.
So  Wedding Day.

60. Mating Invariant If G has rejected B, then she has
Stable Marriage
Mating Invariant
If G has rejected B, then she has
a better current favorite.

61. Mating Invariant If G has rejected B, then she has
Stable Marriage
Mating Invariant
If G has rejected B, then she has
a better current favorite.
Proof: favorite(G) is weakly increasing.

62. Everyone is Married on the Wedding Day
Stable Marriage
Everyone is Married on the Wedding Day
Proof by contradiction:

63. Everyone is Married on the Wedding Day
Stable Marriage
Everyone is Married on the Wedding Day
Proof by contradiction:
If B is not married, can’t be serenading, so list is empty.

64. Everyone is Married on the Wedding Day
Stable Marriage
Everyone is Married on the Wedding Day
Proof by contradiction:
If B is not married, can’t be serenading, so list is empty. By Invariant all girls have favorites better than B.

65. Everyone is Married on the Wedding Day
Stable Marriage
Everyone is Married on the Wedding Day
Proof by contradiction:
If B is not married, can’t be serenading, so list is empty. By Invariant all girls have favorites better than B.
So all girls are married.

66. Everyone is Married on the Wedding Day
Stable Marriage
Everyone is Married on the Wedding Day
Proof by contradiction:
If B is not married, can’t be serenading, so list is empty. By Invariant all girls have favorites better than B.
So all girls are married.
So all boys are married.

67. Stable Marriage
Stability:
Similar easy argument using
the Invariant.

68. In-class Problem
Problem 4

69. Who does better, boys or girls?
Stable Marriage
Who does better, boys or girls?
Girls’ suitors get better, and boy’s sweethearts get worse, so girls do better?

70. Who does better, boys or girls?
Stable Marriage
Who does better, boys or girls?
Girls’ suitors get better, and boy’s sweethearts get worse, so girls do better? No!

71. Mating Algorithm is Optimal for all Boys at once.
Stable Marriage
Mating Algorithm is Optimal for
all Boys at once.

72. Mating Algorithm is Optimal for all Boys at once. Pessimal for all
Stable Marriage
Mating Algorithm is Optimal for
all Boys at once. Pessimal for all
Girls.

73. More questions, rich theory:
Stable Marriage
More questions, rich theory:
Other stable marriages possible?
Can be many.
Can a boy do better by lying? – No!
Can a girl do better by lying – Yes!