Mathematics for Computer Science MIT 6 042 J18

Mathematics for Computer Science MIT 6. 042 J/18. 062 J Computational Processes II Copyright © Albert Meyer, 2002. Prof. Albert Meyer & Dr. Radhika Nagpal March 6, 2002 1

Computational Processes Last lecture: Invariants & Partial Correctness. March 6, 2002 2

Computational Processes Today: Termination, Derived Variables, Getting Married. March 6, 2002 3

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. March 6, 2002 4

Derived variables For GCD, already have variables x, y. March 6, 2002 5

Derived variables Proof of GCD termination: variable y is strictly decreasing and natural number valued. March 6, 2002 6

Derived variables Termination follows by least number principle: y must take a least value, and thus the algorithm is stuck. March 6, 2002 7

Strictly Decreasing Variable 16 12 N Goes down at every step 8 4 0 State March 6, 2002 8

Weakly Decreasing Variable 16 12 N Down or constant after each step 8 4 0 State March 6, 2002 9

Weakly Decreasing Variable (We used to call weakly decreasing variables “nonincreasing” variables. ) March 6, 2002 10

Weakly Decreasing Variable OK terminology but remember: nonincreasing is NOT SAME as “not increasing. ” March 6, 2002 11

In-class Problem 1 March 6, 2002 12

Stable Marriage A Marriage Problem 1 2 3 4 5 Boys Girls A B C D E March 6, 2002 13

Stable Marriage Preferences: Boys 1: CBEAD 2 : ABECD 3 : DCBAE 4 : ACDBE 5 : ABDEC Girls A : 35214 B : 52143 C : 43512 D : 12345 E : 23415 March 6, 2002 14

Stable Marriage Preferences 1: CBEAD 2 : ABECD 3 : DCBAE 4 : ACDBE 5 : ABDEC Try “greedy” strategy for boys March 6, 2002 15

Stable Marriage Preferences 1: CBEAD 2 : ABECD 3 : DCBAE 4 : ACDBE 5 : ABDEC Marry Boy 1 with Girl C (his 1 st choice) 1 C March 6, 2002 16

Stable Marriage Preferences 2 : ABE D 3 : D BAE 4 : A DBE 5 : ABDE Marry Boy 1 with Girl C (his 1 st choice) 1 C March 6, 2002 17

Stable Marriage Preferences 2 : ABED 3 : DBAE 4 : ADBE 5 : ABDE Marry Boy 1 with Girl C (his 1 st choice) 1 C March 6, 2002 18

Stable Marriage Preferences 1 C 2 : ABED 3 : DBAE 4 : ADBE 5 : ABDE Next: Marry Boy 2 with Girl A: (best remaining choice) 2 A March 6, 2002 19

Stable Marriage Final “boy greedy” marriages 1 C 3 D 2 A 4 B 5 E March 6, 2002 20

Stable Marriage Trouble! 1 C 4 B March 6, 2002 21

Stable Marriage Boy 4 likes Girl C better than his wife. 1 C 4 B March 6, 2002 22

Stable Marriage and vice-versa 1 C 4 B March 6, 2002 23

Stable Marriage Rogue Couple 1 C 4 B March 6, 2002 24

Stable Marriage Problem: Marry everyone without any rogue couples! March 6, 2002 25

Stable Marriage Do it! PROBLEM 2 March 6, 2002 26

Stable Marriage More than a puzzle: • College Admissions (original Gale & Shapley paper, 1962) • Matching Hospitals & Residents. • Matching Dance Partners. March 6, 2002 27

Stable Marriage March 6, 2002 28

Stable Marriage The Mating Algorithm March 6, 2002 29

Stable Marriage The Mating Algorithm: day by day March 6, 2002 30

Stable Marriage Morning: boy serenades favorite girl Ted Alice Bob March 6, 2002 31

Stable Marriage Morning: boy serenades favorite girl Afternoon: girl rejects all but favorite Ted Alice Bob March 6, 2002 32

Stable Marriage Morning: boy serenades favorite girl Afternoon: girl rejects all but favorite Evening: rejected boy writes off girl … Alice Ted … March 6, 2002 33

Stable Marriage Stop when no rejects. Girl marries her favorite suitor. March 6, 2002 34

In-class Problem 3 March 6, 2002 35

Stable Marriage Partial Correctness: • Everyone is married. • Marriages are stable. Termination: there exists a Wedding Day. March 6, 2002 36

Stable Marriage Model as State Machine State q: Each boy’s set of eligible girls q(Bob)={Carole, Alice, …} March 6, 2002 37

Stable Marriage Derived Variable: serenading Bob’s favorite eligible girl. March 6, 2002 38

Stable Marriage Derived Variable: suitors all boys serenading Alice. March 6, 2002 39

Stable Marriage Derived Variable: favorite Carole’s preferred suitor. March 6, 2002 40

Stable Marriage Different girls have different favorites, because boys serenade one girl at a time. March 6, 2002 41

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. March 6, 2002 42

Stable Marriage Lemma: A boy’s love tomorrow will be no more desirable than today’s. serenading(B) is weakly decreasing for each B. March 6, 2002 43

Stable Marriage Derived Variable: total-girls-names : : = total number of names not crossed off boy’s lists Strictly decreasing & N-valued. So Wedding Day. March 6, 2002 44

Stable Marriage Mating Invariant If G has rejected B, then she has a better current favorite. March 6, 2002 45

Stable Marriage Mating Invariant If G has rejected B, then she has a better current favorite. Proof: favorite(G) is weakly increasing. March 6, 2002 46

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. March 6, 2002 47

Stable Marriage Stability: Similar easy argument using the Invariant. March 6, 2002 48

In-class Problem 4 March 6, 2002 49

Stable Marriage Who does better, boys or girls? Girls’ suitors get better, and boy’s sweethearts get worse, so girls do better? No! March 6, 2002 50

Stable Marriage Mating Algorithm is Optimal for all Boys at once. Pessimal for all Girls. March 6, 2002 51

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! March 6, 2002 52