Derived Variables A derived variable v is simply

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Derived Variables A derived variable v is simply a function that associate value with

Derived Variables A derived variable v is simply a function that associate value with state. Die Hard Example: total amount of water in the two jugs v : : = b + l March 3, 2002

Derived Variables For GCD, already have variables x, y. March 3, 2002

Derived Variables For GCD, already have variables x, y. March 3, 2002

Derived Variables Proof of GCD termination: Show that variable y is strictly decreasing and

Derived Variables Proof of GCD termination: Show that variable y is strictly decreasing and natural number valued. March 3, 2002

Derived Variables Termination follows by least number principle: y must take a least value,

Derived Variables Termination follows by least number principle: y must take a least value, and thus the algorithm is stuck. March 3, 2002

Strictly Decreasing Variable 16 N 12 8 4 0 State March 3, 2002

Strictly Decreasing Variable 16 N 12 8 4 0 State March 3, 2002

Weakly Decreasing Variable 16 N 12 8 4 0 State March 3, 2002

Weakly Decreasing Variable 16 N 12 8 4 0 State March 3, 2002

Weakly Decreasing Variable We used to call weakly decreasing variables “nonincreasing” variables. OK terminology

Weakly Decreasing Variable We used to call weakly decreasing variables “nonincreasing” variables. OK terminology but remember: nondecreasing is NOT EQUAL to “not decreasing. ” March 3, 2002

Stable Marriage There are 5 girls and 5 boys 1 2 3 4 5

Stable Marriage There are 5 girls and 5 boys 1 2 3 4 5 A B C D E March 3, 2002

Stable Marriage Preferences: Boys 1: CBEAD 2 : ABECD 3 : DCBAE 4 :

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 3, 2002

Stable Marriage Preferences: Boys 1: CBEAD 2 : ABECD 3 : DCBAE 4 :

Stable Marriage Preferences: Boys 1: CBEAD 2 : ABECD 3 : DCBAE 4 : ACDBE 5 : ABDEC Marry 1 with C: C 1 March 3, 2002

Stable Marriage Boys 2 : ABED 3 : DBAE 4 : ADBE 5 :

Stable Marriage Boys 2 : ABED 3 : DBAE 4 : ADBE 5 : ABDE Only 4 boys and 4 girls left. 2 3 4 5 A B D E March 3, 2002

Stable Marriage A problem: Rogue couples 1 4 C B Girl C likes like

Stable Marriage A problem: Rogue couples 1 4 C B Girl C likes like boy 4 better Boy 4 like girl C better March 3, 2002

Stable marriage The mating algorithm – day by day Morning: boys serenade favorite girls

Stable marriage The mating algorithm – day by day Morning: boys serenade favorite girls Afternoon: girl rejects all but best Serenader Evening: rejected boys cross off girls March 3, 2002

Stable marriage Termination condition: stop when no rejection is possible. • The algorithm terminates.

Stable marriage Termination condition: stop when no rejection is possible. • The algorithm terminates. • Everybody ends up married. • The resulting marriages are stable. March 3, 2002

Infinitely Wide Trees … … March 3, 2002

Infinitely Wide Trees … … March 3, 2002

Infinitely Wide Trees … … March 3, 2002

Infinitely Wide Trees … … March 3, 2002

Infinitely Wide Trees March 3, 2002

Infinitely Wide Trees March 3, 2002

Infinitely Wide Trees March 3, 2002

Infinitely Wide Trees March 3, 2002

Tutorial Problem 1 March 3, 2002

Tutorial Problem 1 March 3, 2002