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CPSC 320: Intermediate Algorithm Design and Analysis Schedule transformation example Schedule transformation example Theorem: If S j is a reduced schedule that is the same as S FF for the first requests, then there is a reduced schedule S j+1

  1. CPSC 320: Intermediate Algorithm Design and Analysis Schedule transformation example Schedule transformation example Theorem: If S j is a reduced schedule that is the same as S FF for the first requests, then there is a reduced schedule S j+1 that is the same as S FF for the first j+1 requests, and incurs no more misses than S j . CPSC 320 – 2013W T2 2

  2. Initial state Request S FF S 0 S 1 Initial Cache a f c e a f c e a f c e c a f c e a f c e x x f c e x f c e i x f c i x i c e c x f c i x i c e f x f c i x i c f g x g c i x i g f x x g c i x i g f i x g c i x i g f y x g y i x i g y z z g y i x i g z i z g y i x i g z y z g y i x y g z CPSC 320 – 2013W T2 3 Case 1: item is already in the cache Request S FF S 0 S 1 Initial Cache a f c e a f c e a f c e c a f c e a f c e a f c e x x f c e x f c e x f c e i x f c i x i c e x i c e c x f c i x i c e x i c e f x f c i x i c f x i c f g x g c i x i g f x i g f x x g c i x i g f x i g f i x g c i x i g f x i g f y x g y i x i g y x i g y z z g y i x i g z x i g z i z g y i x i g z x i g z y z g y i x y g z x y g z CPSC 320 – 2013W T2 4

  3. Case 2: S evicts the same item as S FF Request S FF S 1 S 2 Initial Cache a f c e a f c e a f c e c a f c e a f c e a f c e x x f c e x f c e x f c e i x f c i x i c e x i c e c x f c i x i c e x i c e f x f c i x i c f x i c f g x g c i x i g f x i g f x x g c i x i g f x i g f i x g c i x i g f x i g f y x g y i x i g y x i g y z z g y i x i g z x i g z i z g y i x i g z x i g z y z g y i x y g z x y g z CPSC 320 – 2013W T2 5 Case 3a: S evicts a different item from S FF Request S FF S 2 S 3 Initial Cache a f c e a f c e a f c e c a f c e a f c e a f c e x x f c e x f c e x f c e i x f c i x i c e x i c e c x f c i x i c e x i c e f x f c i x i c f x i c f g x g c i x i g f x i g f x x g c i x i g f x i g f i x g c i x i g f x i g f y x g y i x i g y x i g y z z g y i x i g z x i g z i z g y i x i g z x i g z y z g y i x y g z x y g z CPSC 320 – 2013W T2 6

  4. Case 3a: S evicts a different item from S FF Request S FF S 2 S 3 Initial Cache a f c e a f c e a f c e c a f c e a f c e a f c e x x f c e x f c e x f c e i x f c i x i c e x f c i c x f c i x i c e x i c e f x f c i x i c f x i c f g x g c i x i g f x i g f x x g c i x i g f x i g f i x g c i x i g f x i g f y x g y i x i g y x i g y z z g y i x i g z x i g z i z g y i x i g z x i g z y z g y i x y g z x y g z CPSC 320 – 2013W T2 7 Case 3a: S evicts a different item from S FF Request S FF S 2 S 3 Initial Cache a f c e a f c e a f c e c a f c e a f c e a f c e x x f c e x f c e x f c e i x f c i x i c e x f c i c x f c i x i c e x f c i f x f c i x i c f x i c f g x g c i x i g f x i g f x x g c i x i g f x i g f i x g c i x i g f x i g f y x g y i x i g y x i g y z z g y i x i g z x i g z i z g y i x i g z x i g z y z g y i x y g z x y g z CPSC 320 – 2013W T2 8

  5. Case 3a: S evicts a different item from S FF Request S FF S 2 S 3 Initial Cache a f c e a f c e a f c e c a f c e a f c e a f c e x x f c e x f c e x f c e i x f c i x i c e x f c i c x f c i x i c e x f c i f x f c i x i c f x f c i g x g c i x i g f x i g f x x g c i x i g f x i g f i x g c i x i g f x i g f y x g y i x i g y x i g y z z g y i x i g z x i g z i z g y i x i g z x i g z y z g y i x y g z x y g z CPSC 320 – 2013W T2 9 Case 3a: S evicts a different item from S FF Request S FF S 2 S 3 Initial Cache a f c e a f c e a f c e c a f c e a f c e a f c e x x f c e x f c e x f c e i x f c i x i c e x f c i c x f c i x i c e x f c i f x f c i x i c f x f c i g x g c i x i g f x i g f x x g c i x i g f x i g f i x g c i x i g f x i g f y x g y i x i g y x i g y z z g y i x i g z x i g z i z g y i x i g z x i g z y z g y i x y g z x y g z CPSC 320 – 2013W T2 10

  6. Case 1 again. Request S FF S 3 S 4 Initial Cache a f c e a f c e a f c e c a f c e a f c e a f c e x x f c e x f c e x f c e i x f c i x f c i x f c i c x f c i x f c i x f c i f x f c i x f c i x f g i g x g c i x f g i x f g i x x g c i x f g i x f g i i x g c i x f g i x y g i y x g y i x y g i x z g i z z g y i x z g i x z g i i z g y i x z g i x z g i y z g y i x z g y x z g y CPSC 320 – 2013W T2 11 Case 1 again. Request S FF S 4 S 5 Initial Cache a f c e a f c e a f c e c a f c e a f c e a f c e x x f c e x f c e x f c e i x f c i x f c i x f c i c x f c i x f c i x f c i f x f c i x f c i x f c i g x g c i x f g i x f g i x x g c i x f g i x f g i i x g c i x f g i x y g i y x g y i x y g i x z g i z z g y i x z g i x z g i i z g y i x z g i x z g i y z g y i x z g y x z g y CPSC 320 – 2013W T2 12

  7. Case 3b: S evicts a different item from S FF Request S FF S 5 S 6 Initial Cache a f c e a f c e a f c e c a f c e a f c e a f c e x x f c e x f c e x f c e i x f c i x f c i x f c i c x f c i x f c i x f c i f x f c i x f c i x f c i g x g c i x f g i x f g i x x g c i x f g i x f g i i x g c i x f g i x y g i y x g y i x y g i x z g i z z g y i x z g i x z g i i z g y i x z g i x z g i y z g y i x z g y x z g y CPSC 320 – 2013W T2 13 Case 3b: S evicts a different item from S FF Request S FF S 5 S 6 Initial Cache a f c e a f c e a f c e c a f c e a f c e a f c e x x f c e x f c e x f c e i x f c i x f c i x f c i c x f c i x f c i x f c i f x f c i x f c i x f c i g x g c i x f g i x g c i x x g c i x f g i x f g i i x g c i x f g i x y g i y x g y i x y g i x z g i z z g y i x z g i x z g i i z g y i x z g i x z g i y z g y i x z g y x z g y CPSC 320 – 2013W T2 14

  8. Case 3b: S evicts a different item from S FF Request S FF S 5 S 6 Initial Cache a f c e a f c e a f c e c a f c e a f c e a f c e x x f c e x f c e x f c e i x f c i x f c i x f c i c x f c i x f c i x f c i f x f c i x f c i x f c i g x g c i x f g i x g c i x x g c i x f g i x g c i i x g c i x f g i x y g i y x g y i x y g i x z g i z z g y i x z g i x z g i i z g y i x z g i x z g i y z g y i x z g y x z g y CPSC 320 – 2013W T2 15 Case 3b: S evicts a different item from S FF Request S FF S 5 S 6 Initial Cache a f c e a f c e a f c e c a f c e a f c e a f c e x x f c e x f c e x f c e i x f c i x f c i x f c i c x f c i x f c i x f c i f x f c i x f c i x f c i g x g c i x f g i x g c i x x g c i x f g i x g c i i x g c i x f g i x g c i y x g y i x y g i x z g i z z g y i x z g i x z g i i z g y i x z g i x z g i y z g y i x z g y x z g y CPSC 320 – 2013W T2 16

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