Burnt Pancake Problem : New Results (New Lower Bounds on the Diameter and New Experimental Optimality Ratios)

Transcription

1 Burnt Pancake Problem : New Results (New Lower Bounds on the Diameter and New Experimental Optimality Ratios) Bruno Bouzy Paris Descartes University SoCS 06 Tarrytown July 6-8, 06

2 Outline The Burnt Pancake Problem ; Known hard positions : -I n, J n, Y n Exact Solving : New values of g(-i n ) ; Seeking for Hard Positions H n,m Approximate Solving : New Experimental Optimality Ratios Conclusions Burnt Pancake Problem: New Results

3 Pancake Problem : example A stack of pancakes. Each pancake has a size. Goal : sort the stack with the greatest pancake at the bottom. Unburnt case : L == Burnt case : Each pancake has a burnt side. L == Goal : the burnt sides have to be oriented downward Burnt == signed Burnt Pancake Problem: New Results

4 linked classes of problems #spatulas signed yes no Burnt Pancake? (Cohen & Blum 995) (Cibulka 0) Unburnt Pancake NP-hard (Bulteau & al 0) Signed Genome : Polynomial (Pevzner & Hennenhalli 995) Linear * (Bader & al 00) Quadratic (Bergeron 005) Unsigned Genome NP-hard (Caprara 997) Burnt Pancake Problem: New Results

5 Breakpoint (bp), anti-adjacency (aa) Unburnt Burnt Breakpoints Breakpoints Anti-adjacencies Size difference!= +- Size difference!= Size difference == - #bp == #bp == #aa = #bp : number of breakpoints admissible heuristic (Gates Papadimitriou 979 ; Helmert 00) #aa : number of anti-adjacencies interesting feature (Cibulka 0) Burnt Pancake Problem: New Results 5

6 -I n Take the identity stack I n and reverse the pancakes one by one : I -I g(-i n ) was known for n <= 0 n Our contribution 5 : g(-i... n ) known 7 8for 9 n <= g(-i n ) How? IDA* + #bp Burnt Pancake Problem: New Results 6

7 J n and Y n Reverse all the pancakes one by one, except one pancake : J Y J n Y n : reverse all the pancakes except the topmost pancake : reverse all the pancakes except the second pancake Cohen and Blum conjecture : -I n is «maximal» for any n The conjecture is false (Cibulka 0) For n=5 : J 5 and Y 5 are «maximal» but not -I 5 Burnt Pancake Problem: New Results 7

8 H n,m Reverse some pancakes one by one : H,0 = -I H, = H,0 H, =Y H, H,8 =J H,0 H, H,5 =I - m : non negative n bit integer Each pancake corresponds to one bit of m The topmost pancake corresponds to the most significant bit of m H n,m : Starting with -In, reverse the pancakes corresponding to bit== Burnt Pancake Problem: New Results 8

9 Seeking for hard positions (/) B n : set of hard stacks of size n Our light and uncomplete process : B = { -I } For n= to do For any b in B n do For r=,,n new = reverse r th pancake (b) new = add pancake -(n+) at the bottom solve (new) Update B n+ Burnt Pancake Problem: New Results 9

10 Seeking for hard positions (/) N d N T B N 0s -I 6 0s -I J 8 0s -I J 5 0 0s -I 5 J 5 6 0s -I 6 7 0s -I s -I s -I 9 Burnt Pancake Problem: New Results 0

11 Seeking for hard positions (/) N d N T B N 0 8 m -I 0 9 m -I Y J m -I 6m -I Y J 0m -I Y H, J 5 5 h Y 5 J h0 -I 6 H 6, J h -I 7 Burnt Pancake Problem: New Results

12 Seeking for hard positions (/) N d N (*) T B N 8 9 5h -I 8 Y 8 J 8 H 8, 9 0 6h -I 9 Y 9 J 9 H 9, H 9,8 H 9,0 H 8 9, + H 8 9, + H 8 9, +8 0 d -I 0 H 0,8 J 0 8d -I Y J H, H,6 H,8 H,0 H 0, + H 0, + H 0, +6 5 >5d -I Y J... Burnt Pancake Problem: New Results

13 Speeding up IDA* Make use of breakpoints and anti-adjacencies H A = #bp H B = #bp + λ #aa λ such that IDA* using H B remains optimal but faster than IDA* using H A Burnt Pancake Problem: New Results

14 End of part one Part one : optimal solving Heuristic Search Part two : approximate solving Monte-Carlo Search Burnt Pancake Problem: New Results

15 R-approximation Sub-optimal and polynomial time algorithm A L A (p) : length of a solution of problem p obtained with algorithm A L*(p) : length of an optimal solution to problem p R A (p) = L A (p) / L*(p) R-approximation(A) = max p R A (p) Theoretical and worst case analysis Unburnt pancakes : (Fischer 005) R= Burnt pancakes : (Cohen Blum 995) R= Burnt Pancake Problem: New Results 5

16 Experimental Optimality Ratios In practice, average case analysis L*(p) is unknown and replaced by the best lower bound known so far L*(p) = #bp Experimental Optimality Ratio (EOR) EOR A (p) = L A (p) / #bp EOR(A) = mean p EOR A (p) Unburnt pancakes : (Fischer 005) EOR =. (Bouzy 05) EOR =.0 Burnt Pancake Problem: New Results 6

17 Experimental Optimality Ratios Nested Monte Carlo Search (NMCS) (Cazenave 009) Simulations == (Cohen and Blum 995) L*(p) = #bp calibration IDA* optimal but EOR(IDA*). NMCS + Cohen Blum N L EOR T level s s s s s 6.9 0s s s s s 0 Burnt Pancake Problem: New Results 7

18 Conclusions and future work The burnt pancake problem New lower bounds on the diameter with g(-i n ) New hard stacks H n,m IDA* and preliminary heuristic function EOR with NMCS + Cohen & Blum algorithm as simulator Future work : Burnt pancake problem : IDA* and an admissible heuristic function including anti-adjacency Unburnt pancake problem Find new results on the diameter Generate complex stacks Burnt Pancake Problem: New Results 8

19 Thank you for your attention! Questions? Burnt Pancake Problem: New Results 9