Dynamic Programming

Transcript

1. Dynamic Programming Gefei Zhang Celonis GmbH 2014-02-18 Gefei Zhang: Dynamic Programming 1/13
2. Gefei Zhang Dipl.-Inform. and Dr.rer.nat. from LMU Software developer at Celonis GmbH Gefei Zhang: Dynamic Programming 2/13
3. Celonis “The leading vendor for the analysis of operative process data created by IT systems” [IBM] Interactive, process-based Business Intelligence Next level of transparency in enterprise process mining http://www.celonis.de Gefei Zhang: Dynamic Programming 3/13
4. Celonis Gefei Zhang: Dynamic Programming 4/13
5. Content Dynamic Programming Warming up: Fibonacci Introduction: Maximum weighted independent set in a path Application: Knapsack Gefei Zhang: Dynamic Programming 5/13
6. Fibonacci f (n) = f (n − 1) + f (n − 2) simple, but horrible complexity Gefei Zhang: Dynamic Programming 6/13
7. Fibonacci f (n) = f (n − 1) + f (n − 2) simple, but horrible complexity better: bottom up a = 1; b = 1 for i in 3..n a, b = b, a+b return b Gefei Zhang: Dynamic Programming 6/13
8. Maximum weighted independent set in a path Example Index set I = 1, · · · n Array arri > 0, i ∈ I Gefei Zhang: Dynamic Programming 7/13
9. Maximum weighted independent set in a path Example Index set I = 1, · · · n Array arri > 0, i ∈ I Wanted: set S ⊆ I, such that ∀x, y ∈ S, x > y, x − y > 1 Gefei Zhang: Dynamic Programming 7/13
10. Maximum weighted independent set in a path Example Index set I = 1, · · · n Array arri > 0, i ∈ I Wanted: set S ⊆ I, such that ∀x, y ∈ S, x > y, x − y > 1 s∈S arrs max Gefei Zhang: Dynamic Programming 7/13
11. Maximum weighted independent set in a path Example Index set I = 1, · · · n Array arri > 0, i ∈ I Wanted: set S ⊆ I, such that ∀x, y ∈ S, x > y, x − y > 1 s∈S arrs max Gefei Zhang: Dynamic Programming 7/13
12. Maximum weighted independent set in a path Sn optimal Gefei Zhang: Dynamic Programming 8/13
13. Maximum weighted independent set in a path Sn optimal n / ∈ Sn ⇔ Sn = Sn−1 Gefei Zhang: Dynamic Programming 8/13
14. Maximum weighted independent set in a path Sn optimal n / ∈ Sn ⇔ Sn = Sn−1 n ∈ Sn ⇔ Sn = Sn−2 ∪ {n} Gefei Zhang: Dynamic Programming 8/13
15. Maximum weighted independent set in a path Sn optimal n / ∈ Sn ⇔ Sn = Sn−1 n ∈ Sn ⇔ Sn = Sn−2 ∪ {n} An the sum for optimal Sn Gefei Zhang: Dynamic Programming 8/13
16. Maximum weighted independent set in a path Sn optimal n / ∈ Sn ⇔ Sn = Sn−1 n ∈ Sn ⇔ Sn = Sn−2 ∪ {n} An the sum for optimal Sn n ∈ Sn ⇔ An = An−2 + arrn > An−1 Gefei Zhang: Dynamic Programming 8/13
17. Maximum weighted independent set in a path Sn optimal n / ∈ Sn ⇔ Sn = Sn−1 n ∈ Sn ⇔ Sn = Sn−2 ∪ {n} An the sum for optimal Sn n ∈ Sn ⇔ An = An−2 + arrn > An−1 Recursion Ak = max(Ak−1, Ak−2 + arrk) Gefei Zhang: Dynamic Programming 8/13
18. Maximum weighted independent set in a path Sn optimal n / ∈ Sn ⇔ Sn = Sn−1 n ∈ Sn ⇔ Sn = Sn−2 ∪ {n} An the sum for optimal Sn n ∈ Sn ⇔ An = An−2 + arrn > An−1 Recursion Ak = max(Ak−1, Ak−2 + arrk) Bottom up for i in 3..n A[i] = max(A[i-1], A[i-2]+arr[i]) Gefei Zhang: Dynamic Programming 8/13
19. Maximum weighted independent set in a path Sn optimal n / ∈ Sn ⇔ Sn = Sn−1 n ∈ Sn ⇔ Sn = Sn−2 ∪ {n} An the sum for optimal Sn n ∈ Sn ⇔ An = An−2 + arrn > An−1 Recursion Ak = max(Ak−1, Ak−2 + arrk) Bottom up for i in 3..n A[i] = max(A[i-1], A[i-2]+arr[i]) Backtracking: from A to S Ai > Ai−1 ⇒ i ∈ S and i − 1 / ∈ S Ai < Ai−1 ⇒ i / ∈ S Gefei Zhang: Dynamic Programming 8/13
20. Knapsack Problem Gefei Zhang: Dynamic Programming 9/13
21. Knapsack Problem Index set I = 1, · · · , n Given: {vi }, {wi }, i ∈ I, K > 0 Gefei Zhang: Dynamic Programming 9/13
22. Knapsack Problem Index set I = 1, · · · , n Given: {vi }, {wi }, i ∈ I, K > 0 Wanted: set S ⊂ I, such that i∈S wi ≤ K Gefei Zhang: Dynamic Programming 9/13
23. Knapsack Problem Index set I = 1, · · · , n Given: {vi }, {wi }, i ∈ I, K > 0 Wanted: set S ⊂ I, such that i∈S wi ≤ K i∈S vi max Gefei Zhang: Dynamic Programming 9/13
24. Knapsack Problem Index set I = 1, · · · , n Given: {vi }, {wi }, i ∈ I, K > 0 Wanted: set S ⊂ I, such that i∈S wi ≤ K i∈S vi max Gefei Zhang: Dynamic Programming 9/13
25. Knapsack Problem Index set I = 1, · · · , n Given: {vi }, {wi }, i ∈ I, K > 0 Wanted: set S ⊂ I, such that i∈S wi ≤ K i∈S vi max Example Gefei Zhang: Dynamic Programming 9/13
26. Knapsack Problem Sn,k optimal n / ∈ Sn,k ⇔ Sn,k = Sn−1,k n ∈ Sn,k ⇔ Sn,k = Sn−1,k−wn ∪ {n} Vn,k the sum n ∈ Sn,k ⇔ Vn−1,k > Vn−1,k−wn + vn Recursion Vn,k = max(Vn−1,k, Vn−1,k−wn + vn) Gefei Zhang: Dynamic Programming 10/13
27. Knapsack Problem Bottom up for i in 1..n for c in 1..K V[i,c] = max(V[i-1,c], V[i-1, c-w[i]] + v[i]) Backtracking Vn,k = Vn−1,k iff n / ∈ S Gefei Zhang: Dynamic Programming 11/13
28. Summary Dynamic Programming Recursion bottom up Evil mostly in finding recursion Applications in Graph Theory, Computer Linguistics, Bioinformatics, etc. Gefei Zhang: Dynamic Programming 12/13
29. Summary Dynamic Programming Recursion bottom up Evil mostly in finding recursion Applications in Graph Theory, Computer Linguistics, Bioinformatics, etc. Challenge Egg Dropping Gefei Zhang: Dynamic Programming 12/13
30. Thanks Danke Gefei Zhang: Dynamic Programming 13/13