Algorithm Design Techniques: Dynamic Programming

Introduction Dynamic Programming –Divide-and-conquer solution to a problem with overlapping subproblems –Similar to bottom-up recursion where results are saved in a table as they are computed –Typically applied to optimization problems –Typically best used when objects have a linear ordering and cannot be rearranged

Example: Fibonacci Basic recursive solution F(N) = F(N-1)+F(N-2) F1 calculated three times Instead, calculate numbers 1->n, store values as they are calculated Use space instead of time F4 F3 F2 F1 F0 F1F0

Steps (CLR pg323) 1.Characterize the structure of an optimal solution (optimal substructure) 2.Recursively define the value of an optimal solution 3.Compute the value of an optimal solution in a bottom-up fashion 4.Construct an optimal solution from computed information

Ordering Matrix Multiplications Matrices –A (50x10), B (10x40), C (40x30), D (30x5) Compute ABCD –Associative (not commutative) Fits the ordering property –Decide how to parenthesize to achieve fewest operations

Ordering Matrix Multiplications (A((BC)D) –BC = 10x40x30 = 12,000 operations –(BC)D = 12, x30x5 = 13,500 –A((BC)D) = 13, x10x5 = 16,000 (A(B(CD)) = 10,500 ((AB)(CD)) = 36,000 (((AB)C)D) = 87,500 ((A(BC))D) = 34,500

Ordering Matrix Multiplications N-1 T(N) = Σ T(i)T(N-i) i=1 Basic recursive solution is exponential M Left, Right = min Left<=i<=Right {M Left,i +M i+1,Right +c Left-1 c i c Right }

Ordering Matrix Multiplications A B CD A = 50x10 B = 10x40 C = 40x30 D = 30x5 20,00012,0006,000 27,0008,000 10, N 2 /2 values are calculated O(N 3 ) running time

Optimal Binary Search Tree Create binary search tree to minimize N Σ p i (1+d i ) i=1 WordProbability a.22 am.18 and.20 egg.05 if.25 the.02 two.08 if atwo theand amegg

Optimal Binary Search Tree Basic greedy strategy does not work But, problem has the ordering property and optimal substructure property i-1 Right C Left, Right = min{p i + C Left, i-1 +C i+1, Right +  p j +  p j } j=Left j=i+1

Optimal Binary Search Tree (pg431) a..a.22 a am..am.18 am and..and.20 and egg..egg.05 egg if..if.25 if the..the.02 the two..two.08 two a..am.58 a am..and.56 am and..egg.30 and egg..if.35 if if..the.29 if the..two.12 two a..and 1.02 am am..egg.66 and and..if.80 if egg..the.39 if if..two.46 if a..egg 1.17 am am..if 1.21 and and..the.84 if egg..two.57 if a..if 1.83 and am..the 1.27 and and..two 1.02 if a..the 1.89 and am..two 1.53 and a..two 2.15 and

Memoization Maintain top-down recursion, but memoize (keep "memos" of) results allocate array for memo; initialize memo[1] and memo[2] to 1; fib(n) { if memo[n] is not zero, return memo[n]; memo[n] = fib(n-1) + fib(n-2); return memo[n]; }