Design and Analysis of Algorithms - Chapter 81 Dynamic Programming Dynamic Programming is a general algorithm design techniqueDynamic Programming is a general algorithm design technique Invented by American mathematician Richard Bellman in the 1950s to solve optimization problemsInvented by American mathematician Richard Bellman in the 1950s to solve optimization problems “Programming” here means “planning”“Programming” here means “planning” Main idea:Main idea: solve several smaller (overlapping) subproblems solve several smaller (overlapping) subproblems record solutions in a table so that each subproblem is only solved once record solutions in a table so that each subproblem is only solved once final state of the table will be (or contain) solution final state of the table will be (or contain) solution

Design and Analysis of Algorithms - Chapter 82 Example: Fibonacci numbers Recall definition of Fibonacci numbers: f(0) = 0 f(1) = 1 f(n) = f(n-1) + f(n-2) Compute the n th Fibonacci number recursively (top- down) f(n) f(n-1) + f(n-2) f(n-2) + f(n-3) f(n-3) + f(n-4)...

Design and Analysis of Algorithms - Chapter 83 Example: Fibonacci numbers (2) Compute the n th Fibonacci number using bottom-up iteration: f(0) = 0 f(1) = 1 f(2) = 0+1 = 1 f(3) = 1+1 = 2 f(4) = 1+2 = 3 f(n-2) = f(n-1) = f(n) = f(n-1) + f(n-2)

Design and Analysis of Algorithms - Chapter 84 Examples of Dynamic Programming Algorithms Computing binomial coefficientsComputing binomial coefficients Optimal chain matrix multiplicationOptimal chain matrix multiplication Constructing an optimal binary search treeConstructing an optimal binary search tree Warshall’s algorithm for transitive closureWarshall’s algorithm for transitive closure Floyd’s algorithms for all-pairs shortest pathsFloyd’s algorithms for all-pairs shortest paths Some instances of difficult discrete optimization problems:Some instances of difficult discrete optimization problems: travelling salesman travelling salesman knapsack knapsack

Design and Analysis of Algorithms - Chapter 85 Binomial coefficients Algorithm based on identity Algorithm Binomial(n,k)Algorithm Binomial(n,k) for i  0 to n do for j  0 to min(j,k) do if j=0 or j=i then C[i,j]  1 else C[i,j]  C[i-1,j-1]+C[i-1,j] return C[n,k] Pascal’s TrianglePascal’s Triangle ExampleExample Space and Time efficiencySpace and Time efficiency

Design and Analysis of Algorithms - Chapter 86 Warshall’s Algorithm: Transitive Closure Computes the transitive closure of a relation Computes the transitive closure of a relation Alternatively: all paths in a directed graph Alternatively: all paths in a directed graph Example of transitive closure: Example of transitive closure:

Design and Analysis of Algorithms - Chapter 87 Warshall’s Algorithm (2) Main idea: a path exists between two vertices i, j, iff Main idea: a path exists between two vertices i, j, iff there is an edge from i to j; orthere is an edge from i to j; or there is a path from i to j going through vertex 1; orthere is a path from i to j going through vertex 1; or there is a path from i to j going through vertex 1 and/or 2; orthere is a path from i to j going through vertex 1 and/or 2; or there is a path from i to j going through any of the other verticesthere is a path from i to j going through any of the other vertices R R R R R

Design and Analysis of Algorithms - Chapter 88 Warshall’s Algorithm (3) In the k th stage find if a path exists between two vertices i, j using just vertices among 1,…,k R (k-1) [i,j] (path using just 1,…,k-1) R (k) [i,j] = or R (k) [i,j] = or (R (k-1) [i,k] & R (k-1) [k,j]) (path from i to k and from (R (k-1) [i,k] & R (k-1) [k,j]) (path from i to k and from k to i using just 1,…,k-1) k to i using just 1,…,k-1) i j k k th stage {

Design and Analysis of Algorithms - Chapter 89 Warshall’s Algorithm (4) Algorithm Warshall(A[1..n,1..n]) R (0)  A for k  1 to n do for i  1 to n do for i  1 to n do for j  1 to n do for j  1 to n do R (k) [i,j]  R (k-1) [i,j] or R (k-1) [i,k] and R (k-1) [k,j] R (k) [i,j]  R (k-1) [i,j] or R (k-1) [i,k] and R (k-1) [k,j] return R (k) Space and Time efficiency

Design and Analysis of Algorithms - Chapter 810 Floyd’s Algorithm: All pairs shortest paths In a weighted graph, find shortest paths between every pair of verticesIn a weighted graph, find shortest paths between every pair of vertices Same idea: construct solution through series of matrices D(0), D(1), … using an initial subset of the vertices as intermediaries.Same idea: construct solution through series of matrices D(0), D(1), … using an initial subset of the vertices as intermediaries. Example:Example:

Design and Analysis of Algorithms - Chapter 811 Floyd’s Algorithm (2) Algorithm Floyd(W[1..n,1..n]) D  W for k  1 to n do for i  1 to n do for i  1 to n do for j  1 to n do for j  1 to n do D[i,j]  min(D[i,j],D[i,k]+D[k,j]) D[i,j]  min(D[i,j],D[i,k]+D[k,j]) return D Space and Time efficiency When does it not work? Principle of optimality

Design and Analysis of Algorithms - Chapter 812 Optimal Binary Search Trees Keys are not searched with equal probabilityKeys are not searched with equal probability Suppose keys A, B, C, D with probabilities 0.1, 0.2, 0.3, 0.4Suppose keys A, B, C, D with probabilities 0.1, 0.2, 0.3, 0.4 What is the average search cost in each possible structure?What is the average search cost in each possible structure? How many different structures can be produced with n nodes ?How many different structures can be produced with n nodes ? Catalan number C(n)=comb(2n,n)/(n+1)Catalan number C(n)=comb(2n,n)/(n+1)

Design and Analysis of Algorithms - Chapter 813 Optimal Binary Search Trees (2) C[i,j] denotes the smallest average search cost in a tree including items i through j Based on the principle of optimality, the optimal search tree is constructed by using the recurrence C[i,j] = min{C[i,k-1]+C[k+1,j]} + Σp s for 1≤i ≤ j ≤ n and i ≤ k ≤ j C[i,i] = p i

Design and Analysis of Algorithms - Chapter 814 Optimal Binary Search Trees (3) Algorithm OptimalBST(P[1..n]) for i  1 to n do C[i,i-1]  0; C[i,i]  P[i], R[i,i]  I C[n+1,n]  0 for d  1 to n-1 do for i  1 to n-d do j  i+d; minval  ∞ for k  i to j do if C[i,k-1]+C[k+1,j]<minval minval  C[I,k-1]+C[k+1,j]; kmin  k R[i,j]  kmin; sum  P[i]; for s  i+1 to j do sum  sum+P[s] C[i,j]  minval+sum return C[1,n], R

Design and Analysis of Algorithms - Chapter 815 The Knapsack problem Problem: given n items with weight w 1,w 2, …, w n, values v 1, v 2, …, v n and capacity W. Find the most valuable subset that fit into the knapsack. Solution with exhaustive search Let V[i,j] denote the optimal solution for the first i items and capacity j. Purpose to find V[n,W] Recurrence max{V[i-1,j],v i +V[i-1,j-w i ]} if j-w i ≥0 V[i,j] = V[i-1,j] if j-w i <0 {

Design and Analysis of Algorithms - Chapter 816 The Knapsack problem (2) itemweightvalue Example W=5 Space and Time efficiencyi,j ?

Design and Analysis of Algorithms - Chapter 817 Memory functions [1] A disadvantage of the dynamic programming approach is that is solves subproblems not finally needed. An alternative technique is to combine a top- down and a bottom-up approach (recursion plus a table for temporary results)

Design and Analysis of Algorithms - Chapter 818 Memory functions [2] Algorithm MFKnapsack(i,j) if V[i,j]<0 if j<Weights[i] value  MFKnapack[i-1,j] else value  max{MFKnapsack(i-1,j), Values[i]+MFKnapsack[i-1,j-Weights[i]]} V[i,j]  value return V[i,j]

Design and Analysis of Algorithms - Chapter 819 Memory functions (3) itemweightvalue Example W=5 Space and Time efficiencyi,j ?