Chapter 21
Dynamic Programming Algorithms

21.1 Fibonnaci Numbers

A top-down approach of computing, say, f(5) is inefficient do to repeated subcomputations.

A bottom-up approach computes f(0), f(1), f(2), f(3), f(4), f(5) in the listed order.

21.2 0/1 Knapsack Problem

21.3 All Pairs Shortest Paths (Floyd-Warshall)

A dynamic programming algorithm.

FOR k=1 TO n 
  FOR i=1 TO n 
    FOR j=1 TO n 
      c(i,j,k) = min(  c(i,j,k-1), 
                       c(i,k,k-1)+c(k,j,k-1) 
                    ) 

k = Ø	a b c d e f a 1 7 5 b 7 3 c 1 2 d 8 e 4 f
k = {a}	a b c d e f a 1 7 5 b 7,b-a-c ,b-a-d 3,b-a-e ,b-a-f c ,c-a-d 1,c-a-e 2,c-a-f d 8,d-a-e ,d-a-f e 4,e-a-f f
k = {a, b}
k = {a, b, c}
k = {a, b, c, d}
k = {a, b, c, d, e}
k = {a, b, c, d, e, f}

21.4 Demo Applets

• Floyd’s algorithm

21.5 Assignment #9 (due We, Dec 1)

1. Show the tree of memory states traversed by our recursive permutation algorithm for input “a,b,c,d”.
2. Provide a recursive backtracking algorithm for the vertex cover problem.
3. Provide a branch-and-bound algorithm for the 0/1 knapsack problem.
4. Provide a dynamic programming algorithm for the coin exchange problem.