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Dynamic Programming for 0-1 Knapsack Problem

Exercise 17.2-2

For this algorithm, let c[i,w] = value of solution for items 1..i and maximum weight w.

c[i,w] = $\left\{ \begin{array}{ll} 0 & {\rm if} \; i = 0 \; {\rm or} \; w = 0 \\ c[i... ...1,w]) & {\rm if} \; i > 0 \; {\rm and} \; w \geq w_i \\ \end{array} \right.$

DP-01K(v, w, n, W)
1 $\;\;\;\;\;$ for w = 0 to W
2 $\;\;\;\;\;$ $\;\;\;\;\;$ c[0,w] = 0
3 $\;\;\;\;\;$ for i = 1 to n
4 $\;\;\;\;\;$ $\;\;\;\;\;$ c[i,0] = 0
5 $\;\;\;\;\;$ $\;\;\;\;\;$ for w = 1 to W
6 $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ if w[i] $\leq$ w
7 $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ then if v[i] + c[i-1,w-w[i]] > c[i-1,w]
8 $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ then c[i,w] = v[i] + c[i-1,w-w[i]]
9 $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ else c[i,w] = c[i-1,w]
10 $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ else c[i,w] = c[i-1,w]

The run time performance of this algorithm is $\Theta(nW)$ .

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