What is the time complexity of the following dynamic programming implementation of the Knapsack problem with n items and a maximum weight of W?
#include<stdio.h>
int find_max(int a, int b)
{
if(a > b)
return a;
return b;
}
int knapsack(int W, int *wt, int *val,int n)
{
int ans[n + 1][W + 1];
int itm,w;
for(itm = 0; itm <= n; itm++)
ans[itm][0] = 0;
for(w = 0;w <= W; w++)
ans[0][w] = 0;
for(itm = 1; itm <= n; itm++)
{
for(w = 1; w <= W; w++)
{
if(wt[itm - 1] <= w)
ans[itm][w] = find_max(ans[itm - 1][w-wt[itm - 1]]+val[itm - 1], ans[itm - 1][w]);
else
ans[itm][w] = ans[itm - 1][w];
}
}
return ans[n][W];
}
int main()
{
int w[] = {10,20,30}, v[] = {60, 100, 120}, W = 50;
int ans = knapsack(W, w, v, 3);
printf("%d",ans);
return 0;
}
#include<stdio.h>
int find_max(int a, int b)
{
if(a > b)
return a;
return b;
}
int knapsack(int W, int *wt, int *val,int n)
{
int ans[n + 1][W + 1];
int itm,w;
for(itm = 0; itm <= n; itm++)
ans[itm][0] = 0;
for(w = 0;w <= W; w++)
ans[0][w] = 0;
for(itm = 1; itm <= n; itm++)
{
for(w = 1; w <= W; w++)
{
if(wt[itm - 1] <= w)
ans[itm][w] = find_max(ans[itm - 1][w-wt[itm - 1]]+val[itm - 1], ans[itm - 1][w]);
else
ans[itm][w] = ans[itm - 1][w];
}
}
return ans[n][W];
}
int main()
{
int w[] = {10,20,30}, v[] = {60, 100, 120}, W = 50;
int ans = knapsack(W, w, v, 3);
printf("%d",ans);
return 0;
}A. O(n)
B. O(n + w)
C. O(nW)
D. O(n2)
Answer: Option C

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