knapsack problem time complexity code example

Example 1: knapsack problem

// memory efficient and iterative approach to the knapsack problem

#include <bits/stdc++.h>
using namespace std;

// n is the number of items
// w is the knapsack's capacity
int n, w;

int main() {
/*
input format:
n w
value_1 cost_1
value_2 cost_2
.
.
value_n cost_n
*/
    cin >> n >> w;
  	vector<long long> dp(w + 1, 0);

    for (int i = 0; i < n; ++i) {
        int value, cost;
        cin >> value >> cost;
        for (int j = w; j >= cost; --j)
            dp[j] = max(dp[j], value + dp[j - cost]);
    }

    // the answer is dp[w]
    cout << dp[w];
}

Example 2: knapsack

#include<bits/stdc++.h>
using namespace std;
vector<pair<int,int> >a;
//dp table is full of zeros
int n,s,dp[1002][1002];
void ini(){
    for(int i=0;i<1002;i++)
        for(int j=0;j<1002;j++)
            dp[i][j]=-1;
}
int f(int x,int b){
	//base solution
	if(x>=n or b<=0)return 0;
	//if we calculate this before, we just return the answer (value diferente of 0)
	if(dp[x][b]!=-1)return dp[x][b];
	//calculate de answer for x (position) and b(empty space in knapsack)
	//we get max between take it or not and element, this gonna calculate all the
	//posible combinations, with dp we won't calculate what is already calculated.
	return dp[x][b]=max(f(x+1,b),b-a[x].second>=0?f(x+1,b-a[x].second)+a[x].first:INT_MIN);
}
int main(){
	//fast scan and print
	ios_base::sync_with_stdio(0);cin.tie(0);
	//we obtain quantity of elements and size of knapsack
	cin>>n>>s;
	a.resize(n);
	//we get value of elements
	for(int i=0;i<n;i++)
		cin>>a[i].first;
	//we get size of elements
	for(int i=0;i<n;i++)
		cin>>a[i].second;
	//initialize dp table
	ini();
	//print answer
	cout<<f(0,s);
	return 0;
}