Table of contents

• Problem
• Explanation
• Computational complexity
• Solution 1
• Solution 2

# Problem

http://judge.u-aizu.ac.jp/onlinejudge/description.jsp

# Explanation

0	10001	10001	10001	10001	10001	10001	10001	10001	10001	10001	10001	10001	10001	10001	10001	
0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	
0	1	1	2	2	3	3	4	4	5	5	6	6	7	7	8	
0	1	1	2	2	3	3	1	2	2	3	3	4	4	2	3	
0	1	1	2	2	3	3	1	1	2	2	3	3	4	2	2	
0	1	1	2	2	3	3	1	1	2	2	3	1	2	2	2	
0	1	1	2	2	3	3	1	1	2	2	3	1	2	2	2

# Computational complexity

$O(MN)$

# Solution 1

Less space ( $O(N)$ )required.

// C++ 14
#include <iostream>
#include <string>
#include <vector>
#include <list>
#include <algorithm>
#include <queue>
#include <stack>
#include <set>
#include <map>
#include <unordered_map>
#include <math.h>

#define ll long long
#define Int int
#define loop(x, start, end) for(Int x = start; x < end; x++)
#define loopdown(x, start, end) for(int x = start; x > end; x--)
#define rep(n) for(int x = 0; x < n; x++)
#define span(a,x,y) a.begin()+x,a.begin()+y
#define span_all(a) a.begin(),a.end()
#define len(x) (x.size())
#define last(x) (*(x.end()-1))

using namespace std;

#define W_MAX 10000
#define INF 10001
#define N_MAX 50001
#define M_MAX 21

Int N, M;
Int COINS[M_MAX+1];
Int table[N_MAX+1];

Int min_coin() {
  loop(n,0,N+1) {
    table[n] = INF;
  }
  table[0] = 0;
  
  loop(m,1,M+1) {
    Int coin = COINS[m];
    loop(n,coin,N+1) {
      table[n] = min(table[n], table[n-coin] + 1);
    }
  }
  
  return table[N];
}

int main(void) {
  cin >> N >> M;
  COINS[0] = 0;
  loop(i,1,M+1) {
    cin >> COINS[i];
  }
  
  cout << min_coin() << endl;
}

# Solution 2

Require $O(MN)$ space

// C++ 14
#include <iostream>
#include <string>
#include <vector>
#include <list>
#include <algorithm>
#include <queue>
#include <stack>
#include <set>
#include <map>
#include <unordered_map>
#include <math.h>

#define ll long long
#define Int int
#define loop(x, start, end) for(Int x = start; x < end; x++)
#define loopdown(x, start, end) for(int x = start; x > end; x--)
#define rep(n) for(int x = 0; x < n; x++)
#define span(a,x,y) a.begin()+x,a.begin()+y
#define span_all(a) a.begin(),a.end()
#define len(x) (x.size())
#define last(x) (*(x.end()-1))

using namespace std;

#define W_MAX 10000
#define INF 10001
#define N_MAX 50001
#define M_MAX 21

Int N, M;
Int COINS[M_MAX+1];
Int table[M_MAX+1][N_MAX+1];

Int min_coin() {
  loop(n,0,N+1) {
    table[0][n] = INF;
  }
  loop(m,0,M+1) {
    table[m][0] = 0;
  }
  
  loop(m,1,M+1) {
    Int coin = COINS[m];
    loop(n,1,N+1) {
      if (n - coin < 0) {
        table[m][n] = table[m-1][n];
        continue;
      }
      
      table[m][n] = min(table[m-1][n], table[m][n-coin] + 1);
    }
  }
  
  return table[M][N];
}

int main(void) {
  cin >> N >> M;
  COINS[0] = 0;
  loop(i,1,M+1) {
    cin >> COINS[i];
  }
  
  cout << min_coin() << endl;
}