动态规划习题代码题解

最长上升子序列

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

int main()
{
    int n;
    cin >> n;
    vector<int> a(n + 1);
    for (int i = 1; i <= n; i ++ )
        cin >> a[i];
    
    vector<int> dp(n + 1, 1);
    for (int i = 1; i <= n; i ++ )
        for (int j = 1; j < i; j ++ )
            if (a[i] > a[j])
                dp[i] = max(dp[i], dp[j] + 1);
    int maxx = 0;
    for (int i = 1; i <= n; i ++ ) maxx = max(maxx, dp[i]);
    cout << maxx << endl;
    return 0;
}

 P1048 [NOIP 2005 普及组] 采药

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

int main()
{
    int T, M;
    cin >> T >> M;
    vector<int> a(M + 1), b(M + 1);

    for (int i = 1; i <= M; i ++ ) cin >> a[i] >> b[i];

    vector<vector<int>> dp(M + 1, vector<int> (T + 1, 0));
    for (int i = 1; i <= M; i ++ )
        for (int j = 1; j <= T; j ++ )
            {
                dp[i][j] = dp[i - 1][j];
            if (a[i] <= j)
            {
                dp[i][j] = max(dp[i - 1][j], dp[i - 1][j - a[i]] + b[i]);
            }
            }
    int ans = 0;
    for (int i = 1; i <= T; i ++ )
    {
        ans = max(ans, dp[M][i]);
    }
    cout << ans << endl;
    return 0;

    
}

P1060 [NOIP 2006 普及组] 开心的金明

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

int main()
{
    int n, m;
    cin >> m >> n;

    vector<long long> v(n + 1), w(n + 1);
    vector<vector<long long>> dp(n + 1, vector<long long> (m + 1));
    for (int i = 1; i <= n; i ++ )
    {
        long long a, b;
        cin >> a >> b;
        v[i] = a;
        w[i] = a * b;
    }
    long long ans = 0;
    for (int i = 1; i <= n; i ++ )
        for (int j = 1; j <= m; j ++ )
        {
            dp[i][j] = dp[i - 1][j];
            if (j >= v[i])
                dp[i][j] = max(dp[i][j], dp[i - 1][j - v[i]] + w[i]);
            ans = max(ans, dp[i][j]);
        }
    cout << ans << endl;
    
    return 0;
}

P1002 [NOIP 2002 普及组] 过河卒

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

int a[100][100];
long long dp[100][100];
int dx[8] = {1, 2, 2, 1, -1, -2, -2, -1}, dy[8] = {2, 1, -1, -2, -2, -1, 1, 2};
int main()
{
    int n, m, c, d;
    cin >> n >> m >> c >> d;
    memset(a, 0, sizeof a);
    a[c][d] = 1;
    for (int i = 0; i < 8; i ++ )
    {
        int x = dx[i] + c, y = dy[i] + d;
        // cout << x << " " << y << endl;
        if (x >= 0 && x <= n && y >= 0 && y <= m) a[x][y] = 1;
    }
    memset(dp, 0, sizeof dp);
    dp[0][0] = 1;
    
    for (int i = 0; i <= n; i ++ )
    {
        for (int j = 0; j <= m; j ++ )
        {
            
            if (i) dp[i][j] += dp[i - 1][j];
            if (j) dp[i][j] += dp[i][j - 1];
            if (a[i][j]) dp[i][j] = 0;
            // if (a[i][j]) cout << "h" << " ";
            // else cout << dp[i][j] << " ";
        }
        // cout << endl;
    }
    cout << dp[n][m] << endl;
    return 0;
}

P1806 跑步 

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

long long dp[510][510];
int main()
{
    int n;
    cin >> n;
    dp[0][0] = 1;
    for (int i = 1; i <= n; i ++ )
    {
        for (int j = 1; j <= i; j ++ )
        {
            for (int k = 0; k < j; k ++ )
            {
                dp[i][j] += dp[i - j][k];
            }
            // cout << dp[i][j] << " ";
        }
        // cout << endl;
    }
    long long ans = 0;
    for (int i = 1; i <= n; i ++ )
        ans += dp[n][i];
    cout << ans - 1 << endl;
    return 0;
}

最长正方形

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

const int N = 110;

int a[N][N], dp[N][N];

int main()
{
    int n, m;
    cin >> n >> m;
    for (int i = 1; i <= n; i ++ )
    {
        for (int j = 1; j <= m; j ++ )
            cin >> a[i][j];
    }
    int ans = 0;
    for (int i = 1; i <= n; i ++ )
    {
        for (int j = 1; j <= m; j ++ )
        {
            if (a[i][j] == 1)
            {
                dp[i][j] = min(dp[i - 1][j], min(dp[i - 1][j - 1], dp[i][j - 1])) + 1;
            }
            ans = max(ans, dp[i][j]);
            // cout << dp[i][j] << " ";
        }
        // cout << endl;
    }
    cout << ans << endl;
    return 0;
}

P11248 [GESP202409 七级] 矩阵移动

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

const int N = 510;

int dp[N][N][N];

void solve()
{
    int n, m, x;
    cin >> n >> m >> x;
    vector<string> s(n + 1);
    for (int i = 1; i <= n; i ++ ){
        cin >> s[i];
        s[i] = " " + s[i];
        // cout << s[i] << endl;
    }
    
    for (int i = 1; i <= n; i ++ )
        for (int j = 1; j <= m; j ++ )
        {
            for (int k = 0; k <= x; k ++ )
            {
                dp[i][j][k] = max(dp[i][j - 1][k], dp[i - 1][j][k]) + (s[i][j] == '1');
                if (s[i][j] == '?' && k) dp[i][j][k] = max(dp[i][j][k], max(dp[i][j - 1][k - 1], dp[i - 1][j][k - 1]) + 1);
            }
        }
    int mx = 0;
    for (int i = 0; i <= x; i ++ )
        mx = max(mx, dp[n][m][i]);
    cout << mx << endl;
}

int main()
{
    int t;
    cin >> t;
    while (t -- )
    {
        solve();
    }
    return 0;
}