FUNDAMENTAL PART1

Part1

+++

1.快排quick_sort()

分而治之

#include <iostream>

using namespace std;

const int N = 1e6 + 10;

int q[N], n;

void quick_sort(int q[], int l, int r)
{
    if(l >= r) return;
    
    int x = q[l + r >> 1], i = l - 1, j = r + 1;
    
    while(i < j)
    {
        do i ++ ; while(q[i] < x);
        do j -- ; while(q[j] > x);
        
        if(i < j) swap(q[i], q[j]);
    }
    
    quick_sort(q, l, j);
    quick_sort(q, j + 1, r);
}

int main()
{
    scanf("%d", &n);
    for(int i = 0; i < n; i ++ ) scanf("%d", q + i);
    
    quick_sort(q, 0, n - 1);
    
    for(int i = 0; i < n; i ++ ) printf("%d ", q[i]);
    cout << endl;
    
    return 0;
}

2.归并排序merge_sort()

#include <iostream>

using namespace std;

const int N = 1e6 + 10;

int q[N], temp[N], n;

void merge_sort(int q[], int l, int r)
{
    if(l >= r) return;
    
    int mid = l + r >> 1;
    
    merge_sort(q, l, mid), merge_sort(q, mid + 1, r);
    
    int k = 0, i = l, j = mid + 1;
    while(i <= mid && j <= r)
        if(q[i] <= q[j]) temp[k ++ ] = q[i ++ ];
        else temp[k ++ ] = q[j ++ ];
    
    while(i <= mid) temp[k ++ ] = q[i ++ ];
    while(j <= r) temp[k ++ ] = q[j ++ ];
    
    for(i = l, j = 0; i <= r; i ++ , j ++ ) q[i] = temp[j];
}

int main()
{
    scanf("%d", &n);
    for (int i = 0; i < n; i ++ ) scanf("%d", q + i);
    
    merge_sort(q, 0, n - 1);
    
    for (int i = 0; i < n; i ++ ) printf("%d ", q[i]);
    cout << endl;
    
    return 0;
}

3.高精度加法

(不压位)

#include <iostream>
#include <vector>

using namespace std;

vector<int> add(vector<int> &A, vector<int> &B)
{
    vector<int> C;
    
    int t = 0;
    for(int i = 0; i < A.size() || i < B.size(); i ++ )
    {
        if(i < A.size()) t += A[i];
        if(i < B.size()) t += B[i];
        
        C.push_back(t % 10);
        t /= 10;
    }
    
    if(t) C.push_back(1);
    
    return C;
}

int main()
{
    string a, b;
    vector<int> A, B;
    
    cin >> a >> b;
    
    for (int i = a.size() - 1; i >= 0; i -- ) A.push_back(a[i] - '0');
    for (int i = b.size() - 1; i >= 0; i -- ) B.push_back(b[i] - '0');
    
    auto C = add(A, B);
    
    for(int i = C.size() - 1; i >= 0; i -- ) printf("%d", C[i]);
    
    return 0;
}

4.高精度减法

#include <iostream>
#include <vector>

using namespace std;

bool cmp(vector<int> &A, vector<int> &B)
{
    if(A.size() != B.size()) return A.size() > B.size();
    
    for(int i = A.size() - 1; i >= 0; i -- )
        if(A[i] != B[i]) return A[i] > B[i];
    
    return true;
}

vector<int> sub(vector<int> &A, vector<int> &B)
{
    vector<int> C;
    for(int i = 0, t = 0; i < A.size(); i ++ )
    {
        t = A[i] - t;
        if(i < B.size()) t -= B[i];
        
        C.push_back((t + 10) % 10);
        if(t < 0) t = 1;
        else t = 0;
    }
    
    while(C.size() > 1 && C.back() == 0) C.pop_back();//消除前置0
    
    return C;
}

int main()
{
    string a, b;
    vector<int> A, B;
    
    cin >> a >> b;
    for(int i = a.size() - 1; i >= 0; i -- ) A.push_back(a[i] - '0');
    for(int i = b.size() - 1; i >= 0; i -- ) B.push_back(b[i] - '0');
    
    if(cmp(A, B))
    {
        auto C = sub(A, B);
        
        for(int i = C.size() - 1; i >= 0; i -- ) printf("%d", C[i]);
    }
    else
    {
        auto C = sub(B, A);
        
        printf("-");
        for(int i = C.size() - 1; i >= 0; i -- ) printf("%d", C[i]);
    }
    
    return 0;
}

5.高精度乘法

#include <iostream>
#include <vector>

using namespace std;

vector<int> mul(vector<int> &A, int b)
{
    vector<int> C;
    
    for(int i = 0, t = 0; i < A.size() || t; i ++ )
    {
        if(i < A.size()) t += A[i] * b;
        C.push_back(t % 10);
        t /= 10;
        
    }
    
    while(C.size() > 1 && C.back() == 0) C.pop_back();
    
    return C;
}

int main()
{
    string a;
    int b;
    
    cin >> a >> b;
    
    vector<int> A;
    for(int i = a.size() - 1; i >= 0 ; i -- ) A.push_back(a[i] - '0');
    
    auto C = mul(A, b);
    
    for(int i = C.size() - 1; i >= 0; i -- ) printf("%d", C[i]);
    
    return 0;
}

6.高精度除法

#include <iostream>
#include <algorithm>
#include <vector>

using namespace std;

vector<int> div(vector<int> &A, int b, int &r)
{
    vector<int> C;
    r = 0;
    for(int i = A.size() - 1; i >= 0; i -- )
    {
        r = r * 10 + A[i];
        C.push_back(r / b);
        r %= b;
    }
    
    reverse(C.begin(), C.end());
    
    while(C.size() > 1 && C.back() == 0) C.pop_back();
    
    return C;
}

int main()
{
    string a;
    int b;
    
    cin >> a >> b;
    
    vector<int> A;
    for(int i = a.size() - 1; i >= 0; i -- ) A.push_back(a[i] - '0');
    
    int r;
    auto C = div(A, b, r);
    
    for(int i = C.size() - 1; i >= 0; i -- ) printf("%d", C[i]);
    cout << endl << r << endl;
    
    return 0;
}

7.前缀和

one way

#include <iostream>

using namespace std;

const int N = 1e5 + 10;

int a[N], s[N];

int main()
{
    int n, m;
    scanf("%d%d", &n, &m);
    
    for (int i = 1; i <= n; i ++ )
    {
        scanf("%d", a + i);
        s[i] = s[i - 1] + a[i];
    }
    
    int l, r;
    while(m -- )
    {
        scanf("%d%d", &l, &r);
        printf("%d
", s[r] - s[l - 1]);
    }
    
    return 0;
}

two way

#include <iostream>

using namespace std;

const int N = 1010;

int a[N][N], s[N][N];
int n, m, q;

int main()
{
    scanf("%d%d%d", &n, &m, &q);
    
    for (int i = 1; i <= n; i ++ )
        for (int j = 1; j <= m; j ++ )
        {
            scanf("%d", &a[i][j]);
            s[i][j] = s[i - 1][j] + s[i][j - 1] - s[i - 1][j - 1] + a[i][j];
        }
    
    int x1, y1, x2, y2;
    while(q -- )
    {
        scanf("%d%d%d%d", &x1, &y1, &x2, &y2);
        printf("%d
", s[x2][y2] - s[x2][y1 - 1] - s[x1 - 1][y2] + s[x1 - 1][y1 - 1]);
    }
    
    return 0;
}

8.差分

one way

#include <iostream>

using namespace std;

const int N = 1e5 + 10;

int a[N], b[N];
int n, m;

void insert(int l, int r, int c)
{
    b[l] += c, b[r + 1] -= c;
}

int main()
{
    scanf("%d%d", &n, &m);
    
    for (int i = 1; i <= n; i ++ )
    {
        scanf("%d", a + i);
        insert(i, i, a[i]);
    }
    
    int l, r, c;
    while(m -- )
    {
        scanf("%d%d%d", &l, &r, &c);
        insert(l, r, c);
    }
    
    for (int i = 1; i <= n; i ++ ) b[i] += b[i - 1];

    for (int i = 1; i <= n; i ++ ) printf("%d ", b[i]);

    return 0;
}

two way

#include <iostream>

using namespace std;

const int N = 1010;

int a[N][N], b[N][N];
int n, m, q;

void insert(int x1, int y1, int x2, int y2, int c)
{
    b[x1][y1] += c;
    b[x2 + 1][y1] -= c;
    b[x1][y2 + 1] -= c;
    b[x2 + 1][y2 + 1] += c;
}

int main()
{
    scanf("%d%d%d", &n, &m, &q);
    for (int i = 1; i <= n; i ++ )
        for (int j = 1; j <= m; j ++ )
        {
            scanf("%d", &a[i][j]);
            insert(i, j, i, j, a[i][j]);
        }
    
    int x1, x2, y1, y2, c;
    while(q -- )
    {
        cin >> x1 >> y1 >> x2 >> y2 >> c;
        insert(x1, y1, x2, y2, c);
    }
    
    for (int i = 1; i <= n; i ++ )
        for (int j = 1; j <= m; j ++ )
            b[i][j] += b[i - 1][j] + b[i][j - 1] - b[i - 1][j - 1];
    
    for (int i = 1; i <= n; i ++ )
    {
        for(int j = 1; j <= m; j ++ )
            printf("%d ", b[i][j]);
        
        puts("");
    }
    
    return 0;
}

+++

+++

1.第k个数(quick_sort)

#include <iostream>
#include <algorithm>

using namespace std;

const int N = 1e5 + 10;

int a[N], n, k;

int quick_sort(int l, int r, int k)
{
    if(l == r) return a[l];
    
    int x = a[l + r >> 1], i = l - 1, j = r + 1;
    while(i < j)
    {
        do i ++ ; while(a[i] < x);
        do j -- ; while(a[j] > x);
        
        if(i < j) swap(a[i], a[j]);
    }
    
    int sl = j - l + 1;
    if(k <= sl) return quick_sort(l, j, k);
    
    return quick_sort(j + 1, r, k - sl);
}

int main()
{
    cin >> n >> k;
    for (int i = 0; i < n; i ++ ) cin >> a[i];
    
    cout << quick_sort(0, n - 1, k) << endl;
    
    return 0;
}

2.逆序对的数量

#include <iostream>

using namespace std;

typedef long long LL;

const int N = 1e5 + 10;

int q[N], temp[N], n;

LL merge_sort(int l, int r)
{
    if(l >= r) return 0;
    
    int mid = l + r >> 1;
    
    LL res = merge_sort(l, mid) + merge_sort(mid + 1, r);
    
    int k = 0, i = l, j = mid + 1;
    while(i <= mid && j <= r)
        if(q[i] <= q[j]) temp[k ++ ] = q[i ++ ];
        else
        {
            temp[k ++ ] = q[j ++ ];
            res += mid - i + 1;
        }
    
    while(i <= mid) temp[k ++ ] = q[i ++ ];
    while(j <= r) temp[k ++ ] = q[j ++ ];
    
    for (i = l, j = 0; i <= r; i ++ , j ++ ) q[i] = temp[j];
    
    return res;
}

int main()
{
    scanf("%d", &n);
    for (int i = 0; i < n; i ++ ) scanf("%d", q + i);
    
    printf("%lld
", merge_sort(0, n - 1));
    
    return 0;
}

+++

+++

9.双指针(最长连续不重复子序列)

先想暴力做法，然后根据单调关系优化复杂度

#include <iostream>

using namespace std;

const int N = 1e5 + 10;

int a[N], s[N], n;

int main()
{
    scanf("%d", &n);
    for (int i = 0; i < n; i ++ ) cin >> a[i];
    
    int res = 0;
    for (int i = 0, j = 0; i < n; i ++ )
    {
        s[a[i]] ++ ;
        while(j <= i && s[a[i]] > 1)
        {
            s[a[j]] -- ;
            j ++ ;
        }
        
        res = max(res, i - j + 1);
    }
    
    cout << res << endl;
    
    return 0;
}

10.位运算（1的个数）

#include <iostream>

using namespace std;

int lowbit(int x)
{
    return x & -x;
}

int main()
{
    int n;
    
    cin >> n;
    while(n -- )
    {
        int x;
        cin >> x;
        
        int res = 0;
        while(x) x -= lowbit(x), res ++ ;
        
        cout << res << ' ';
    }
    
    return 0;
}

11.离散化（区间和）

#include <iostream>
#include <vector>
#include <algorithm>

using namespace std;

typedef pair<int, int> PII;

const int N = 3e5 + 10;

int n, m;
int a[N], s[N];

vector<int> alls;
vector<PII> add, query;

int find(int x)
{
    int l = 0, r = alls.size() - 1;
    while(l < r)
    {
        int mid = l + r >> 1;
        if(alls[mid] >= x) r = mid;
        else l = mid + 1;
    }
    
    return r + 1;
}

int main()
{
    cin >> n >> m;
    
    int x, c;
    for (int i = 0; i < n; i ++ )
    {
        cin >> x >> c;
        
        add.push_back({x, c});
        alls.push_back(x);
    }
    
    int l, r;
    for (int i = 0; i < m; i ++ )
    {
        cin >> l >> r;
        
        query.push_back({l, r});
        alls.push_back(l);
        alls.push_back(r);
    }
    
    //去重
    sort(alls.begin(), alls.end());
    alls.erase(unique(alls.begin(), alls.end()), alls.end());
    
    for (auto item : add)
    {
        int x = find(item.first);
        a[x] += item.second;
    }
    
    for (int i = 1; i <= alls.size(); i ++ ) s[i] = s[i - 1] + a[i];
    
    for (auto item : query)
    {
        int l = find(item.first), r = find(item.second);
        
        cout << s[r] - s[l - 1] << endl;
    }
    
    return 0;
}

12.区间和并

#include <iostream>
#include <vector>
#include <algorithm>

using namespace std;

typedef pair<int, int> PII;

vector<PII> segs;
int n;

void merge(vector<PII> &segs)
{
    vector<PII> res;
    
    sort(segs.begin(), segs.end());
    
    int st = -2e9, ed = -2e9;
    for (auto seg : segs)
        if(ed < seg.first)
        {
            if(ed != -2e9) res.push_back({st, ed});
            st = seg.first, ed = seg.second;
        }
        else ed = max(ed, seg.second);
    
    if(st != -2e9) res.push_back({st, ed});
    
    segs = res;
}

int main()
{
    scanf("%d", &n);
    int l, r;
    while(n -- )
    {
        scanf("%d%d", &l, &r);
        segs.push_back({l, r});
    }
    
    merge(segs);
    
    cout << segs.size() << endl;
    
    return 0;
}