1.火柴排队

思路

1.求最小值的时候，可以直接按升序排序，这样得到的值就是最小值
2.求最小交换次数的时候，不能直接排序，因为只能交换相邻的数，只需要知道他们的相对大小，所以可以先用离散化，把火柴高度映射成 1 到 n，然后用一个中间数组 c，让 b 数组按照 a 数组的顺序归并排序，交换相邻两个元素，最多只会使得逆序对数量减一

#include<iostream>
#include<algorithm>
using namespace std;
const int N = 1e5 + 10, mod = 99999997;
int a[N], b[N], c[N], d[N];
int n;

// 离散化 a 和 b 数组
void init(int q[]){
    for(int i = 1; i <= n; i++) d[i] = i;
    // d 数组根据 q 数组的大小关系排序
    sort(d + 1, d + n + 1, [&](int x, int y){
        return q[x] < q[y];
    });
    for(int i = 1; i <= n; i++) q[d[i]] = i;
}

int merge_sort(int l, int r){
    if(l >= r) return 0;
    int mid = (l + r) / 2;
    int res = (merge_sort(l, mid) + merge_sort(mid + 1, r)) % mod;
    int x = 0, i = l, j = mid + 1;
    while(i <= mid && j <= r){
        if(b[i] <= b[j]) d[x++] = b[i++];
        else{
            d[x++] = b[j++];
            res = (res + mid - i + 1) % mod;
        }
    }
    while(i <= mid) d[x++] = b[i++];
    while(j <= r) d[x++] = b[j++];
    for(int i = l, j = 0; j < x; i++, j++) b[i] = d[j];
    return res;
}

int main(){
    scanf("%d", &n);
    for(int i = 1; i <= n; i++) scanf("%d", &a[i]);
    for(int i = 1; i <= n; i++) scanf("%d", &b[i]);
    init(a), init(b);
    
    //for(int i = 1; i <= n; i ++ ) cout<<a[i]<<" ";
    //cout << "a" << endl;
    //for(int i = 1; i <= n; i ++ ) cout<<b[i]<<" ";
    //cout << "b" << endl;
    
    // c 数组做为中间数组，使得 a 数组是 "有序的"，让 b 数组按照 a 数组的顺序进行归并排序
    for(int i = 1; i <= n; i++) c[a[i]] = i;
    for(int i = 1; i <= n; i++) b[i] = c[b[i]];
    
    //for(int i = 1; i <= n; i ++ ) cout<<b[i]<<" ";
    //cout << "b" << endl;
    
    // 让 b 数组按照 a 数组的顺序进行归并排序
    int res = merge_sort(1, n);
    printf("%d", res);
    return 0;
}

2.归并排序

思路

模板题

#include<iostream>
using namespace std;
const int N = 1e5 + 10;
int a[N], b[N];
int n;

void merge_sort(int l, int r){
    if(l >= r) return;
    int mid = (l + r) / 2;
    merge_sort(l, mid), merge_sort(mid + 1, r);
    int x = 0, i = l, j = mid + 1;
    while(i <= mid && j <= r){
        if(a[i] <= a[j]) b[x++] = a[i++];
        else b[x++] = a[j++];
    }
    while(i <= mid) b[x++] = a[i++];
    while(j <= r) b[x++] = a[j++];
    for(int i = l, j = 0; j < x; i++, j++) a[i] = b[j];
}

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

3.逆序对的数量

思路

模板题

#include<iostream>
using namespace std;
const int N = 1e5 + 10;
int a[N], b[N];
int n;
long long res;

void merge_sort(int l, int r){
    if(l >= r) return;
    int mid = (l + r) / 2;
    merge_sort(l, mid), merge_sort(mid + 1, r);
    int x = 0, i = l, j = mid + 1;
    while(i <= mid && j <= r){
        if(a[i] <= a[j]) b[x++] = a[i++];
        else{
            res += 1ll * mid - i + 1;
            b[x++] = a[j++];
        }
    }
    while(i <= mid) b[x++] = a[i++];
    while(j <= mid) b[x++] = a[j++];
    for(int i = l, j = 0; j < x; i++, j++) a[i] = b[j];
}

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

4.小朋友排队

思路

k 队逆序对，最少交换次数就是 k，对于每个数，k1 表示左边有多少个比它大的，k2 表示右边有多少个比它小的，所有数的 k1 和 k2 加起来 >= 2 * k，最小就是 2 * k，也就是逆序对数量的两倍，所以一共交换 1 + 2 + 3 + … + k1 + k2，那么不高兴程度之和就是每个位置的 (1 + k1 + k2) * (k1 + k2) / 2 之和

#include<iostream>
using namespace std;
const int N = 1e5 + 10;
pair<int, int> a[N], b[N]; // 存储值和下标
long long sum[N];
int n;

void merge_sort(int l, int r){
    if(l >= r) return;
    int mid = (l + r) / 2;
    merge_sort(l, mid), merge_sort(mid + 1, r);
    int x = 0, i = l, j = mid + 1;
    while(i <= mid && j <= r){
        // 加上后面比 a[i] 小的数
        if(a[i].first <= a[j].first){
            sum[a[i].second] += j - mid - 1;
            b[x++] = a[i++];
        }else{
            // 加上前面比 a[j] 大的数
            sum[a[j].second] += mid - i + 1;
            b[x++] = a[j++];
        }
    }
    while(i <= mid){
        sum[a[i].second] += j - mid - 1;
        b[x++] = a[i++];
    }
    while(j <= r) b[x++] = a[j++];
    for(int i = l, j = 0; j < x; i++, j++) a[i] = b[j];
}

int main(){
    scanf("%d", &n);
    int x;
    for(int i = 0; i < n; i++){
        scanf("%d", &x);
        a[i] = make_pair(x, i);
    }
    merge_sort(0, n - 1);
    long long res = 0;
    for(int i = 0; i < n; i++) res += (1 + sum[i]) * sum[i] / 2;
    printf("%lld", res);
    return 0;
}

5.超快速排序

思路

逆序对的数量模板题

#include<iostream>
using namespace std;
const int N = 5e5 + 10;
int a[N], b[N];
int n;
long long res;

void merge_sort(int l, int r){
    if(l >= r) return;
    int mid = (l + r) / 2;
    merge_sort(l, mid), merge_sort(mid + 1, r);
    int x = 0, i = l, j = mid + 1;
    while(i <= mid && j <= r){
        if(a[i] <= a[j]) b[x++] = a[i++];
        else{
            res += 1ll * mid - i + 1;
            b[x++] = a[j++];
        }
    }
    while(i <= mid) b[x++] = a[i++];
    while(j <= mid) b[x++] = a[j++];
    for(int i = l, j = 0; j < x; i++, j++) a[i] = b[j];
}

int main(){
    while(scanf("%d", &n) && n){
        for(int i = 0; i < n; i++) scanf("%d", &a[i]);
        merge_sort(0, n - 1);
        printf("%lld\n", res);
        res = 0;
    }
    return 0;
}