快排的思想 近似O(n)

调用降序快排的partition函数，设区间为[low,high]，返回index，则index左边都是大于data[index]的。
1. 若index及index左边数字有k个则data[index]就是第k大，index及其左边元素为Top K元素
2. 左边数字大于k个则继续在[low,index]里找
3. 左边数字小于k个则去右边[index+1,high]找 k - 左边数字个数

#include 
#include 
using namespace std;
const int maxn = 1e5 + 5;
//改为 data[high] >= key 和 data[low] <= key 则为第k小
int part(int *data, int low, int high) {
    int key = data[low];
    while (low < high) {
        while (low < high && data[high] <= key) high--;
        data[low] = data[high];
        while (low < high && data[low] >= key) low++;
        data[high] = data[low] ;
    }
    data[low] = key;
    return low;
}
int k_th(int *data, int k, int low, int high) {
    if (high - low + 1 < k || k < 1) return -1;
    int pos = part(data, low, high);
    int cnt = pos - low + 1;
    if (cnt == k) return data[pos];
    else if (cnt < k) return k_th(data, k - cnt, pos + 1, high);
    else return k_th(data, k, low, pos);
}
int k_th(int *data, int n, int k) {
    return k_th(data, k, 0, n - 1); //闭区间
    //遍历data[0,k)即可获得top K，且是有序的
}

int main() {

    // int data[] = {1, 5, 6, 7, 3, 2, 10, 9, 0, 231, 3214, 61};
    // int n = sizeof(data) / sizeof(int);
    // int k = 2;
    // cout << k_th(data, n, k) << endl;

    // freopen("in.txt","r",stdin);
    // freopen("out.txt","w",stdout);
    int n, k, data[maxn];
    std::ios::sync_with_stdio(false);
    while (cin >> n >> k) {
        for (int i = 0; i < n; ++i) {
            cin >> data[i];
        }
        cout << k_th(data, n, k) << endl;
    }
    return 0;
}

小根堆 O(nlogk)

维护一个k个元素的小根堆，保持堆顶为第k大，扫一遍数据，若堆里个数小于k则插入，否则看新的数和堆顶数大小关系：
1. 若新来的数小于等于堆顶，即新元素比Top K里最小的还小，则新来的数显然不可能是前k大
2. 若新来的数大于堆顶，则删掉堆顶，将新数字放到堆里且调整堆来保持堆的属性

由于实现堆代码量较多，我们可以用C++的优先队列、set等代替手工堆偷跑，当然这里也提供了手动实现版。

#include 
#include 
#include 
#include 
using namespace std;
const int maxn = 1e5 + 5;
//维持一个k大小的最小堆，根据新元素和堆顶大小决定要不要加入堆且删堆顶
// O(nlogk)
int biggest_k_th(int *data, int n, int k) {
    priority_queue, greater >q;   //小根堆
    while (!q.empty()) q.pop();

    for (int i = 0; i < n; ++i) {
        if (q.size() < k) {
            q.push(data[i]);
        } else if (data[i] > q.top()) {
            q.pop();
            q.push(data[i]);
        }
    }
    //取k次q.top()且pop()k次即为有序的前K大
    return q.top();
}

int smallest_k_th(int *data, int n, int k) {
    priority_queueq;   //大根堆
    while (!q.empty()) q.pop();

    for (int i = 0; i < n; ++i) {
        if (q.size() < k) {
            q.push(data[i]);
        } else if (data[i] < q.top()) {
            q.pop();
            q.push(data[i]);
        }
    }
    return q.top();
}

int main() {
    // freopen("in.txt","r",stdin);
    // freopen("out.txt","w",stdout);
    std::ios::sync_with_stdio(false);
    int n, k, data[maxn];
    while (cin >> n >> k) {
        for (int i = 0; i < n; ++i) {
            cin >> data[i];
        }
        cout << biggest_k_th(data, n, k) << endl;
    }
    return 0;
}

手动实现版

#include 
#include 
#include 
using namespace std;
const int maxn = 1e5 + 5;
const int maxK = 1e5 + 5;

int heapCnt = 0;
int heap[maxK];

void adjust(int *heap, int begin, int end) {    //[begin,end)
    int cur = begin;
    int son = 2 * cur + 1;
    while (son < end) {
        if (son + 1 < end && heap[son] > heap[son + 1]) son++;
        if (heap[cur] < heap[son]) return;
        swap(heap[son], heap[cur]);
        cur = son;
        son = 2 * cur + 1;
    }
}

void buildHeap(int *heap, int k) {  //[data,data+k) 开区间
    for (int i = k / 2; i >= 0;  --i) {
        adjust(heap, i, k);
    }
}

int k_th(int *data, int n, int k) {
    heapCnt = 0;
    for (int i = 0; i < n; ++i) {
        if (heapCnt < k) {
            heap[heapCnt++] = data[i];
            if (heapCnt == k) {
                buildHeap(heap, k); //data[0,k)共k个
            }
        } else {
            if (data[i] > heap[0]) {
                heap[0] = data[i];
                adjust(heap, 0, heapCnt);
            }
        }
    }
    return heap[0];
}

int main() {
    // freopen("in.txt", "r", stdin);
    // freopen("out.txt", "w", stdout);
    int n, k, data[maxn];
    std::ios::sync_with_stdio(false);
    while (cin >> n >> k) {
        for (int i = 0; i < n; ++i) {
            cin >> data[i];
        }
        cout << k_th(data, n, k) << endl;
    }
    return 0;
}

计数排序 O(n)

按照计数排序思想给数据的值计数，再从数据的最大值往最小值遍历，则总次数大于等于k的那个数为第k大，见代码一目了然。
优点：速度快且不用库也代码量少，妥妥的O(n)
缺点：只适用于数值不大的情况，当然你用hashmap这类库计数的话就没这问题了。

#include <iostream>
#include <cstdio>
#include <cstring>
using namespace std;
const int maxn = 1e5 + 5;
const int maxVal = 1e5 + 5; //O(n) 适用于数据值不大的情况

int k_th(int *data, int n, int k) {
    int mmin = data[0], mmax = data[0];
    int times[maxVal];
    memset(times,0,sizeof(times));

    for (int i = 0; i < n; ++i) {
        mmin = min(mmin, data[i]);
        mmax = max(mmax, data[i]);
        times[data[i]]++;
    }

    int cnt = 0;
    for (int i = mmax; i >= mmin; --i) {
        cnt += times[i];
        if (cnt >= k) { // >= 是因为第k大的数可能有若干个
            return i;
        }
        //反过来遍历则为第k小
        //每次输出times[i]次i则为有序前k大
    }
    return -1;
}

int main() {
    // freopen("in.txt","r",stdin);
    // freopen("out.txt","w",stdout);
    int n, k, data[maxn];
    std::ios::sync_with_stdio(false);
    while (cin >> n >> k) {
        for (int i = 0; i < n; ++i) {
            cin >> data[i];
        }
        cout<< k_th(data, n, k) <<endl; pre="" return="">

二分 O(nlogn)

二分答案值区间[l,r]，最开始l=所有数的最小值，r=最大值，假设当前值是mid，如果所有数据中大于等于mid的数字至少k个，说明当前数值可能是答案（若mid存在的情况则将区间调为[mid,r]，mid不存在的话就改为[mid+1,r]），否则mid偏大，在[l,mid-1]里查找；二分不会的可见这篇文章。

#include 
#include 
#include 
using namespace std;
const int maxn = 1e5 + 5;
const int maxVal = 1e5 + 5;

bool ok(int *data, int n, int k, int mid) {
    int cnt = 0;
    for (int i = 0; i < n; ++i) {
        if (data[i] >= mid) cnt++;
    }
    return cnt >= k;
}
int k_th(int *data, int n, int k) {
    int mmin = data[0], mmax = data[0];
    bool vis[maxVal];
    memset(vis, false, sizeof(vis));

    for (int i = 0; i < n; ++i) {
        mmin = min(mmin, data[i]);
        mmax = max(mmax, data[i]);
        vis[data[i]] = true;
    }

    int l = mmin, r = mmax;
    while (l < r) {
        int mid = (l + r + 1) / 2;
        if (ok(data, n, k, mid)) {
            if (!vis[mid]) l = mid + 1;
            else l = mid;
        } else {
            r = mid - 1;
        }
    }
    return l;
}

int main() {
    // freopen("in.txt", "r", stdin);
    // freopen("out.txt", "w", stdout);
    int n, k, data[maxn];
    std::ios::sync_with_stdio(false);
    while (cin >> n >> k) {
        for (int i = 0; i < n; ++i) {
            cin >> data[i];
        }
        cout << k_th(data, n, k) << endl;
    }
    return 0;
}

暴力式选择/冒泡排序 O(kn)

特慢做法：排序k个，每次遍历n个元素，O(k*n)

#include 
#include 
#include 
using namespace std;
const int maxn = 1e5 + 5;

int k_th(int *data, int n, int k) {
    for (int i = 0; i < k; ++i) {
        for (int j = 0; j < n - i - 1; ++j) {
            if (data[j] > data[j + 1]) {
                swap(data[j], data[j + 1]);
            }
        }
    }
    return data[n-k];
}

int main() {
    // freopen("in.txt", "r", stdin);
    // freopen("out.txt", "w", stdout);
    int n, k, data[maxn];
    std::ios::sync_with_stdio(false);
    while (cin >> n >> k) {
        for (int i = 0; i < n; ++i) {
            cin >> data[i];
        }
        cout << k_th(data, n, k) << endl;
    }
    return 0;
}

真暴力排序O(nlogn)

排完取 data[k]，这么暴力就不说了。

数据生成代码

生成10组数据，每组一个n（范围：[a_n,b_n]），然后n个数 [a,b]。

#include 
#include 
#include 
using namespace std;
int rand_ab(int a, int b) { //[a,b]
    return a + rand() % (b + 1 - a);
}
void make(){    
    int a_n = 10000, b_n = 100000;
    int a = 1, b = 10000;
    for (int i = 0; i < 10; ++i) {
        int n = rand_ab(a_n, b_n);
        printf("%d ", n);
        int a_k = 1, b_k = n;
        printf("%d\n", rand_ab(a_k,b_k));
        printf("%d", rand_ab(a, b));
        for (int i = 1; i < n; ++i) {
            printf(" %d", rand_ab(a, b));
        }
        printf("\n");
    }
}
int main() {
    // freopen("out.txt","w",stdout);
    make();
    return 0;
}