Merge Sort及其对一类问题的应用

1.归并排序 O(nlogn) stable

#include <iostream>
#include <vector>
using namespace std;

void merge(vector<int>& arr, int l, int mid, int r){
    int n1 = mid - l + 1, n2 = r - mid;
    vector<int> left(n1);
    vector<int> right(n2);
    for(int i = 0; i < n1; ++i) left[i] = arr[l + i];
    for(int i = 0; i < n2; ++i) right[i] = arr[mid + 1 + i];
    int i = 0, j = 0, k = l;
    while(i < n1 && j < n2){
        if(left[i] <= right[j]) arr[k++] = left[i++];
        else arr[k++] = right[j++];
    }
    while(i < n1) arr[k++] = left[i++];
    while(j < n2) arr[k++] = right[j++];
}


void mergeSort(vector<int>& arr, int l, int r){
    if(l < r){
        int mid = (l + r) / 2;
        mergeSort(arr, l, mid);
        mergeSort(arr, mid + 1, r);
        merge(arr, l, mid, r);
    }
}


int main(){
    vector<int> input = {3, 4, 6, 1, 9, 5, 2, 7, 0, 8};
    mergeSort(input, 0, input.size() - 1);
    for(int i : input)
        cout << i << " ";
    return 0;
}

2.数组中逆序对个数Count Inversions

#include <iostream>
#include <vector>
using namespace std;

int merge(vector<int>& arr, int l, int mid, int r);

int mergeSort(vector<int>& arr, int l, int r){
    int invCount = 0;
    if(l < r){
        int mid = (l + r) / 2;
        invCount = mergeSort(arr, l, mid);
        invCount += mergeSort(arr, mid + 1, r);
        invCount += merge(arr, l, mid, r); 
    }
    return invCount;
}


int merge(vector<int>& arr, int l, int mid, int r){
    int n1 = mid - l + 1, n2 = r - mid;
    vector<int> left(n1);
    vector<int> right(n2);
    for(int i = 0; i < n1; ++i) left[i] = arr[l + i];
    for(int i = 0; i < n2; ++i) right[i] = arr[mid + 1 + i];
    int i = 0, j = 0, k = l;
    int invCount = 0;
    while(i < n1 && j < n2){
        if(left[i] > right[j]){
            invCount += mid - i + 1;
            arr[k++] = right[j++];
        }
        else
            arr[k++] = left[i++];
    }
    while(i < n1) arr[k++] = left[i++];
    while(j < n2) arr[k++] = right[j++];
    return invCount;
}


int main(){
    vector<int> input = {1, 3, 5, 2, 4};
    int ans = mergeSort(input, 0, input.size() - 1);
    for(int i : input)
        cout << i << " ";
    cout << endl;
    cout << ans;
    return 0;
}

3.Leetcode 493 Reverse Pairs

Given an array nums, we call (i, j) an important reverse pair if i < j and nums[i] > 2 * nums[j]. You need to return the number of important reverse pairs in the given array.

class Solution {
public:
    vector<int> helper;
    
    int reversePairs(vector<int>& nums) {
        helper.resize(nums.size());
        return mergeSort(nums, 0, nums.size() - 1);
    }
    
    int mergeSort(vector<int>& nums, int s, int e){
        if(s >= e) return 0;
        int mid = s + (e - s) / 2;
        int cnt = mergeSort(nums, s, mid) + mergeSort(nums, mid + 1, e);
        for(int i = s, j = mid + 1; i <= mid; ++i){
            while(j <= e && nums[i] / 2.0 > nums[j])
                j++;
            cnt += j - (mid + 1);
        }
        merge(nums, s, mid, e);
        return cnt;
    }
    
    void merge(vector<int>& nums, int s, int mid, int e){
        for(int i = s; i <= e; ++i) helper[i] = nums[i];
        int p1 = s;
        int p2 = mid + 1;
        int i = s;
        while(p1 <= mid || p2 <= e){  //注意这个merge的逻辑
            if(p1 > mid || p2 <= e && helper[p1] >= helper[p2])
                nums[i++] = helper[p2++];
            else
                nums[i++] = helper[p1++];
        }
    }
    
};

第二次写:

class Solution {
public:
    vector<int> a;
    vector<int> t;
    int ans = 0;
    // [l, r)
    void merge(int l, int r) {
        if (r - l <= 1) return;
        int mid = (l + r) >> 1;
        merge(l, mid);
        merge(mid, r);
        for (int i = l, j = mid; i < mid; ++i) {
            while (j < r && a[i] / 2.0 > a[j]) j++;
            ans += j - mid;
        }
        int p = l, q = mid, s = l;
        while (s < r) {
            // 注意不能和求逆序对一样，在此处if内统计个数，而要单独开个循环统计
            // 否则 [2,4],[1,3,5]这里面的4,1就没法统计到
            if (p >= mid || q < r && a[p] > a[q]) t[s++] = a[q++];
            else t[s++] = a[p++];
        }
        for (int i = l; i < r; ++i) a[i] = t[i];
    }
    int reversePairs(vector<int>& nums) {
        a = nums;
        t = nums;
        merge(0, a.size());
        return ans;
    }
};

4.Leetcode 315 Count  of Smaller Numbers After self

You are given an integer array nums and you have to return a new counts array. The counts array has the porperty where counts[i] is the number o f smaller elements to the right of nums[i].

用 pair 记录每个数字在数组中的索引，对 {nums[i], i} 归并排序的同时计算 nums[i] 右侧小于它的数的个数，将个数加到 res[i] 中。

class Solution {
public:
    typedef pair<int, int> pii;
    typedef vector<pii>::iterator pit;
    
    void merge(pit l, pit r, vector<int>& res) {
        if (r - l <= 1) return;
        auto mid = l + (r - l) / 2; 
        merge(l, mid, res);
        merge(mid, r, res);
        for (auto i = l, j = mid; i != mid; ++i) {
            while (j != r && i->first > j->first) ++j;
            res[i->second] += j - mid; 
        }
        inplace_merge(l, mid, r);
    }
    vector<int> countSmaller(vector<int>& nums) {
        int n = nums.size();
        vector<int> res(n);
        vector<pii> nu(n);
        for (int i = 0; i < n; ++i) nu[i] = make_pair(nums[i], i);
        merge(nu.begin(), nu.end(), res);
        return res;
    }
};

 

5.Leetcode 327 Count of Range Sum

Given an interger array nums, return the number of range sums that lie in [lower, upper] inclusive. Range sum S(i, j) is defined as the sum of the elements in nums between indices i and j (i <= j), inclusive.

Note: A naive algorithm of $O(n_2)$ is trivial. You MUST do better than that.

class Solution {
public:
    int countRangeSum(vector<int>& nums, int lower, int upper) {
        int size = nums.size();
        if(size == 0) return 0;
        vector<long> sums(size + 1, 0);
        for(int i = 0; i < size; ++i)
            sums[i + 1] = sums[i] + nums[i];
        return help(sums, 0, size + 1, lower, upper);
    }
    
    int help(vector<long>& sums, int start, int end, int lower, int upper){
        if(end - start <= 1) return 0;
        int mid = (start + end) / 2;
        int cnt = help(sums, start, mid, lower, upper) + help(sums, mid, end, lower, upper);
        int m = mid, n = mid, t = mid, len = 0;
        vector<long> cache(end - start, 0);
        for(int i = start, s = 0; i < mid; ++i, ++s){
            while(m < end && sums[m] - sums[i] < lower) ++m;
            while(n < end && sums[n] - sums[i] <= upper) ++n;
            cnt += n - m;
            while(t < end && sums[t] < sums[i]) cache[s++] = sums[t++];
            cache[s] = sums[i];
            len = s;
        }
        for(int i = 0; i <= len; ++i) sums[start + i] = cache[i];
        return cnt;
    }
};

 用迭代器和 inplace_merge 实现：

class Solution {
public:
    typedef vector<long>::iterator it;
    int cnt = 0;
    int lower, upper;
    void merge(it l, it r) {
        if (r - l <= 1) return;
        auto mid = l + (r - l) / 2;
        merge(l, mid);
        merge(mid, r);
        for (auto i = l, j = mid, k = mid; i != mid; ++i) {
            while (j != r && *j - *i < lower) ++j;
            while (k != r && *k - *i <= upper) ++k;
            cnt += k - j;
        }
        inplace_merge(l, mid, r);
    }
    int countRangeSum(vector<int>& nums, int lower, int upper) {
        this->lower = lower;
        this->upper = upper;
        int n = nums.size();
        vector<long> sums(n + 1, 0);
        for (int i = 0; i < n; ++i) sums[i + 1] = sums[i] + nums[i];
        merge(sums.begin(), sums.end());
        return cnt;
    }
};