(栈，数组)

反向思维：如果知道最小值，那如何求出这个最小值所在的子数组, 即找到这个最小值左边比它更小的元素的位置left 和 右边比它更小的元素的位置right，res += A[i] * ( i - left ) * ( right - i )。

class Solution {
public:
    int sumSubarrayMins(vector<int>& A) {
       // find one element : right : next smaller element, left : smaller/equal element
        int n = A.size();
        vector<int> preSmaller(n, -1);  //preSmaller[i] = j : i的前面第一个比i小的元素的位置j
        vector<int> nextSmaller(n, n);  //nextSmaller[i] = j : i的后面第一个比i小的元素的位置j
        //单调栈：(非严格)增序，存index
        stack<int> Stack;
        
        //找nextSmaller
        for(int i=0; i<n; i++){
            while(!Stack.empty() && A[Stack.top()] > A[i]){
                    //遇到一个比栈顶元素小的，退栈
                    nextSmaller[Stack.top()] = i;
                    Stack.pop();
            }
            Stack.push(i);
        }
        
        while(!Stack.empty())
            Stack.pop();     //把栈清空
        
        //找preSmaller
        //注意做记录的位置和之前的不同
        for(int i=0; i<n; i++){  
            while(!Stack.empty() && A[Stack.top()] > A[i])
                //栈顶元素大于A[i]时，退栈
                Stack.pop();
            if(!Stack.empty())
                preSmaller[i] = Stack.top();
            Stack.push(i);
        }
        
        long result = 0;
        long M = 1e9+7;
        for(int i = 0; i<n; i++){
            //以A[i]为最小元素的subarray个数
            long times = (i-preSmaller[i])*(nextSmaller[i] - i);
            result += A[i] * times;
            result %= M;
        }
        return result;
    }
};

 解法一：记忆化递归：

class Solution {
public:
    vector<long> sums;
    vector<vector<int>> mem;
    int splitArray(vector<int>& nums, int m) {
        //记忆化递归
        int n = nums.size();
        sums = vector<long>(n);
        mem = vector<vector<int>>(m+1, vector<int>(n, INT_MAX));    //将个数为n得数组分成m个非空连续子数组
        sums[0] = nums[0];
        for(int i=1; i<n; i++){
            sums[i] = sums[i-1] + nums[i];   // 前序和
        }
        return splitArray(nums, n-1, m);
    }
    
    int splitArray(vector<int>& nums, int k, int m){
        //min of largest sum of splitting nums[0] - nums[k] into m groups
        if(m==1)
            //划分为1个数组
            return sums[k];
        if(m > k+1)
            return INT_MAX;
        if(mem[m][k] != INT_MAX)
            return mem[m][k];
        int ans = INT_MAX;
        for(int i=0; i<k; i++)
            ans = min(ans, max(splitArray(nums, i, m-1), int(sums[k]-sums[i])) );
        return mem[m][k] = ans;
    }
};

解法二：动态规划

class Solution {
public:
    int splitArray(vector<int>& nums, int m) {
        int n = nums.size();
        vector<long> sums(n);
        vector<vector<int>> dp(m+1, vector<int>(n, INT_MAX));
        
        //前缀和
        sums[0] = nums[0];
        for(int i=1; i<n; i++){
            sums[i] = sums[i-1] + nums[i];
        }
        
        //将0-i个数分为1组就是其前缀和
        for(int i=0; i<n; i++){
            dp[1][i] = sums[i];  
        }
        
        // dp[i][j]：把nums[0] - nums[j] 分为i组
        for(int i=2; i<=m; i++){
            //将数组划分为i组
            for(int j=i-1; j<n; j++){
                //末尾元素位置为j
                for(int k=0; k<j; k++){
                    //起始元素位置为k
                    // 将0-j分为i组的问题 转化为 将0-k分为i-1组的问题 + sum(k+1...j)
                    dp[i][j] = min(dp[i][j], max(dp[i-1][k], int(sums[j]-sums[k]) ) );
                }
            }
        }
        return dp[m][n-1];
    }
};

解法三：二分法

l = max(nums) : 将nums中的每个元素分为1组中的最大值 为 range的下界；

r = sum(nums) : 将nums整体分为1组为range的上界；

range为左闭右开区间，所以 r+1

问题转化为=> 找到一个最小的candidate C ，使得将nums划分为m组的每一组都不大于C

accumulate()

This function returns the sum of all the values lying in a range between [first, last) with the variable sum.

1. Syntax 1:

   accumulate(first, last, sum);
   first, last : first and last elements of range 
                 whose elements are to be added
   sum :  initial value of the sum
   

2. Syntax 2: This function returns the sum of all the values lying between [first, last) with the variable sum.

   accumulate(first, last, sum, myfun); 
   myfun : a function for performing any 
           specific task. For example, we can
           find product of elements between
           first and last.

// C++ program to demonstrate working of accumulate() 
#include <iostream> 
#include <numeric> 
using namespace std; 

// User defined function 
int myfun(int x, int y) 
{ 
    // for this example we have taken product 
    // of adjacent numbers 
    return x * y ; 
} 

int main() 
{ 
    // Initialize sum = 1 
    int sum = 1; 
    int a[] = {5 , 10 , 15} ; 
    
    // Simple default accumulate function 
    cout << "
Result using accumulate: "; 
    cout << accumulate(a , a+3 , sum); 
    
    // Using accumulate function with 
    // defined function 
    cout << "
Result using accumulate with"
            "user-defined function: "; 
    cout << accumulate(a, a+3, sum, myfun); 
    
    // Using accumulate function with 
    // pre-defined function 
    cout << "
Result using accumulate with "
            "pre-defined function: "; 
    cout << accumulate(a, a+3, sum, std::minus<int>()); 
    
    return 0; 
} 

class Solution {
public:
    int splitArray(vector<int>& nums, int m) {
        long l = *max_element(begin(nums), end(nums));   //得到nums中的最大值
        //统计nums中元素的和 并加1
        long r = accumulate(begin(nums), end(nums), 0L) +1;  //  0L表示long型的0
        while(l<r){
            long limit = l + (r-l)/2;
            if(min_groups(nums, limit) > m)
                //子数组的数量超过m,将每个子数组的和：candidate C分配的大一点
                l = limit+1;
            else
                r = limit;
        }
        return l;
    }
    
    int min_groups(vector<int>& nums, long limit){
        long sum = 0;     //子数组中元素的和
        int groups = 1;   //子数组的数量
        for(int num : nums){
            if(sum + num > limit){
                sum = num;
                groups ++;
            }
            else
                sum += num;
        }
        return groups;
    }
};

约瑟夫环问题：

https://www.nowcoder.com/practice/f78a359491e64a50bce2d89cff857eb6?tpId=13&tqId=11199&rp=1&ru=/ta/coding-interviews&qru=/ta/coding-interviews/question-ranking

参考：剑值offer 面试题62

解法一：链表

class Solution {
public:
    int LastRemaining_Solution(int n, int m)
    {
        if(n<1 || m<1)
            return -1;
        int i=0; 
        list<int> numbers;    //用std::list来模拟一个环形链表
        for(i=0; i<n; i++)
            numbers.push_back(i);
        
        list<int>::iterator current = numbers.begin();
        while(numbers.size() >1){
            for(int i=1; i<m; i++){
                current ++;
                if(current == numbers.end())
                    current = numbers.begin();   //因为list不是一个环形结构，故每当current扫描到链表末尾时就将其移到头部
            }
            //list<int>::iterator next = ++current;
            current++;  //current加1后指向被杀掉的人的下一个人
            list<int>::iterator next = current;
            if(next == numbers.end())
                next = numbers.begin();
            current--; //current-1 后指向被杀掉的人
            numbers.erase(current);
            current = next;
        }
        return *current;
    }
};

解法二：循环/递归，用数学找规律法找到一个递归式：

 

循环写法：

class Solution {
public:
    int LastRemaining_Solution(int n, int m)
    {
        if(n<1 || m<1)
            return -1;
        int last = 0;
        for(int i=2; i<=n; i++){
            last = (last+m)%i;
        }
        return last;
    }
};

hulu 2019笔试第一题：

 

1. 通过约瑟夫环找到由第一个节点开始时获胜的节点的index

2. 归纳总结得到　第i个节点开始时获胜的节点为(i+index)%array_len

3. 线性获得权值

// pch.cpp: 与预编译标头对应的源文件；编译成功所必需的

#include "pch.h"

// 一般情况下，忽略此文件，但如果你使用的是预编译标头，请保留它。
#include <vector>
#include <list>
#include <map>
#include <set>
#include <queue>
#include <deque>
#include <stack>
#include <bitset>
#include <algorithm>
#include <functional>
#include <numeric>
#include <utility>
#include <sstream>
#include <iostream>
#include <iomanip>
#include <cstdio>
#include <cmath>
#include <cstdlib>
#include <ctime>
#include <string>
#include <cstring>
#include <math.h> 

using namespace std;

int yuesefu(int N, int M) {
    if (N == 1) {
        return 0; //这里返回下标,从0开始，只有一个元素就是剩余的元素0
    }
    else {
        return (yuesefu(N - 1, M) + M) % N; //我们传入的n是总共多少个数
    }
}
int main() {
    int n, m;
    cin >> n >> m;
    vector<int> a(n);
    vector<int> w(n);
    vector<double> v(n);
    
    for (int i = 0; i < n; i++) {
        cin >> a[i]; //1 good ; 0 bad
    }
    double sum = 0;
    for (int i = 0; i < n; i++) {
        cin >> w[i]; //1 good ; 0 bad
        sum += w[i];
    }
    for (int i = 0; i < n; i++) {
        v[i] = w[i] / sum;
    }

    double x = 0;
    int r = yuesefu(n, m);
    if(a[r] == 1)
        x += v[0];
    for (int i = 1; i < n; i++) {
        if (a[(i + r) % n] == 1)
            x += v[i];
    }

    cout << setiosflags(ios::fixed) << setprecision(5) << x << endl;
    return 0;
}