leetCode 53. maximum subarray

Find the contiguous subarray within an array (containing at least one number) which has the largest sum.

For example, given the array [-2,1,-3,4,-1,2,1,-5,4],
the contiguous subarray [4,-1,2,1] has the largest sum = 6.

贪婪算法找每个当前位置对应的最大的subarray, 

class Solution {
public:
    int maxSubArray(vector<int>& nums) {
        
        int n=nums.size();
        vector<int>  eachnum;
        int sum=0;
        for(int i=0;i<n;i++)
        {
            if(sum<0)
            {
                sum=0;
            }
            sum+=nums[i];
            eachnum.push_back(sum);
            cout<<sum<<endl;
          //  eachnum[i]=sum;
        }
        vector<int>:: iterator biggest=max_element(eachnum.begin(),eachnum.end());
        return *biggest;
    }
};

　　

更简单的方法，记录当前最大的结果，因为每遇到一个数，可能有两种可能，一是加这个数，二是从这个数开始。 那我们就要找当前的最优情况与历史最优情况比较。

和可能的结果：

class Solution {
public:
    int maxSubArray(vector<int>& nums) {
        int adj_sum = 0;
        int cont_sum = nums[0];
        
        for (vector<int>::iterator it = nums.begin(); it<nums.end(); it++)
        {
           
            adj_sum+=*it;
            adj_sum = max(adj_sum, *it);//记录加当前数与从当前数开始的最大值
            cont_sum = max(cont_sum, adj_sum);// 比较当前的最大值与历史最大值，记录最大值
        }
        return cont_sum;
    }
};

　　

这是一个最优化问题，最优化问题一般都可以用DP（动态规划）解决。 对于动态规划，要考虑子问题是什么（形式的子问题或状态的子问题）子问题就可以用recursive solution。

1. 对于 maxSubArray( int a[], int i,int j), is searching for the maxSubArray for a[i:j]
2. the goal is to figure out what maxSubArray(A,0,A.length()-1) is.
3. 对于maxSubArray(int a[], int i, int j) is difficult ot connect this sub problem to the original, so we change the format of the sub problem to maxSubArray(int a[], int i), which means the maxSubArray for A[0:i]. which must has A[i] as the end. so we should keep track of each solution of the sub problem to update the global optimal value.

maxSubArray(A,i)=maxSubArray(A, i-1)>0?maxSubArray(A, i-1):0+A[i];

so the solution is same as the previous one:

    int maxSubArray(vector<int> & nums)
    {
        int n=nums.size();
        int num=0;
        int* dp=new int[n];
        dp[0]=nums[0];
        int maxsum=nums[0];
        for (int i=1;i<n;i++)
        {
            if(dp[i-1]<0)
            dp[i]=nums[i];
            else
                dp[i]=dp[i-1]+nums[i];
            cout<<dp[i]<<"  "<<dp[i-1]<<endl;
            maxsum=max(maxsum,dp[i]);
            cout<<maxsum<<endl;
        }
        return maxsum; 
    }

　　开始写 dp[i]=nums[i]+dp[i-1]>0?dp[i-1]:0;  有错哈哈， 忘了带括号

                   dp[i]=nums[i]+(dp[i-1]>0?dp[i-1]:0);