5403. Find the Kth Smallest Sum of a Matrix With Sorted Rows

You are given an m * n matrix, mat, and an integer k, which has its rows sorted in non-decreasing order.

You are allowed to choose exactly 1 element from each row to form an array. Return the Kth smallest array sum among all possible arrays.

Example 1:

Input: mat = [[1,3,11],[2,4,6]], k = 5
Output: 7
Explanation: Choosing one element from each row, the first k smallest sum are:
[1,2], [1,4], [3,2], [3,4], [1,6]. Where the 5th sum is 7.  

Example 2:

Input: mat = [[1,3,11],[2,4,6]], k = 9
Output: 17

Example 3:

Input: mat = [[1,10,10],[1,4,5],[2,3,6]], k = 7
Output: 9
Explanation: Choosing one element from each row, the first k smallest sum are:
[1,1,2], [1,1,3], [1,4,2], [1,4,3], [1,1,6], [1,5,2], [1,5,3]. Where the 7th sum is 9.  

Example 4:

Input: mat = [[1,1,10],[2,2,9]], k = 7
Output: 12

Constraints:

• m == mat.length
• n == mat.length[i]
• 1 <= m, n <= 40
• 1 <= k <= min(200, n ^ m)
• 1 <= mat[i][j] <= 5000
• mat[i] is a non decreasing array.

题意：

　　给出一个二维数组，数组的每一行按照非减的顺序进行排列，在每一行中选出一个数字，求出所选数字的和，将这些数字排序，并求出第k小的那个。

思路：

　　本来想的是用DFS进行求解，如果这样做的话，最欢情况下的时间复杂度会是O(40^40)，用DFS遍历求解的话，有可能会出现后面出现的数字比前面出现数字要小的情况，好像没办法剪枝，这种方法行不通。换种思路，我们可以用相邻两行进行相加，求得的和进行排序，截取前面的k个数字。然后再将所求的和与下一行元素相加。

Code：

class Solution {
public:
    int kthSmallest(vector<vector<int>>& mat, int k) {
        vector<int> ans = {0}, temp; 
        for (int i = 0; i < mat.size(); ++i) {
            for (int j = 0; j < mat[i].size(); ++j) {
                for (int v = 0; v < ans.size(); ++v) {
                    temp.push_back(mat[i][j] + ans[v]);
                }
            }
            sort(temp.begin(), temp.end());
            ans.clear();
            int len = min(k, (int)temp.size());
            for (int j = 0; j < len; ++j) {
                ans.push_back(temp[j]);
            }
            temp.clear();
        }
        return ans[k-1];
    }
};

参考：

　　https://leetcode.com/problems/find-the-kth-smallest-sum-of-a-matrix-with-sorted-rows/discuss/609707/c%2B%2B-solution-with-explanation