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# 1458. Max Dot Product of Two Subsequences

Given two arrays `nums1` and `nums2`.

Return the maximum dot product between **non-empty** subsequences of nums1 and nums2 with the same length.

A subsequence of a array is a new array which is formed from the original array by deleting some (can be none) of the characters without disturbing the relative positions of the remaining characters. (ie, `[2,3,5]` is a subsequence of `[1,2,3,4,5]` while `[1,5,3]` is not).

**Example 1:**

```
Input: nums1 = [2,1,-2,5], nums2 = [3,0,-6]
Output: 18
Explanation: Take subsequence [2,-2] from nums1 and subsequence [3,-6] from nums2.
Their dot product is (2*3 + (-2)*(-6)) = 18.
```

**Example 2:**

```
Input: nums1 = [3,-2], nums2 = [2,-6,7]
Output: 21
Explanation: Take subsequence [3] from nums1 and subsequence [7] from nums2.
Their dot product is (3*7) = 21.
```

**Example 3:**

```
Input: nums1 = [-1,-1], nums2 = [1,1]
Output: -1
Explanation: Take subsequence [-1] from nums1 and subsequence [1] from nums2.
Their dot product is -1.
```

**Constraints:**

* `1 <= nums1.length, nums2.length <= 500`
* `-1000 <= nums1[i], nums2[i] <= 1000`

对两个数组问从中各选一个子串，满足它俩积最大且长度一样（并>=1）时积是多少。优化 + subseq => DP。一共M \* N步，每步action为1. 选i和j 2. 选i或选j。dp\[i]\[j]表前i和前j元素组成的子串满足条件时最大积。都选时最大为dp\[i]\[j]+=max(dp\[i -1]\[j -1],0)，选一个时最大为dp\[i -1]\[j])或dp\[i]\[j -1])。

代码参考了[这](https://leetcode.com/problems/max-dot-product-of-two-subsequences/discuss/648420/JavaC%2B%2BPython-the-Longest-Common-Sequence)。

```python
class Solution:
    def maxDotProduct(self, nums1: List[int], nums2: List[int]) -> int:
        M, N = len(nums1), len(nums2)
        dp = [[0] * N for _ in range(M)]
        for i in range(M):
            for j in range(N):
                dp[i][j] = nums1[i] * nums2[j]
                if i and j: dp[i][j] += max(dp[i - 1][j - 1], 0)
                if i: dp[i][j] = max(dp[i][j], dp[i - 1][j])
                if j: dp[i][j] = max(dp[i][j], dp[i][j - 1])
        return dp[-1][-1]
    
```