1569 Max Dot Product Of Two Subsequences

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

class Solution:
    def maxDotProduct(self, nums1: List[int], nums2: List[int]) -> int:
        
        memo = {}

        # max dot product of two subsequences starting at i,j
        def dp(i,j):

            if i == len(nums1) or j == len(nums2):
                return float("-inf") # if we are passed the boundary, dont pick anything from there.

            if (i,j) in memo:
                return memo[(i,j)]

            take = nums1[i] * nums2[j]
            res = max(
            # take i,j. move forward
                take + dp(i+1, j+1),

            # take this subsequence and just end.
                take,

            # skip i: i+1,j
                dp(i+1,j),

            # skip j: i,j+1
                dp(i,j+1),
            )

            memo[(i,j)] = res

            return memo[(i,j)]

        return dp(0,0)