3290. Maximum Multiplication Score
Description
You are given an integer array a
of size 4 and another integer array b
of size at least 4.
You need to choose 4 indices i0
, i1
, i2
, and i3
from the array b
such that i0 < i1 < i2 < i3
. Your score will be equal to the value a[0] * b[i0] + a[1] * b[i1] + a[2] * b[i2] + a[3] * b[i3]
.
Return the maximum score you can achieve.
Example 1:
Input: a = [3,2,5,6], b = [2,-6,4,-5,-3,2,-7]
Output: 26
Explanation:
We can choose the indices 0, 1, 2, and 5. The score will be 3 * 2 + 2 * (-6) + 5 * 4 + 6 * 2 = 26
.
Example 2:
Input: a = [-1,4,5,-2], b = [-5,-1,-3,-2,-4]
Output: -1
Explanation:
We can choose the indices 0, 1, 3, and 4. The score will be (-1) * (-5) + 4 * (-1) + 5 * (-2) + (-2) * (-4) = -1
.
Constraints:
a.length == 4
4 <= b.length <= 105
-105 <= a[i], b[i] <= 105
Solutions
Solution 1: Memoization
We design a function $\textit{dfs}(i, j)$, which represents the maximum score that can be obtained starting from the $i$-th element of array $a$ and the $j$-th element of array $b$. Then the answer is $\textit{dfs}(0, 0)$.
The function $\textit{dfs}(i, j)$ is calculated as follows:
- If $j \geq \text{len}(b)$, it means array $b$ has been completely traversed. At this point, if array $a$ has also been completely traversed, return $0$; otherwise, return negative infinity.
- If $i \geq \text{len}(a)$, it means array $a$ has been completely traversed. Return $0$.
- Otherwise, we can either skip the $j$-th element of array $b$ and move to the next element, i.e., $\textit{dfs}(i, j + 1)$; or we can choose the $j$-th element of array $b$, in which case the score is $a[i] \times b[j]$ plus $\textit{dfs}(i + 1, j + 1)$. We take the maximum of these two values as the return value of $\textit{dfs}(i, j)$.
We can use memoization to avoid redundant calculations.
The time complexity is $O(m \times n)$, and the space complexity is $O(m \times n)$. Here, $m$ and $n$ are the lengths of arrays $a$ and $b$, respectively.
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