#3290

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Maximum Multiplication Score

The problem requires selecting four indices from an array to maximize a dynamic score with a transition approach.

Problem Statement

Given an integer array a of size 4 and another integer array b with at least 4 elements, select four indices i0, i1, i2, and i3 from array b such that i0 < i1 < i2 < i3. Your goal is to maximize the score computed as a[0] * b[i0] + a[1] * b[i1] + a[2] * b[i2] + a[3] * b[i3].

Return the maximum score achievable by choosing these indices.

Examples

Example 1

Input: a = [3,2,5,6], b = [2,-6,4,-5,-3,2,-7]

Output: 26

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

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

Solution Approach

Dynamic Programming with State Transitions

The key to solving this problem is applying dynamic programming to track the maximum score by transitioning through the elements of array b. Use a dynamic programming table to store the best score achievable at each step of the selection process. Iterate through array b and update the score progressively.

Greedy Selection with Optimal Indices

While dynamic programming ensures correctness, a greedy approach can help in determining the best indices in a way that maximizes the score. After defining the DP transitions, a greedy approach helps refine which indices contribute best to the score.

Efficient Iteration and Pruning

To optimize performance, prune unnecessary iterations by skipping indices that won't contribute to maximizing the score. By reducing the number of iterations, the solution can scale efficiently to larger arrays, ensuring a solution that works within the problem's constraints.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the dynamic programming solution. A basic approach might involve iterating through array b in a nested loop, leading to a time complexity of O(n^2). However, optimizations could reduce this to O(n), where n is the length of array b. The space complexity is determined by the DP table, typically O(n) for storing intermediate results.

What Interviewers Usually Probe

Candidate recognizes the importance of dynamic programming for state transitions.
Candidate correctly identifies the need for greedy selection in conjunction with dynamic programming.
Candidate efficiently manages the large input sizes with optimizations to reduce complexity.

Common Pitfalls or Variants

Common pitfalls

Overcomplicating the approach with unnecessary loops.
Failing to efficiently transition states in the DP table.
Misunderstanding the problem by not considering array b’s ordered indices.

Follow-up variants

Consider variations in the size of array b, which might require different optimization strategies.
Test the solution with varying values in array a and how it influences score maximization.
Alter the problem to handle multiple sets of arrays for more complex scoring scenarios.

FAQ

What is the best approach to solving the Maximum Multiplication Score problem?

The best approach is to use dynamic programming combined with greedy selection to efficiently calculate the maximum score by selecting indices from array b.

How can dynamic programming be applied to this problem?

Dynamic programming is applied by creating a table that tracks the maximum score for selecting indices progressively from array b, ensuring all transitions are optimal.

What is the time complexity of solving this problem?

The time complexity varies depending on the approach. A straightforward solution may have O(n^2) complexity, but with optimizations, it can be reduced to O(n).

How does the greedy approach help in this problem?

The greedy approach helps by ensuring that, once the dynamic programming transitions are defined, we can quickly select the best indices that contribute to the score.

What are some common mistakes when solving the Maximum Multiplication Score problem?

Common mistakes include overcomplicating the solution with unnecessary loops, failing to optimize transitions in the DP table, or not considering the order of indices in array b.

terminal

Solution

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 $\textit{dfs}(0, 0)$.

1

2

3

4

5

6

7

8

9

10

11

class Solution:
    def maxScore(self, a: List[int], b: List[int]) -> int:
        @cache
        def dfs(i: int, j: int) -> int:
            if j >= len(b):
                return 0 if i >= len(a) else -inf
            if i >= len(a):
                return 0
            return max(dfs(i, j + 1), a[i] * b[j] + dfs(i + 1, j + 1))

        return dfs(0, 0)

Continue Practicing

#3291 minimum number of valid strings to form target i #3292 minimum number of valid strings to form target ii #3287 find the maximum sequence value of array