#1770

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Maximum Score from Performing Multiplication Operations

Solve the Maximum Score from Performing Multiplication Operations problem using dynamic programming and state transitions.

array dynamic programming

Problem Statement

You are given two integer arrays: nums (size n) and multipliers (size m), with n >= m. You start with a score of 0 and are to perform exactly m operations. On the ith operation, you must pick an element either from the start or the end of the nums array, multiply it by the ith element from multipliers, and add the result to your score.

Your task is to determine the maximum possible score after performing all m operations. The order of operations and which elements to select are critical for achieving the optimal score.

Examples

Example 1

Input: nums = [1,2,3], multipliers = [3,2,1]

Output: 14

An optimal solution is as follows:

Choose from the end, [1,2,3], adding 3 * 3 = 9 to the score.
Choose from the end, [1,2], adding 2 * 2 = 4 to the score.
Choose from the end, [1], adding 1 * 1 = 1 to the score. The total score is 9 + 4 + 1 = 14.

Example 2

Input: nums = [-5,-3,-3,-2,7,1], multipliers = [-10,-5,3,4,6]

Output: 102

An optimal solution is as follows:

Choose from the start, [-5,-3,-3,-2,7,1], adding -5 * -10 = 50 to the score.
Choose from the start, [-3,-3,-2,7,1], adding -3 * -5 = 15 to the score.
Choose from the start, [-3,-2,7,1], adding -3 * 3 = -9 to the score.
Choose from the end, [-2,7,1], adding 1 * 4 = 4 to the score.
Choose from the end, [-2,7], adding 7 * 6 = 42 to the score. The total score is 50 + 15 - 9 + 4 + 42 = 102.

Constraints

n == nums.length
m == multipliers.length
1 <= m <= 300
m <= n <= 105
-1000 <= nums[i], multipliers[i] <= 1000

Solution Approach

Dynamic Programming (DP) Setup

The problem can be approached using a state transition dynamic programming technique. We define a DP array, where each entry represents the maximum score achievable by performing a set number of operations, considering selections from both ends of the array.

State Transitions

At each step, we decide whether to pick the current element from the start or the end of the nums array, using the corresponding multiplier. This requires tracking the state transitions as we move through the problem, ensuring the maximum score is maintained.

Space Optimization

While the naive solution uses a 2D DP table, this can be optimized to a 1D array. The solution involves calculating the possible scores by iterating through the multipliers and updating the states accordingly to reduce space complexity.

Complexity Analysis

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

The time complexity of the solution is O(m^2), where m is the length of the multipliers array, due to the need to process each element of nums in relation to each multiplier. The space complexity is O(m), as the space can be optimized to store only the current and previous states in the DP array.

What Interviewers Usually Probe

Look for the candidate's ability to recognize dynamic programming patterns.
Assess how well the candidate manages state transitions between operations.
Evaluate their ability to optimize space complexity while maintaining correctness.

Common Pitfalls or Variants

Common pitfalls

Choosing elements greedily from the start or end can lead to suboptimal solutions.
Not properly managing the dynamic programming table may result in incorrect score calculations.
Failing to optimize space complexity when using a 2D DP table can lead to unnecessary memory usage.

Follow-up variants

Modify the problem to include negative multipliers, affecting the strategy for selecting elements.
Increase the size of the nums array to test the scalability of the solution.
Change the problem to allow multiple operations with different ranges of multipliers.

FAQ

What is the key pattern used to solve Maximum Score from Performing Multiplication Operations?

The key pattern is state transition dynamic programming, where we decide between selecting elements from the start or the end of the array to maximize the score.

How does dynamic programming help in this problem?

Dynamic programming helps by tracking the maximum score achievable at each step, considering choices from both ends of the nums array.

Why can't we solve this problem using a greedy approach?

A greedy approach, such as always choosing the largest element from the start or end, may not lead to the optimal score due to the interaction between multipliers and selections.

What is the space complexity of the solution?

The space complexity of the optimized solution is O(m), where m is the length of the multipliers array, achieved by using a 1D dynamic programming table.

How can I optimize the time complexity of this problem?

The time complexity can be optimized by carefully managing the state transitions and reducing the number of redundant calculations in the dynamic programming approach.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def maximumScore(self, nums: List[int], multipliers: List[int]) -> int:
        @cache
        def f(i, j, k):
            if k >= m or i >= n or j < 0:
                return 0
            a = f(i + 1, j, k + 1) + nums[i] * multipliers[k]
            b = f(i, j - 1, k + 1) + nums[j] * multipliers[k]
            return max(a, b)

        n = len(nums)
        m = len(multipliers)
        return f(0, n - 1, 0)

Solution 2

#### Python3

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class Solution:
    def maximumScore(self, nums: List[int], multipliers: List[int]) -> int:
        @cache
        def f(i, j, k):
            if k >= m or i >= n or j < 0:
                return 0
            a = f(i + 1, j, k + 1) + nums[i] * multipliers[k]
            b = f(i, j - 1, k + 1) + nums[j] * multipliers[k]
            return max(a, b)

        n = len(nums)
        m = len(multipliers)
        return f(0, n - 1, 0)

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