Maximum Score from Performing Multiplication Operations

Maximum Score from Performing Multiplication Operations

1. What is this problem about?

The Maximum Score from Performing Multiplication Operations interview question is a dynamic programming problem involving choices from both ends of an array. You are given two arrays: nums and multipliers. You need to perform m operations. In the i-th operation, you pick one number from either the start or the end of nums, multiply it by multipliers[i], and add the result to your score. The number you pick is removed from nums.

The goal is to maximize the total score after m operations.

2. Why is this asked in interviews?

Google asks the Maximum Score from Performing Multiplication Operations coding problem to test a candidate's ability to handle DP with multiple variables and identify space optimization opportunities. It evaluates whether you can recognize that the state of the problem (which elements are left in nums) can be represented by just two variables (the number of elements picked from the left and the total number of operations performed).

3. Algorithmic pattern used

The Array, Dynamic Programming interview pattern is applied. Let dp[i][j] be the max score after i operations, having picked j elements from the left.

• Total operations = i.
• Left elements picked = j.
• Right elements picked = i - j.
• Index of left element available = j.
• Index of right element available = n - 1 - (i - j).

Transition: dp[i][j] = max( multipliers[i] * nums[j] + dp[i+1][j+1], // Pick left multipliers[i] * nums[n-1-(i-j)] + dp[i+1][j] // Pick right )

4. Example explanation

nums: [1, 2, 3], multipliers: [3, 2] Operation 1 (m=3):

• Pick Left (1): Score = 3*1 = 3. Remaining nums: [2, 3].
• Pick Right (3): Score = 3*3 = 9. Remaining nums: [1, 2]. Operation 2 (m=2):
• From [2, 3], pick 2 (Left): 3 + 22 = 7. Pick 3 (Right): 3 + 23 = 9.
• From [1, 2], pick 1 (Left): 9 + 21 = 11. Pick 2 (Right): 9 + 22 = 13. Max score: 13.

5. Common mistakes candidates make

In the Maximum Score from Performing Multiplication Operations coding problem, a common error is using n (the size of nums) in the DP table dimensions. Since m is usually much smaller than n, using dp[m][m] is much more efficient. Another mistake is trying to use a greedy approach (picking the larger product at each step), which fails because a small current gain might prevent a much larger future gain.

6. Interview preparation tip

Whenever you have a problem where you pick from either end of an array, think about a DP state that tracks how many elements were picked from one end. This automatically tells you how many were picked from the other end if you know the total number of picks. This "reduction of variables" is a common DP optimization.