Problem Description

You are given two arrays nums and multipliers. Array nums is of size n, and multipliers is of size m, with the condition that n >= m. You start with a score of 0 and intend to perform exactly m operations to maximize your score.

Each operation consists of the following steps:

1. Select an integer x from either the start or end of the array nums.
2. Multiply x by the corresponding multipliers[i] (where i is the index of the current operation, starting at 0) and add the result to your score.
3. Remove x from nums.

The challenge is to figure out the sequence of operations that leads to the maximum possible score after performing m operations, and return this maximum score.

Intuition

To solve this problem, we use dynamic programming to keep track of the highest score we can achieve after a certain number of operations. Specifically, we create a 2D array f where f[i][j] represents the maximum score we can achieve by selecting i elements from the start and j elements from the end of nums. The value of k in the code represents the current operation we're on, and ans will store our final answer.

Here's the intuition behind the solution step by step:

1. We prepare a 2D DP array f with dimensions (m + 1) x (m + 1) where each cell is initially filled with negative infinity (-inf) because we want to calculate the maximum. The +1 ensures we have space for zero selections from both ends.

2. Set f[0][0] to 0 as the base case, which represents that no operation has been performed yet and the score is zero.

3. For each possible set of selections from the start and end (represented by i and j respectively), calculate the score for that state and store it in f[i][j].

4. For the state represented by f[i][j], we consider two possibilities:

   • f[i - 1][j] + multipliers[k] * nums[i - 1]: This corresponds to selecting the element x from the start of the array.
   • f[i][j - 1] + multipliers[k] * nums[n - j]: This corresponds to selecting the element x from the end of the array.

5. Take the maximum of these two values to find the optimal score for that particular operation.

6. If the sum of i and j is equal to m (meaning all operations have been used), update ans to track the highest score found up to that point.

7. After evaluating all possible combinations of selections from nums, return the maximum score ans.

This approach ensures that at every step, we are considering all possible choices and computing the optimal score that can be achieved given the previous decisions, thereby solving the problem optimally.

Learn more about Dynamic Programming patterns.

Solution Approach

The solution approach uses dynamic programming, a method for solving a complex problem by breaking it down into simpler subproblems. It is applicable here because the decision at each step depends on the results of previous steps, and there are overlapping subproblems.

The implementation details are as follows:

• Initialization: Create a 2D list f of size (m + 1) x (m + 1), where m is the length of the multipliers array, and set all elements to -inf. This represents the maximum possible scores. Additionally, set f[0][0] to 0 as a starting point since no operations mean no score.

• Nested Loops: Two nested loops iterate over all possibilities where i represents the number of elements selected from the start, and j represents the number of elements selected from the end. Since the problem requires exactly m operations, the outer loop ranges from 0 to m, inclusive, ensuring i + j <= m.

• State Update: For each (i, j) pair:

  • Calculate the index k = i + j - 1, which represents the operation number.
  • Update f[i][j] according to the dynamic programming state transition equation:
    • If i > 0, meaning we can select from the start, update f[i][j] with the maximum of its current value or the value from choosing the start (f[i - 1][j] + multipliers[k] * nums[i - 1]).
    • If j > 0, meaning we can select from the end, update f[i][j] with the maximum of its current value or the value from choosing the end (f[i][j - 1] + multipliers[k] * nums[n - j]).
  • Check if i + j equals m, that is, if we have performed m operations. If so, update ans with the maximum of its current value and f[i][j].

• Answer Calculation: After populating the f table with all possible scores, ans, which has been tracking the maximum, will contain the maximum score possible. Hence, return ans.

By storing and updating the maximum score at each state, the solution efficiently computes the maximum score possible after m operations. This dynamic programming table eliminates the need to recompute overlapping subproblems and ensures each subproblem is solved only once. The final answer is found by considering all possible ways to perform the m operations.

Example Walkthrough

Let's illustrate the dynamic programming solution approach with a small example.

Suppose we have:

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

Here, n = 3 (size of nums) and m = 2 (size of multipliers). As per the problem, we have to perform m operations, so in this case, we need to perform 2 operations.

Following the solution approach:

1. Initialization: We create a 2D list f with dimensions (3 x 3) (since m + 1 = 2 + 1 = 3), initializing all elements to -inf. Then, set f[0][0] to 0. The table looks like this:

f = [
  [0, -inf, -inf],
  [-inf, -inf, -inf],
  [-inf, -inf, -inf]
]

2. Nested Loops: We iterate through all combinations of i (selections from the start) and j (selections from the end). Our loops will cover (i, j) pairs such as (0, 0), (1, 0), (0, 1), and so on, but will not exceed i + j = m:

For i = 0 to 2
    For j = 0 to 2
        If i + j <= 2 // This is our operation limit `m`
            Perform state updates

3. State Update: We update f[i][j] for each (i, j) pair:
   • When i = 1 and j = 0, k = 0:
     • We can take from the start: f[1][0] gets updated to max(-inf, 0 + 3 * nums[0]) which is 3.
   • When i = 0 and j = 1, k = 0:
     • We can take from the end: f[0][1] gets updated to max(-inf, 0 + 3 * nums[2]) which is 9.

We continue doing this to fill out our table f. After the first set of operations, part of our table looks like this:

f = [
  [0, 9, -inf],
  [3, -inf, -inf],
  [-inf, -inf, -inf]
]

4. Answer Calculation: We check f[i][j] values where i + j = m to update ans, the maximum score so far. Since m = 2, we will check f[2][0], f[1][1], and f[0][2] to find our answer.

Continuing the update process with i and j values:

• When i = 1 and j = 1, k = 1:
  • Take from the start: f[1][1] can become max(-inf, f[0][1] + 2 * nums[0]) which is 9 + 2 * 1 = 11.
  • Take from the end: f[1][1] can become max(-inf, f[1][0] + 2 * nums[2]) which is 3 + 2 * 3 = 9. Since 11 is greater, f[1][1] becomes 11.

After updating all values, our table f looks like this:

f = [
  [0, 9, -inf],
  [3, 11, -inf],
  [-inf, -inf, -inf]
]

Our maximum score ans is the maximum value in f where i + j = m, which in this case is f[1][1] = 11.

Thus, the final answer is 11; it's the maximum score we can get by performing 2 operations using the given nums and multipliers.

Solution Implementation

Time and Space Complexity

The provided Python code aims to find the maximum score from multiplying elements of nums with multipliers based on specific rules. The function maximumScore utilizes dynamic programming with a 2D array f to store intermediate results.

Time Complexity:

The time complexity depends on two loops that iterate over the range (m + 1). The outer loop variable i runs from 0 to m, and for each value of i, the inner loop variable j runs up to (m - i). This essentially results in a total of roughly m/2 * m/2 iterations, which can be approximated to O(m^2) where m is the length of the multipliers list.

Moreover, the updates within the if conditions inside the nested loops also operate in constant time. Therefore, no additional complexity is added there.

Hence, the overall time complexity of this code is O(m^2).

Space Complexity:

The space complexity is determined by the size of the 2D array f, which has dimensions (m + 1) by (m + 1). So the total space used is (m + 1) * (m + 1), resulting in a space complexity of O(m^2) as well.

Thus, the overall space complexity of the algorithm is O(m^2).

Learn more about how to find time and space complexity quickly using problem constraints.