#2321

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Maximum Score Of Spliced Array

Maximize the score of two arrays by splicing and swapping a subarray using dynamic programming.

Problem Statement

You are given two integer arrays, nums1 and nums2, both of length n. Your goal is to maximize the score by swapping a contiguous subarray from nums1 with the same subarray from nums2. You may choose indices left and right, where 0 <= left <= right < n, to define the subarray to swap. The operation may or may not be applied.

After performing the swap, you are to compute the maximum score, defined as the maximum of the sums of the modified arrays nums1 and nums2. The challenge is to find the optimal indices for the subarray to swap, or to determine that no swap is better.

Examples

Example 1

Input: nums1 = [60,60,60], nums2 = [10,90,10]

Output: 210

Choosing left = 1 and right = 1, we have nums1 = [60,90,60] and nums2 = [10,60,10]. The score is max(sum(nums1), sum(nums2)) = max(210, 80) = 210.

Example 2

Input: nums1 = [20,40,20,70,30], nums2 = [50,20,50,40,20]

Output: 220

Choosing left = 3, right = 4, we have nums1 = [20,40,20,40,20] and nums2 = [50,20,50,70,30]. The score is max(sum(nums1), sum(nums2)) = max(140, 220) = 220.

Example 3

Input: nums1 = [7,11,13], nums2 = [1,1,1]

Output: 31

We choose not to swap any subarray. The score is max(sum(nums1), sum(nums2)) = max(31, 3) = 31.

Constraints

n == nums1.length == nums2.length
1 <= n <= 105
1 <= nums1[i], nums2[i] <= 104

Solution Approach

Dynamic Programming Setup

Set up a dynamic programming solution where we track the maximum possible score after swapping a subarray. Use two cumulative sums, one for each array, to calculate the score after each potential swap.

Prefix and Suffix Computations

For each possible subarray swap, calculate prefix and suffix sums to determine the effect of the swap on the score. These calculations help in optimizing the solution by reducing redundant recalculations.

Iterating Over Possible Swaps

Iterate through all possible subarrays to swap, calculating the potential scores and comparing them to find the maximum score. Dynamic programming helps us efficiently explore all potential swaps without redundant calculations.

Complexity Analysis

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

Time complexity depends on the number of possible subarrays, which can be O(n^2) without optimization. However, by using dynamic programming and cumulative sums, the solution can be optimized to O(n), reducing unnecessary recalculations. Space complexity depends on the storage needed for the cumulative sums and intermediate states, typically O(n).

What Interviewers Usually Probe

Evaluating knowledge of dynamic programming with state transitions.
Ability to optimize through cumulative sums and subarray manipulation.
Assessment of problem-solving efficiency, particularly with large input sizes.

Common Pitfalls or Variants

Common pitfalls

Not recognizing the importance of cumulative sums for efficient score computation.
Failing to account for the possibility of no swap being optimal.
Overcomplicating the problem by considering unnecessary subarrays or operations.

Follow-up variants

What if the arrays have different lengths?
How would this change if negative numbers were allowed in the arrays?
What if the swap operation could be applied multiple times?

FAQ

What is the primary approach for solving the Maximum Score Of Spliced Array?

The primary approach is dynamic programming with state transitions, optimizing the search for the optimal subarray swap.

Can this problem be solved without dynamic programming?

While possible, solving this problem without dynamic programming would be inefficient, especially for larger arrays. Dynamic programming ensures optimal performance.

What is the time complexity of the best solution for this problem?

The best solution achieves a time complexity of O(n) by utilizing dynamic programming and cumulative sums to avoid redundant calculations.

How does cumulative summation help in solving this problem?

Cumulative summation helps by precomputing the sum of subarrays, allowing for efficient comparison of potential scores after swapping subarrays.

What happens if no swap results in a higher score?

If no swap results in a higher score, the maximum score will be the sum of the original arrays, indicating that the swap operation is unnecessary.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def maximumsSplicedArray(self, nums1: List[int], nums2: List[int]) -> int:
        def f(nums1, nums2):
            d = [a - b for a, b in zip(nums1, nums2)]
            t = mx = d[0]
            for v in d[1:]:
                if t > 0:
                    t += v
                else:
                    t = v
                mx = max(mx, t)
            return mx

        s1, s2 = sum(nums1), sum(nums2)
        return max(s2 + f(nums1, nums2), s1 + f(nums2, nums1))

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