Maximize Win From Two Segments

Problem Description

You are given an integer array prizePositions that is sorted in non-decreasing order, where prizePositions[i] is the position of the ith prize. You are also given an integer k. You are allowed to select two segments with integer endpoints, each of length k. Your goal is to collect all prizes whose position falls within at least one of the two selected segments. Return the maximum number of prizes you can win if you choose the two segments optimally.

Key Insights

• The problem requires selecting two segments of fixed length k on a sorted array of positions to maximize the number of prizes collected.
• The segments can overlap, which means prizes can be counted in both segments.
• A two-pointer or sliding window technique can be utilized to efficiently calculate the number of prizes within any segment.
• Pre-computing the maximum number of prizes that can be collected from the left segment can optimize the search for the right segment.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

To solve this problem, we can use a two-pointer technique along with a sliding window approach. The idea is to maintain a window for the first segment and find the maximum number of prizes that can be collected. Then, for each potential starting point of the second segment, we can compute the possible number of prizes that can be collected by combining results from the first segment and the second segment.

1. Use a pointer to iterate through the prizePositions to define the end of the first segment.
2. For each end of the first segment, calculate how many prizes are covered.
3. Use another pointer to find the maximum number of prizes covered by the second segment starting from each valid position.
4. Keep track of the maximum number of prizes collected from both segments.

Code Solutions