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Maximize Score of Numbers in Ranges

Number: 3485

Difficulty: Medium

Paid? No

Companies: N/A

Problem Description

Given an array of integers named start and an integer d, you are given n intervals defined as [start[i], start[i] + d] for each index i. You need to choose one integer from each interval such that the score, defined as the minimum absolute difference between any two selected integers, is maximized. Return the maximum possible score.

Key Insights

  • Sort the intervals by their start values to streamline the greedy selection.
  • Use binary search on the possible score (minimum difference) values.
  • Use a greedy approach to check if a candidate score is feasible:
    • For each interval, try to choose the smallest number that is at least the previous selected number plus the candidate difference.
    • Ensure the chosen number is within the current interval.
  • The problem is analogous to the "aggressive cows" problem with intervals instead of fixed positions.

Space and Time Complexity

Time Complexity: O(n log M) where M is the range of possible differences (roughly (max(start)+d - min(start))). Also, sorting takes O(n log n). Space Complexity: O(n) for sorting the intervals.

Solution

We begin by sorting the array start to process the intervals in order. Then, we perform a binary search to determine the maximum feasible minimum difference (score). For each candidate score (mid) in our binary search, we greedily choose numbers from each interval. Starting from the first interval, we select the smallest possible number that is not less than the required minimum (previously chosen number + candidate score). If the chosen number falls within the current interval [start[i], start[i] + d], we update our current selection and continue. If we are able to choose one number per interval under this strategy, the candidate score is feasible; otherwise, it is not. We adjust our binary search accordingly until we find the maximum feasible minimum difference.

Code Solutions

# Python Solution

def maximize_score(start, d):
    # Sort the intervals by their starting point
    start.sort()
    
    # Define a helper function to check if a candidate minimum difference is feasible.
    def can_choose(candidate):
        # Choose the smallest possible number in the first interval.
        # Since the interval is [start[0], start[0]+d], we set current to start[0]
        current = start[0]
        # Process the rest intervals
        for i in range(1, len(start)):
            # The next required number must be at least current + candidate diff
            required = current + candidate
            # The available interval is [start[i], start[i]+d]
            # Choose the maximum between required and start[i]
            chosen = max(required, start[i])
            # If chosen is beyond the allowed interval, candidate is not feasible.
            if chosen > start[i] + d:
                return False
            # Update current with the chosen number
            current = chosen
        return True

    # Determine the search range for binary search.
    lo, hi = 0, start[-1] + d - start[0]  # hi is the max possible difference
    # Binary search to find the maximum candidate feasible score.
    while lo < hi:
        # Try with the middle candidate
        mid = (lo + hi + 1) // 2  # use upper mid to avoid infinite loop.
        if can_choose(mid):
            lo = mid
        else:
            hi = mid - 1
    return lo

# Example usage:
#print(maximize_score([6, 0, 3], 2))  # Output: 4
#print(maximize_score([2, 6, 13, 13], 5))  # Output: 5
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