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Maximize Score of Numbers in Ranges

Difficulty: Medium

Problem Description

You are given an array of integers start and an integer d, representing n intervals [start[i], start[i] + d]. You are asked to choose n integers where the i-th integer must belong to the i-th interval. The score of the chosen integers is defined as the minimum absolute difference between any two integers that have been chosen. Return the maximum possible score of the chosen integers.

Key Insights

  • Each chosen integer must be within a specific interval defined by the start array and the integer d.
  • The goal is to maximize the minimum absolute difference between the selected integers.
  • A greedy approach can be employed by considering the maximum and minimum endpoints of the intervals.
  • Sorting can help to identify the ranges and possible values effectively.

Space and Time Complexity

Time Complexity: O(n log n) - for sorting the intervals
Space Complexity: O(1) - for the variables used in the algorithm (not counting input storage)

Solution

The problem can be approached by first calculating the possible ranges for each interval. After that, we sort these intervals based on their starting points. The main idea is to select the maximum possible integer from each interval while ensuring that the chosen integers maintain the largest possible minimum difference. A binary search can be used to efficiently determine the maximum score achievable.

Code Solutions

def maximizeScore(start, d):
    n = len(start)
    # Create intervals
    intervals = [(start[i], start[i] + d) for i in range(n)]
    
    # Sort intervals by starting points
    intervals.sort()
    
    # Binary search for the answer
    left, right = 0, 10**9
    
    def canAchieve(minDiff):
        last_chosen = intervals[0][0]  # Choose the first interval's start
        for i in range(1, n):
            # Find the next number to choose in the current interval
            next_chosen = last_chosen + minDiff
            
            # Check if next_chosen is within the current interval
            if next_chosen > intervals[i][1]:
                return False
            
            # Update last_chosen to the maximum possible value
            last_chosen = max(last_chosen + minDiff, intervals[i][0])
        
        return True
    
    while left < right:
        mid = (left + right + 1) // 2
        if canAchieve(mid):
            left = mid  # mid is achievable
        else:
            right = mid - 1  # mid is not achievable
    
    return left
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