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Maximum Number of Groups With Increasing Length

Difficulty: Hard

Problem Description

You are given a 0-indexed array usageLimits of length n. Your task is to create groups using numbers from 0 to n - 1, ensuring that each number, i, is used no more than usageLimits[i] times in total across all groups. Each group must consist of distinct numbers, and each group (except the first one) must have a length strictly greater than the previous group. Return an integer denoting the maximum number of groups you can create while satisfying these conditions.

Key Insights

  • Each group must have a length that increases strictly compared to the previous group.
  • The maximum size of groups is limited by the usage limits provided in the usageLimits array.
  • The problem can be approached using a greedy strategy, where we try to form the largest possible groups incrementally.

Space and Time Complexity

Time Complexity: O(n log n) - Sorting the usageLimits array takes O(n log n), and iterating over it to form groups is O(n). Space Complexity: O(1) - We are using a constant amount of extra space.

Solution

To solve this problem, we can use a greedy approach:

  1. Sort the usageLimits array to prioritize the numbers that can be used the least.
  2. Iterate through potential group sizes, starting from 1 (the size of the first group) and increasing it by 1 for each subsequent group.
  3. Count how many times we can use each number based on its usage limits to satisfy the required group size.
  4. Continue creating groups until we can no longer satisfy the next group size.

The algorithm prioritizes using numbers that can form valid groups while adhering to their usage limits.

Code Solutions

def maxGroups(usageLimits):
    usageLimits.sort()  # Sort the usage limits
    group_count = 0
    current_group_size = 1
    
    for limit in usageLimits:
        if limit >= current_group_size:  # If we can form a group of the current size
            group_count += 1  # Increment the group count
            current_group_size += 1  # Increase the required group size for the next group
            
    return group_count
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