To solve the Maximum Number of Groups With Increasing Length problem, we must determine the maximum number of groups that can be formed while adhering to the usage limits of numbers. The problem is efficiently solved by using binary search over the valid answer space, combined with a greedy strategy for forming the groups and checking their validity against the constraints.
Problem Statement
You are given a 0-indexed array usageLimits of length n. Your goal is to form the maximum number of groups such that each group contains integers from 0 to n - 1, where each integer, i, can be used no more than usageLimits[i] times in total across all groups.
Return the maximum number of groups you can create while satisfying these conditions. The solution involves finding the maximum possible value for the number of groups that can be formed using binary search and a greedy strategy to ensure each integer is used appropriately.
Examples
Example 1
Input: usageLimits = [1,2,5]
Output: 3
In this example, we can use 0 at most once, 1 at most twice, and 2 at most five times. One way of creating the maximum number of groups while satisfying the conditions is: Group 1 contains the number [2]. Group 2 contains the numbers [1,2]. Group 3 contains the numbers [0,1,2]. It can be shown that the maximum number of groups is 3. So, the output is 3.
Example 2
Input: usageLimits = [2,1,2]
Output: 2
In this example, we can use 0 at most twice, 1 at most once, and 2 at most twice. One way of creating the maximum number of groups while satisfying the conditions is: Group 1 contains the number [0]. Group 2 contains the numbers [1,2]. It can be shown that the maximum number of groups is 2. So, the output is 2.
Example 3
Input: usageLimits = [1,1]
Output: 1
In this example, we can use both 0 and 1 at most once. One way of creating the maximum number of groups while satisfying the conditions is: Group 1 contains the number [0]. It can be shown that the maximum number of groups is 1. So, the output is 1.
Constraints
- 1 <= usageLimits.length <= 105
- 1 <= usageLimits[i] <= 109
Solution Approach
Binary Search Over Valid Answer Space
The core approach for solving this problem is to apply binary search over the possible number of groups. This binary search is done over the range from 0 to n, where n is the length of the usageLimits array. At each step, the binary search checks if it is possible to form that many groups by leveraging a greedy approach to check if the usage limits can accommodate the groups.
Greedy Group Formation
A greedy approach is used to form groups incrementally, ensuring that each integer is used no more than its corresponding limit from the usageLimits array. This greedy method helps in determining whether the current mid-point number of groups in binary search is achievable under the given constraints.
Validating Group Forming with Sorting
Sorting the usageLimits array can aid in efficiently managing the group formation. Once sorted, the greedy approach becomes more manageable, as the numbers with lower usage limits will be placed in earlier groups, ensuring that we do not exceed any usage limit while maximizing the number of groups.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity depends primarily on the binary search, which operates in O(log n) time, and the group validation step, which requires iterating over the usageLimits array in O(n) time. Therefore, the overall time complexity is O(n log n). The space complexity is O(n) due to the space required for sorting and storing intermediate results during the greedy process.
What Interviewers Usually Probe
- Can the candidate efficiently apply binary search over the possible answer space?
- Is the candidate comfortable using a greedy approach to validate group formation?
- Does the candidate optimize the solution using sorting for easier greedy implementation?
Common Pitfalls or Variants
Common pitfalls
- Failing to correctly implement binary search over the valid answer space can lead to inefficient solutions.
- Not handling edge cases where the group sizes or the usage limits are very small or large.
- Ignoring the importance of sorting the
usageLimitsarray to facilitate the greedy approach.
Follow-up variants
- Handling different ranges of
usageLimitsvalues to see how the solution scales. - Optimizing the solution further using advanced techniques, such as dynamic programming.
- Exploring alternative greedy approaches for edge cases where the binary search may not be optimal.
How GhostInterview Helps
- GhostInterview helps you break down the problem into manageable steps by focusing on binary search and greedy strategies.
- It provides insight into using sorting to optimize the greedy approach, offering hints that improve group formation efficiency.
- GhostInterview also helps in detecting potential pitfalls in the solution by encouraging validation of edge cases and efficient binary search usage.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the optimal solution for the Maximum Number of Groups With Increasing Length problem?
The optimal solution involves using binary search over the number of possible groups and validating each mid-point number of groups using a greedy approach.
Can we use a greedy approach alone to solve the problem?
A greedy approach alone is not enough; binary search is necessary to efficiently explore the valid number of groups and optimize the solution.
Why is sorting necessary for this problem?
Sorting the usageLimits array helps manage the group formation process, as it ensures that smaller limits are handled first, preventing overuse of any integer.
What is the time complexity of the solution?
The time complexity is O(n log n) due to binary search and the greedy validation process, where n is the length of the usageLimits array.
How does binary search apply to this problem?
Binary search is applied to find the maximum number of groups by checking if it's possible to form that number of groups with the given constraints.
Need direct help with Maximum Number of Groups With Increasing Length instead of spending more time grinding it?
Download GhostInterview when you want a LeetCode solver, not another long practice loop. Capture Maximum Number of Groups With Increasing Length from a screenshot, get the answer path and complexity, and use supported stealth workflows that stay outside captured layers.
Capture the prompt fast instead of rewriting the problem by hand.
Get the solution path, trade-offs, and complexity summary in one pass.
Stay outside captured layers on supported screen-share workflows.
Stay in the same pattern family
Determine the minimum cost to convert an integer array into a palindromic array using allowed element modifications efficiently.
Open problem page#2818 Apply Operations to Maximize ScoreMaximize the score by applying operations on a subarray at most k times, utilizing stack-based state management.
Open problem page#2659 Make Array EmptySolve the "Make Array Empty" problem using binary search to determine the minimum number of operations required.
Open problem page