#3020

Medium

auto_awesomeArray scanning plus hash lookup

LeetCode Problem Workspace

Find the Maximum Number of Elements in Subset

This problem asks you to find the maximum subset size where each number in the subset follows a specific pattern with hash table lookups.

array hash table enumeration

Problem Statement

You are given an array of positive integers nums, and your task is to find the maximum number of elements in a subset that satisfies a given pattern.

The subset should follow a specific rule, and you must return the largest valid subset size. This problem relies heavily on array scanning and hash table lookups to identify the valid subset efficiently.

Examples

Example 1

Input: nums = [5,4,1,2,2]

Output: 3

We can select the subset {4,2,2}, which can be placed in the array as [2,4,2] which follows the pattern and 22 == 4. Hence the answer is 3.

Example 2

Input: nums = [1,3,2,4]

Output: 1

We can select the subset {1}, which can be placed in the array as [1] which follows the pattern. Hence the answer is 1. Note that we could have also selected the subsets {2}, {3}, or {4}, there may be multiple subsets which provide the same answer.

Constraints

2 <= nums.length <= 105
1 <= nums[i] <= 109

Solution Approach

Array Scanning

First, scan through the array to examine all numbers and identify potential elements for the subset. This involves checking the conditions one by one and ensuring the numbers are valid candidates.

Hash Table Lookup

Use a hash table to store the numbers and check the condition for each element efficiently. Hash table lookups allow you to quickly verify if an element can be included in the subset.

Subset Maximization

Once you have validated each number using the hash table, track the size of the largest valid subset. Continue updating the maximum subset size as you iterate over the array.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity of this problem depends on the final approach, particularly the efficiency of scanning the array and performing hash table lookups. If the approach involves linear scans with constant-time hash table lookups, the complexity will be O(n). Space complexity will also depend on the hash table storage, making it O(n) in the worst case.

What Interviewers Usually Probe

Evaluates your ability to implement efficient array scanning techniques.
Assesses your understanding of hash table usage for fast lookups.
Tests your ability to maximize a subset size by iterating over arrays efficiently.

Common Pitfalls or Variants

Common pitfalls

Not considering edge cases where no valid subset is found.
Inefficiently handling hash table lookups, causing slow performance.
Overlooking the requirement to track the largest subset during the array scan.

Follow-up variants

You could be asked to implement the same solution using different data structures like sets or arrays.
The problem might require you to optimize for time or space complexity.
You could encounter variations that involve different subset conditions or constraints.

FAQ

How can I find the maximum subset in the 'Find the Maximum Number of Elements in Subset' problem?

You need to scan the array, use hash table lookups for validation, and track the largest valid subset during the iteration.

What is the time complexity of solving the 'Find the Maximum Number of Elements in Subset' problem?

The time complexity depends on the approach, but if implemented efficiently, it could be O(n) with linear scans and constant-time hash lookups.

How do hash tables improve the efficiency of solving this problem?

Hash tables allow for constant-time lookups, making it faster to verify whether an element can be included in the subset without needing to re-scan the array.

What are common mistakes to avoid in the 'Find the Maximum Number of Elements in Subset' problem?

Common mistakes include inefficient handling of hash table lookups and failing to correctly track the largest subset while scanning the array.

What patterns should I focus on when solving the 'Find the Maximum Number of Elements in Subset' problem?

Focus on array scanning combined with hash table lookups for efficient validation and subset maximization.

terminal

Solution

Solution 1: Hash Table + Enumeration

We use a hash table $cnt$ to record the occurrence count of each element in the array $nums$. For each element $x$, we can keep squaring it until its count in the hash table $cnt$ is less than $2$. At this point, we check if the count of $x$ in the hash table $cnt$ is $1$. If it is, it means that $x$ can still be included in the subset. Otherwise, we need to remove an element from the subset to ensure the subset count is odd. Then we update the answer and continue to enumerate the next element.

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class Solution:
    def maximumLength(self, nums: List[int]) -> int:
        cnt = Counter(nums)
        ans = cnt[1] - (cnt[1] % 2 ^ 1)
        del cnt[1]
        for x in cnt:
            t = 0
            while cnt[x] > 1:
                x = x * x
                t += 2
            t += 1 if cnt[x] else -1
            ans = max(ans, t)
        return ans

Continue Practicing

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