How to Solve X of a Kind in a Deck of Cards

To solve this problem, check if the deck can be partitioned into groups where each group contains the same number of identical cards. The key is to use hash table counting to determine the frequency of each card value, and then ensure that these frequencies are divisible by a common divisor.

Problem Statement

You are given an integer array deck where each element represents the number written on a card. Your task is to determine if it's possible to partition the deck into one or more groups such that each group contains the same number of identical cards.

Return true if such a partition is possible, or false otherwise. You can assume that the deck contains at least one card, and all values in the array are non-negative integers.

Examples

Example 1

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

Output: true

Possible partition [1,1],[2,2],[3,3],[4,4].

Example 2

Input: deck = [1,1,1,2,2,2,3,3]

Output: false

No possible partition.

Constraints

• 1 <= deck.length <= 104
• 0 <= deck[i] < 104

Solution Approach

Count Frequencies

Start by counting the frequency of each card value using a hash table or dictionary. This step is crucial for understanding how often each card appears, which is the foundation for determining valid partitions.

Find Greatest Common Divisor (GCD)

Once you have the frequencies of all the card values, calculate the greatest common divisor (GCD) of these frequencies. For the partition to be possible, the GCD must be greater than 1, ensuring that all card counts can be divided evenly into groups.

Verify Divisibility

If the GCD is greater than 1, it confirms that you can partition the deck into groups with equal counts. Otherwise, return false, as a valid partition cannot be formed.

Complexity Analysis

The time complexity of this solution is O(N log N), where N is the number of cards in the deck. This accounts for the time taken to count the frequencies and compute the GCD. The space complexity is O(N) because we need to store the frequency counts of the cards in a hash table or dictionary.

What Interviewers Usually Probe

• Check if the candidate understands how to utilize hash tables for counting elements efficiently.
• Observe if they can correctly identify the need for computing the GCD to ensure divisibility.
• Pay attention to how the candidate approaches the verification of group divisibility and their understanding of the problem constraints.

Common Pitfalls or Variants

Common pitfalls

• Misunderstanding the need for GCD: Candidates might incorrectly assume that the card counts should simply be equal without considering GCD as a condition for partitioning.
• Failure to handle edge cases: Candidates should be cautious of scenarios where some cards have very few occurrences or all cards are identical.
• Inefficient frequency counting: If a candidate does not choose an efficient way to count card frequencies, such as using a hash table or dictionary, the solution could become too slow for larger inputs.

Follow-up variants

• Consider variations where the deck contains a very large number of cards, requiring an optimized approach for counting and GCD computation.
• What if some card values are missing or are spread out very unevenly? Discuss how the algorithm scales under these conditions.
• Consider a constraint where the deck has a restricted number of card types or unique values, which could lead to different optimization strategies.

How GhostInterview Helps

• GhostInterview helps by guiding you through efficient solutions using array scanning and hash lookup for counting frequencies.
• It assists in optimizing the GCD computation to ensure divisibility is checked correctly, enhancing problem-solving skills for interview contexts.
• By offering targeted hints and reminders about the importance of frequency counting and divisibility, GhostInterview ensures you focus on the key pattern for this problem.

Topic Pages

• Array
• Hash Table
• Math
• Counting
• Number Theory

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FAQ

What is the main algorithmic approach to solve 'X of a Kind in a Deck of Cards'?

The main approach is to use hash tables for frequency counting, followed by computing the GCD of the frequencies to check if they can be divided evenly into groups.

How do I calculate the GCD of multiple numbers in this problem?

You can use the Euclidean algorithm to compute the GCD of the card frequencies, or use a built-in function to simplify this process.

What should I do if the GCD is 1?

If the GCD is 1, it's impossible to partition the deck into groups with the same number of identical cards, so you should return false.

How do I handle large input sizes for this problem?

For large inputs, the algorithm needs to efficiently count card frequencies and calculate the GCD without unnecessary recomputation. Using hash tables or dictionaries ensures this step is performed efficiently.

What edge cases should I consider when solving 'X of a Kind in a Deck of Cards'?

Consider cases where there is only one type of card, or where the cards appear in very uneven frequencies, as these cases can test the correctness of the solution.