#3272

Hard

LeetCode Problem Workspace

Find the Count of Good Integers

Count good integers by rearranging digits to form k-palindromic numbers, leveraging hash tables and math techniques.

hash table math combinatorics enumeration

Problem Statement

You are given two integers, n and k. The task is to count the number of good integers of length n that can be rearranged to form a k-palindromic integer. A k-palindromic integer is one where each digit, when rearranged, forms a number that can be mirrored across its midpoint, considering the parameter k.

For instance, if n = 3 and k = 5, the valid good integers could be rearranged to form valid k-palindromic numbers. Your goal is to calculate how many such integers exist by applying an efficient approach using hash tables and mathematical analysis.

Examples

Example 1

Input: n = 3, k = 5

Output: 27

Some of the good integers are:

Example 2

Input: n = 1, k = 4

Output: 2

The two good integers are 4 and 8.

Example 3

Input: n = 5, k = 6

Output: 2468

Example details omitted.

Constraints

1 <= n <= 10
1 <= k <= 9

Solution Approach

Use Hash Tables for Digit Frequency Tracking

Track the frequency of each digit using a hash table to store counts. By efficiently counting occurrences, we can identify potential good integers and check if they can be rearranged into a k-palindromic integer.

Mathematical Rearrangement Validation

Use combinatorics and modular arithmetic to determine if a set of digits can be rearranged into a k-palindromic number. This involves mathematical checks on the frequency and distribution of digits.

Optimize with Constraints on n and k

Given the constraints (n ≤ 10, k ≤ 9), the problem can be solved efficiently by focusing on valid combinations and skipping unnecessary calculations, ensuring that the time complexity remains manageable.

Complexity Analysis

Metric	Value
Time	O(n \log n \times 10^m)
Space	O(n \times 10^m)

The time complexity is O(n \log n \times 10^m) due to the sorting and digit processing required to generate possible good integers. The space complexity is O(n \times 10^m) because of the storage needed for digit frequencies and combinations.

What Interviewers Usually Probe

Check if the candidate optimizes hash table usage for digit counting.
Evaluate the candidate's approach to applying combinatorics in rearranging digits.
Assess if the candidate can handle the constraints efficiently without brute force.

Common Pitfalls or Variants

Common pitfalls

Not efficiently handling the count of digits in the hash table, leading to unnecessary complexity.
Missing the importance of combinatorial logic in validating k-palindromic numbers.
Overlooking the constraints, attempting brute-force solutions for larger values of n or k.

Follow-up variants

Varying the size of n and k could introduce a need for different optimization strategies.
Expanding the problem to allow for larger k values could affect the solution's performance.
Testing with larger numbers or edge cases like n = 1 or k = 9 can challenge the solution's efficiency.

FAQ

What is the pattern to solve the "Find the Count of Good Integers" problem?

The pattern involves using hash tables to track digit frequencies and applying mathematical combinatorics to validate k-palindromic number formations.

How does combinatorics help in solving this problem?

Combinatorics helps by calculating possible rearrangements of digits to form valid k-palindromic numbers, ensuring an efficient solution without brute force.

What are the constraints for the "Find the Count of Good Integers" problem?

The constraints are 1 <= n <= 10 and 1 <= k <= 9, allowing for efficient solutions without needing excessive computational resources.

How do hash tables assist in solving this problem?

Hash tables store the frequency of each digit, enabling fast lookup and counting, which is essential for identifying valid good integers.

What are common mistakes to avoid in this problem?

Common mistakes include not optimizing the use of hash tables, missing combinatorial logic for checking k-palindromic numbers, and attempting brute-force solutions without considering constraints.

terminal

Solution

Solution 1: Enumeration + Combinatorics

We can consider enumerating all palindromic numbers of length $n$ and checking whether they are $k$-palindromic numbers. Due to the properties of palindromic numbers, we only need to enumerate the first half of the digits and then reverse and append them to form the full number.

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class Solution:
    def countGoodIntegers(self, n: int, k: int) -> int:
        fac = [factorial(i) for i in range(n + 1)]
        ans = 0
        vis = set()
        base = 10 ** ((n - 1) // 2)
        for i in range(base, base * 10):
            s = str(i)
            s += s[::-1][n % 2 :]
            if int(s) % k:
                continue
            t = "".join(sorted(s))
            if t in vis:
                continue
            vis.add(t)
            cnt = Counter(t)
            res = (n - cnt["0"]) * fac[n - 1]
            for x in cnt.values():
                res //= fac[x]
            ans += res
        return ans

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