How do you solve Preimage Size of Factorial Zeroes Function on LeetCode?

Preimage Size of Factorial Zeroes Function (LeetCode #793) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(log(n)) time complexity and O(1) space complexity. Key insight: The number of trailing zeros in a factorial is determined by the number of times 5 is a factor in the numbers leading up to that factorial.

What is the time complexity of Preimage Size of Factorial Zeroes Function?

The optimal solution for Preimage Size of Factorial Zeroes Function runs in O(log(n)) time and uses O(1) space. The time complexity is O(log(n)) due to the binary search, which significantly reduces the number of checks needed compared to the brute force method.

What is the brute force solution for Preimage Size of Factorial Zeroes Function?

The brute force approach for Preimage Size of Factorial Zeroes Function is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute force approach calculates the number of trailing zeros for each factorial from 0! up to some limit and counts how many times this equals k. This is straightforward but inefficient for large k. The optimal Optimal Solution approach improves this to O(log(n)).

What algorithm or data structure is used to solve Preimage Size of Factorial Zeroes Function?

Preimage Size of Factorial Zeroes Function uses the following concepts: Math, Binary Search. The recommended patterns to study are: Binary Search, Mathematical Properties.

What are the interview tips for Preimage Size of Factorial Zeroes Function?

Always clarify the problem statement and ask about edge cases. Think about the mathematical properties involved before jumping into coding. Practice explaining your thought process as you code, as this helps interviewers understand your reasoning.

What are common mistakes when solving Preimage Size of Factorial Zeroes Function?

Not considering the edge cases where k is very large or very small. Confusing the calculation of trailing zeros with other properties of factorials.

#793

Preimage Size of Factorial Zeroes Function

Hard

MathBinary SearchBinary SearchMathematical Properties

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(log(n))
Space	O(1)	O(1)

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Intuition

Time O(log(n))Space O(1)

Instead of calculating trailing zeros for every number, we can use binary search to find the range of numbers whose factorials have exactly k trailing zeros. This drastically reduces the number of calculations needed.

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Algorithm

4 steps

1Step 1: Define a function to calculate the number of trailing zeros in x! as before.
2Step 2: Use binary search to find the smallest x such that f(x) >= k.
3Step 3: Use binary search again to find the smallest x such that f(x) > k.
4Step 4: The difference between the two x values gives the count of integers x such that f(x) = k.

solution.py24 lines

1# Full working Python code
2
3def trailing_zeros(n):
4    count = 0
5    while n > 0:
6        n //= 5
7        count += n
8    return count
9
10def preimage_size_fzf(k):
11    def binary_search(target):
12        left, right = 0, 5 * (target + 1)
13        while left < right:
14            mid = (left + right) // 2
15            if trailing_zeros(mid) < target:
16                left = mid + 1
17            else:
18                right = mid
19        return left
20
21    left = binary_search(k)
22    right = binary_search(k + 1)
23    return right - left
24

ℹ

Complexity note: The time complexity is O(log(n)) due to the binary search, which significantly reduces the number of checks needed compared to the brute force method.

1The number of trailing zeros in a factorial is determined by the number of times 5 is a factor in the numbers leading up to that factorial.
2Using binary search allows us to efficiently find the range of integers that meet the criteria, rather than checking each integer one by one.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.