#1414

Medium

auto_awesomeGreedy choice plus invariant validation

LeetCode Problem Workspace

Find the Minimum Number of Fibonacci Numbers Whose Sum Is K

Find the minimum number of Fibonacci numbers that sum up to a given integer k, using a greedy approach.

Problem Statement

Given a positive integer k, return the minimum number of Fibonacci numbers whose sum equals k. The Fibonacci numbers are defined as: 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. You may reuse the same Fibonacci number multiple times.

The goal is to find the minimal count of Fibonacci numbers that add up to the target k, using a greedy approach. This means selecting the largest Fibonacci number less than or equal to k at each step until k is reduced to zero.

Examples

Example 1

Input: k = 7

Output: 2

The Fibonacci numbers are: 1, 1, 2, 3, 5, 8, 13, ... For k = 7 we can use 2 + 5 = 7.

Example 2

Input: k = 10

Output: 2

For k = 10 we can use 2 + 8 = 10.

Example 3

Input: k = 19

Output: 3

For k = 19 we can use 1 + 5 + 13 = 19.

Constraints

1 <= k <= 109

Solution Approach

Generate Fibonacci Numbers

First, generate all Fibonacci numbers up to k. This ensures that we only consider valid numbers and avoid unnecessary calculations. Fibonacci numbers grow exponentially, so there will only be a small number of Fibonacci numbers up to any given k.

Greedy Selection

Select the largest Fibonacci number that is less than or equal to k. Subtract this number from k and repeat the process until k reaches zero. This approach minimizes the number of numbers used to sum up to k.

Validate the Sum

At each step, ensure that the sum of the selected Fibonacci numbers equals k. Since we use a greedy approach, the validation ensures no number is missed, and the solution is optimal.

Complexity Analysis

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

The time complexity is determined by the number of Fibonacci numbers up to k, which grows logarithmically. In the worst case, the number of Fibonacci numbers needed is small, so the solution is efficient. The space complexity is also logarithmic due to the storage of Fibonacci numbers up to k.

What Interviewers Usually Probe

Assess if the candidate uses a greedy approach and justifies it effectively.
Look for a clear understanding of Fibonacci number generation and efficient use of them.
Check if the candidate ensures correctness by validating the final sum.

Common Pitfalls or Variants

Common pitfalls

Failing to generate the correct Fibonacci sequence up to k.
Choosing Fibonacci numbers that are too small, leading to excessive additions.
Overcomplicating the problem by considering all possible combinations instead of using a greedy approach.

Follow-up variants

Allow only distinct Fibonacci numbers in the sum.
Limit the number of Fibonacci numbers that can be used in the sum.
Extend the problem to larger integers, requiring more advanced optimizations.

FAQ

What is the greedy approach in the problem 'Find the Minimum Number of Fibonacci Numbers Whose Sum Is K'?

The greedy approach selects the largest Fibonacci number less than or equal to k and subtracts it from k, repeating this process until the sum equals k.

How do I generate Fibonacci numbers for this problem?

You generate Fibonacci numbers starting from 1, 1, and then continue adding the previous two numbers until the value exceeds k.

What are the constraints for the problem 'Find the Minimum Number of Fibonacci Numbers Whose Sum Is K'?

The constraint is that 1 <= k <= 10^9.

Can the same Fibonacci number be used multiple times?

Yes, the same Fibonacci number can be used multiple times to reach the sum of k.

What is the time complexity of the solution for the problem 'Find the Minimum Number of Fibonacci Numbers Whose Sum Is K'?

The time complexity is logarithmic in relation to k because the number of Fibonacci numbers is small, and only a few operations are needed to reduce k to zero.

terminal

Solution

Solution 1: Greedy

We can greedily select the largest Fibonacci number that does not exceed $k$ each time, then subtract this number from $k$ and increment the answer by one. This process is repeated until $k = 0$.

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class Solution:
    def findMinFibonacciNumbers(self, k: int) -> int:
        a = b = 1
        while b <= k:
            a, b = b, a + b
        ans = 0
        while k:
            if k >= b:
                k -= b
                ans += 1
            a, b = b - a, a
        return ans

Continue Practicing

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