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509. Fibonacci Number ​

题目 ​

The Fibonacci numbers, commonly denoted F(n) form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,

F(0) = 0, F(1) = 1
F(n) = F(n - 1) + F(n - 2), for n > 1.

Given n, calculate F(n).

 

Example 1:

Input: n = 2
Output: 1
Explanation: F(2) = F(1) + F(0) = 1 + 0 = 1.

Example 2:

Input: n = 3
Output: 2
Explanation: F(3) = F(2) + F(1) = 1 + 1 = 2.

Example 3:

Input: n = 4
Output: 3
Explanation: F(4) = F(3) + F(2) = 2 + 1 = 3.

 

Constraints:

  • 0 <= n <= 30
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    • 思路一 ​

    递归法解决问题，使用 python 时，可以使用 function.tools 中的 cache 缓存计算结果，时间复杂度由原来的 O(2n) 降低到 O(N)

    解法 ​

    python
    from functools import cache


class Solution:
    def fib(self, n: int) -> int:
        @cache
        def fib(self, n: int) -> int:
            if n == 0:
                return 0
            elif n == 1:
                return 1
            else:
                return self.fib(n - 1) + self.fib(n - 2)

    复杂度分析 ​

    • 时间复杂度 O(N)
    • 空间复杂度 O(N)

    解法二 ​

    py
    # leetcode submit region begin(Prohibit modification and deletion)


class Solution:
    def fib(self, n: int) -> int:
        if n < 2:
            return n

        p, q, r = 0, 0, 1
        for i in range(2, n + 1):
            p, q = q, r
            r = p + q

        return r
        
# leetcode submit region end(Prohibit modification and deletion)

    复杂度分析 ​

    • 时间复杂度 O(N)
    • 空间复杂度 O(1)