509. Fibonacci Number

題目

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

題目大意

斐波那契數列, 通常用 F(n) 表示

F(0) = 0, F(1) = 1 F(N) = F(N - 1) + F(N - 2), 其中 N > 1. 給定 N，計算 F(N)。

提示：0 ≤ N ≤ 30 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 ,610, 987……

解題思路

遇到遞迴最好畫出遞迴樹

            f(20)
           /     \ 
        f(19)   f(18)
       ...           ...
      /  \           /  \   
    f(1) f(2)       f(1) f(2)

這一題解法很多，大的分類是四種，遞歸，記憶化搜索(dp)，矩陣快速冪，通項公式。其中記憶化搜索可以寫 3 種方法，自底向上的，自頂向下的，優化空間複雜度版的。通項公式方法實質是求 a^b 這個還可以用快速冪優化時間複雜度到 O(log n) 。

來源

• https://books.halfrost.com/leetcode/ChapterFour/0500~0599/0509.Fibonacci-Number/
• https://leetcode.com/problems/fibonacci-number/

解答

https://github.com/kimi0230/LeetcodeGolang/blob/master/Leetcode/0509.Fibonacci-Number/Fibonacci-Number.go

// --- Directions
// Print out the n-th entry in the fibonacci series.
// The fibonacci series is an ordering of numbers where
// each number is the sum of the preceeding two.
// For example, the sequence
//  [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
// forms the first ten entries of the fibonacci series.
// Example:
//   fib(4) === 3

package fibonaccinumber

import "math"

// Fib : iterative 迴圈  O(n) . 空間複雜  O(n). 自底向上的記憶化搜索
func FibIterative(n int) int {
    var result = []int{0, 1}

    for i := 2; i