6060
6161时间复杂度 $O(n),ドル空间复杂度 $O(1)$。
6262
63+ ** 方法二:矩阵快速幂**
64+ 65+ 我们设 $Fib(n)$ 表示一个 1ドル \times 2$ 的矩阵 $\begin{bmatrix} F_n & F_ {n - 1} \end{bmatrix},ドル其中 $F_n$ 和 $F_ {n - 1}$ 分别是第 $n$ 个和第 $n - 1$ 个斐波那契数。
66+ 67+ 我们希望根据 $Fib(n-1) = \begin{bmatrix} F_ {n - 1} & F_ {n - 2} \end{bmatrix}$ 推出 $Fib(n)$。也即是说,我们需要一个矩阵 $base,ドル使得 $Fib(n - 1) \times base = Fib(n),ドル即:
68+ 69+ $$
70+ \begin{bmatrix}
71+ F_{n - 1} & F_{n - 2}
72+ \end{bmatrix} \times base = \begin{bmatrix} F_n & F_{n - 1} \end{bmatrix}
73+ $$
74+ 75+ 由于 $F_n = F_ {n - 1} + F_ {n - 2},ドル所以矩阵 $base$ 的第一列为:
76+ 77+ $$
78+ \begin{bmatrix}
79+ 1 \\
80+ 1
81+ \end{bmatrix}
82+ $$
83+ 84+ 第二列为:
85+ 86+ $$
87+ \begin{bmatrix}
88+ 1 \\
89+ 0
90+ \end{bmatrix}
91+ $$
92+ 93+ 因此有:
94+ 95+ $$
96+ \begin{bmatrix} F_{n - 1} & F_{n - 2} \end{bmatrix} \times \begin{bmatrix}1 & 1 \\ 1 & 0\end{bmatrix} = \begin{bmatrix} F_n & F_{n - 1} \end{bmatrix}
97+ $$
98+ 99+ 我们定义初始矩阵 $res = \begin{bmatrix} 1 & 1 \end{bmatrix},ドル那么 $F_n$ 等于 $res$ 乘以 $base^{n - 1}$ 的结果矩阵中第一行的第一个元素。使用矩阵快速幂求解即可。
100+ 101+ 时间复杂度 $O(\log n),ドル空间复杂度 $O(1)$。
102+ 63103<!-- tabs:start -->
64104
65105### ** Python3**
@@ -75,6 +115,31 @@ class Solution:
75115 return b
76116```
77117
118+ ``` python
119+ class Solution :
120+ def climbStairs (self n : int ) -> int :
121+ def mul (a : List[List[int ]], b : List[List[int ]]) -> List[List[int ]]:
122+ m, n = len (a), len (b[0 ])
123+ c = [[0 ] * n for _ in range (m)]
124+ for i in range (m):
125+ for j in range (n):
126+ for k in range (len (a[0 ])):
127+ c[i][j] = c[i][j] + a[i][k] * b[k][j]
128+ return c
129+ 130+ def pow (a : List[List[int ]], n : int ) -> List[List[int ]]:
131+ res = [[1 , 1 ], [0 , 0 ]]
132+ while n:
133+ if n & 1 :
134+ res = mul(res, a)
135+ n >>= 1
136+ a = mul(a, a)
137+ return res
138+ 139+ a = [[1 , 1 ], [1 , 0 ]]
140+ return pow (a, n - 1 )[0 ][0 ]
141+ ```
142+ 78143### ** Java**
79144
80145<!-- 这里可写当前语言的特殊实现逻辑 -->
@@ -93,6 +158,40 @@ class Solution {
93158}
94159```
95160
161+ ``` java
162+ class Solution {
163+ public int climbStairs (int n ) {
164+ int [][] a = {{1 , 1 ,}, {1 , 0 }};
165+ return pow(a, n - 1 )[0 ][0 ];
166+ }
167+ 168+ private int [][] mul (int [][] a , int [][] b ) {
169+ int m = a. length, n = b[0 ]. length;
170+ int [][] c = new int [m][n];
171+ for (int i = 0 ; i < m; ++ i) {
172+ for (int j = 0 ; j < n; ++ j) {
173+ for (int k = 0 ; k < a[0 ]. length; ++ k) {
174+ c[i][j] += a[i][k] * b[k][j];
175+ }
176+ }
177+ }
178+ return c;
179+ }
180+ 181+ private int [][] pow (int [][] a , int n ) {
182+ int [][] res = {{1 , 1 }, {0 , 0 }};
183+ while (n > 0 ) {
184+ if ((n & 1 ) == 1 ) {
185+ res = mul(res, a);
186+ }
187+ n >> = 1 ;
188+ a = mul(a, a);
189+ }
190+ return res;
191+ }
192+ }
193+ ```
194+ 96195### ** C++**
97196
98197``` cpp
@@ -110,6 +209,87 @@ public:
110209};
111210```
112211
212+ ```cpp
213+ class Solution {
214+ public:
215+ int climbStairs(int n) {
216+ vector<vector<long long>> a = {{1, 1}, {1, 0}};
217+ return pow(a, n - 1)[0][0];
218+ }
219+
220+ private:
221+ vector<vector<long long>> mul(vector<vector<long long>>& a, vector<vector<long long>>& b) {
222+ int m = a.size(), n = b[0].size();
223+ vector<vector<long long>> res(m, vector<long long>(n));
224+ for (int i = 0; i < m; ++i) {
225+ for (int j = 0; j < n; ++j) {
226+ for (int k = 0; k < a[0].size(); ++k) {
227+ res[i][j] += a[i][k] * b[k][j];
228+ }
229+ }
230+ }
231+ return res;
232+ }
233+
234+ vector<vector<long long>> pow(vector<vector<long long>>& a, int n) {
235+ vector<vector<long long>> res = {{1, 1}, {0, 0}};
236+ while (n) {
237+ if (n & 1) {
238+ res = mul(res, a);
239+ }
240+ a = mul(a, a);
241+ n >>= 1;
242+ }
243+ return res;
244+ }
245+ };
246+ ```
247+ 248+ ### ** Go**
249+ 250+ ``` go
251+ func climbStairs (n int ) int {
252+ a , b := 0 , 1
253+ for i := 0 ; i < n; i++ {
254+ a, b = b, a+b
255+ }
256+ return b
257+ }
258+ ```
259+ 260+ ``` go
261+ type matrix [2 ][2 ]int
262+ 263+ func climbStairs (n int ) int {
264+ a := matrix{{1 , 1 }, {1 , 0 }}
265+ return pow (a, n-1 )[0 ][0 ]
266+ }
267+ 268+ func mul (a , b matrix ) (c matrix ) {
269+ m , n := len (a), len (b[0 ])
270+ for i := 0 ; i < m; i++ {
271+ for j := 0 ; j < n; j++ {
272+ for k := 0 ; k < len (a[0 ]); k++ {
273+ c[i][j] += a[i][k] * b[k][j]
274+ }
275+ }
276+ }
277+ return
278+ }
279+ 280+ func pow (a matrix , n int ) matrix {
281+ res := matrix{{1 , 1 }, {0 , 0 }}
282+ for n > 0 {
283+ if n&1 == 1 {
284+ res = mul (res, a)
285+ }
286+ a = mul (a, a)
287+ n >>= 1
288+ }
289+ return res
290+ }
291+ ```
292+ 113293### ** JavaScript**
114294
115295``` js
@@ -129,15 +309,47 @@ var climbStairs = function (n) {
129309};
130310```
131311
132- ### ** Go**
312+ ``` js
313+ /**
314+ * @param {number} n
315+ * @return {number}
316+ */
317+ var climbStairs = function (n ) {
318+ const a = [
319+ [1 , 1 ],
320+ [1 , 0 ],
321+ ];
322+ return pow (a, n - 1 )[0 ][0 ];
323+ };
133324
134- ``` go
135- func climbStairs (n int ) int {
136- a , b := 0 , 1
137- for i := 0 ; i < n; i++ {
138- a, b = b, a+b
139- }
140- return b
325+ function mul (a , b ) {
326+ const [m , n ] = [a .length , b[0 ].length ];
327+ const c = Array (m)
328+ .fill (0 )
329+ .map (() => Array (n).fill (0 ));
330+ for (let i = 0 ; i < m; ++ i) {
331+ for (let j = 0 ; j < n; ++ j) {
332+ for (let k = 0 ; k < a[0 ].length ; ++ k) {
333+ c[i][j] += a[i][k] * b[k][j];
334+ }
335+ }
336+ }
337+ return c;
338+ }
339+ 340+ function pow (a , n ) {
341+ let res = [
342+ [1 , 1 ],
343+ [0 , 0 ],
344+ ];
345+ while (n) {
346+ if (n & 1 ) {
347+ res = mul (res, a);
348+ }
349+ a = mul (a, a);
350+ n >>= 1 ;
351+ }
352+ return res;
141353}
142354```
143355
@@ -154,6 +366,46 @@ function climbStairs(n: number): number {
154366}
155367```
156368
369+ ``` ts
370+ function climbStairs(n : number ): number {
371+ const = [
372+ [1 , 1 ],
373+ [1 , 0 ],
374+ ];
375+ return pow (a , n - 1 )[0 ][0 ];
376+ }
377+ 378+ function mul(a : number [][], b : number [][]): number [][] {
379+ const = [a .length , b [0 ].length ];
380+ const = Array (m )
381+ .fill (0 )
382+ .map (() => Array (n ).fill (0 ));
383+ for (let i = 0 ; i < m ; ++ i ) {
384+ for (let j = 0 ; j < n ; ++ j ) {
385+ for (let k = 0 ; k < a [0 ].length ; ++ k ) {
386+ c [i ][j ] += a [i ][k ] * b [k ][j ];
387+ }
388+ }
389+ }
390+ return c ;
391+ }
392+ 393+ function pow(a : number [][], n : number ): number [][] {
394+ let res = [
395+ [1 , 1 ],
396+ [0 , 0 ],
397+ ];
398+ while (n ) {
399+ if (n & 1 ) {
400+ res = mul (res , a );
401+ }
402+ a = mul (a , a );
403+ n >>= 1 ;
404+ }
405+ return res ;
406+ }
407+ ```
408+ 157409### ** Rust**
158410
159411``` rust
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