feat: add solutions to lc problem: No.0634

No.0634.Find the Derangement of An Array

Author: yanglbme

solution/0600-0699/0634.Find the Derangement of An Array/README.md

@@ -39,22 +39,143 @@
 
 <!-- 这里可写通用的实现逻辑 -->
 
+**方法一：动态规划**
+
+我们定义 $f[i]$ 表示长度为 $i$ 的数组的错位排列的数量。初始时 $f[0] = 1$, $f[1] = 0$。答案即为 $f[n]$。
+
+对于长度为 $i$ 的数组，我们考虑将数字 $1$ 放在哪个位置，假设放在第 $j$ 个位置，这里有 $i-1$ 种选择，那么接下来数字 $j$ 可以有两种选择：
+
+-   放在第 $1$ 个位置，那么剩下的 $i - 2$ 个位置可以有 $f[i - 2]$ 种错位排列，因此总共有 $(i - 1) \times f[i - 2]$ 种错位排列；
+-   不放在第 $1$ 个位置，那么相当于转化为了长度为 $i - 1$ 的数组的错位排列，因此总共有 $(i - 1) \times f[i - 1]$ 种错位排列。
+
+综上，我们有如下状态转移方程：
+
+$$
+f[i] = (i - 1) \times (f[i - 1] + f[i - 2])
+$$
+
+最终答案即为 $f[n]$。注意答案的取模操作。
+
+我们发现，状态转移方程中只与 $f[i - 1]$ 和 $f[i - 2]$ 有关，因此我们可以使用两个变量 $a$ 和 $b$ 来分别表示 $f[i - 1]$ 和 $f[i - 2]$，从而将空间复杂度降低到 $O(1)$。
+
+时间复杂度 $O(n)$，空间复杂度 $O(1)$。其中 $n$ 为数组的长度。
+
 <!-- tabs:start -->
 
 ### **Python3**
 
 <!-- 这里可写当前语言的特殊实现逻辑 -->
 
 ```python
+class Solution:
+    def findDerangement(self, n: int) -> int:
+        mod = 10**9 + 7
+        f = [1] + [0] * n
+        for i in range(2, n + 1):
+            f[i] = (i - 1) * (f[i - 1] + f[i - 2]) % mod
+        return f[n]
+```
 
+```python
+class Solution:
+    def findDerangement(self, n: int) -> int:
+        mod = 10**9 + 7
+        a, b = 1, 0
+        for i in range(2, n + 1):
+            a, b = b, ((i - 1) * (a + b)) % mod
+        return b
 ```
 
 ### **Java**
 
 <!-- 这里可写当前语言的特殊实现逻辑 -->
 
 ```java
+class Solution {
+    public int findDerangement(int n) {
+        long[] f = new long[n + 1];
+        f[0] = 1;
+        final int mod = (int) 1e9 + 7;
+        for (int i = 2; i <= n; ++i) {
+            f[i] = (i - 1) * (f[i - 1] + f[i - 2]) % mod;
+        }
+        return (int) f[n];
+    }
+}
+```
+
+```java
+class Solution {
+    public int findDerangement(int n) {
+        final int mod = (int) 1e9 + 7;
+        long a = 1, b = 0;
+        for (int i = 2; i <= n; ++i) {
+            long c = (i - 1) * (a + b) % mod;
+            a = b;
+            b = c;
+        }
+        return (int) b;
+    }
+}
+```
+
+### **C++**
+
+```cpp
+class Solution {
+public:
+    int findDerangement(int n) {
+        long long f[n + 1];
+        memset(f, 0, sizeof(f));
+        f[0] = 1;
+        const int mod = 1e9 + 7;
+        for (int i = 2; i <= n; i++) {
+            f[i] = (i - 1LL) * (f[i - 1] + f[i - 2]) % mod;
+        }
+        return f[n];
+    }
+};
+```
+
+```cpp
+class Solution {
+public:
+    int findDerangement(int n) {
+        long long a = 1, b = 0;
+        const int mod = 1e9 + 7;
+        for (int i = 2; i <= n; ++i) {
+            long long c = (i - 1) * (a + b) % mod;
+            a = b;
+            b = c;
+        }
+        return b;
+    }
+};
+```
+
+### **Go**
+
+```go
+func findDerangement(n int) int {
+	f := make([]int, n+1)
+	f[0] = 1
+	const mod = 1e9 + 7
+	for i := 2; i <= n; i++ {
+		f[i] = (i - 1) * (f[i-1] + f[i-2]) % mod
+	}
+	return f[n]
+}
+```
 
+```go
+func findDerangement(n int) int {
+	a, b := 1, 0
+	const mod = 1e9 + 7
+	for i := 2; i <= n; i++ {
+		a, b = b, (i-1)*(a+b)%mod
+	}
+	return b
+}
 ```
 
 ### **...**

solution/0600-0699/0634.Find the Derangement of An Array/README_EN.md

@@ -38,13 +38,113 @@
 ### **Python3**
 
 ```python
+class Solution:
+    def findDerangement(self, n: int) -> int:
+        mod = 10**9 + 7
+        f = [1] + [0] * n
+        for i in range(2, n + 1):
+            f[i] = (i - 1) * (f[i - 1] + f[i - 2]) % mod
+        return f[n]
+```
 
+```python
+class Solution:
+    def findDerangement(self, n: int) -> int:
+        mod = 10**9 + 7
+        a, b = 1, 0
+        for i in range(2, n + 1):
+            a, b = b, ((i - 1) * (a + b)) % mod
+        return b
 ```
 
 ### **Java**
 
 ```java
+class Solution {
+    public int findDerangement(int n) {
+        long[] f = new long[n + 1];
+        f[0] = 1;
+        final int mod = (int) 1e9 + 7;
+        for (int i = 2; i <= n; ++i) {
+            f[i] = (i - 1) * (f[i - 1] + f[i - 2]) % mod;
+        }
+        return (int) f[n];
+    }
+}
+```
+
+```java
+class Solution {
+    public int findDerangement(int n) {
+        final int mod = (int) 1e9 + 7;
+        long a = 1, b = 0;
+        for (int i = 2; i <= n; ++i) {
+            long c = (i - 1) * (a + b) % mod;
+            a = b;
+            b = c;
+        }
+        return (int) b;
+    }
+}
+```
+
+### **C++**
+
+```cpp
+class Solution {
+public:
+    int findDerangement(int n) {
+        long long f[n + 1];
+        memset(f, 0, sizeof(f));
+        f[0] = 1;
+        const int mod = 1e9 + 7;
+        for (int i = 2; i <= n; i++) {
+            f[i] = (i - 1LL) * (f[i - 1] + f[i - 2]) % mod;
+        }
+        return f[n];
+    }
+};
+```
+
+```cpp
+class Solution {
+public:
+    int findDerangement(int n) {
+        long long a = 1, b = 0;
+        const int mod = 1e9 + 7;
+        for (int i = 2; i <= n; ++i) {
+            long long c = (i - 1) * (a + b) % mod;
+            a = b;
+            b = c;
+        }
+        return b;
+    }
+};
+```
+
+### **Go**
+
+```go
+func findDerangement(n int) int {
+	f := make([]int, n+1)
+	f[0] = 1
+	const mod = 1e9 + 7
+	for i := 2; i <= n; i++ {
+		f[i] = (i - 1) * (f[i-1] + f[i-2]) % mod
+	}
+	return f[n]
+}
+```
 
+```go
+func findDerangement(n int) int {
+	a, b := 1, 0
+	const mod = 1e9 + 7
+	for i := 2; i <= n; i++ {
+		a, b = b, (i-1)*(a+b)%mod
+	}
+	return b
+}
 ```
 
 ### **...**

solution/0600-0699/0634.Find the Derangement of An Array/Solution.cpp

@@ -0,0 +1,13 @@
+class Solution {
+public:
+    int findDerangement(int n) {
+        long long a = 1, b = 0;
+        const int mod = 1e9 + 7;
+        for (int i = 2; i <= n; ++i) {
+            long long c = (i - 1) * (a + b) % mod;
+            a = b;
+            b = c;
+        }
+        return b;
+    }
+};

solution/0600-0699/0634.Find the Derangement of An Array/Solution.go

@@ -0,0 +1,8 @@
+func findDerangement(n int) int {
+	a, b := 1, 0
+	const mod = 1e9 + 7
+	for i := 2; i <= n; i++ {
+		a, b = b, (i-1)*(a+b)%mod
+	}
+	return b
+}

solution/0600-0699/0634.Find the Derangement of An Array/Solution.java

@@ -0,0 +1,12 @@
+class Solution {
+    public int findDerangement(int n) {
+        final int mod = (int) 1e9 + 7;
+        long a = 1, b = 0;
+        for (int i = 2; i <= n; ++i) {
+            long c = (i - 1) * (a + b) % mod;
+            a = b;
+            b = c;
+        }
+        return (int) b;
+    }
+}

solution/0600-0699/0634.Find the Derangement of An Array/Solution.py

@@ -0,0 +1,7 @@
+class Solution:
+    def findDerangement(self, n: int) -> int:
+        mod = 10**9 + 7
+        a, b = 1, 0
+        for i in range(2, n + 1):
+            a, b = b, ((i - 1) * (a + b)) % mod
+        return b