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feat: add solutions to lcci problem: No.08.01 #1619

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363 changes: 306 additions & 57 deletions lcci/08.01.Three Steps Problem/README.md
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327 changes: 271 additions & 56 deletions lcci/08.01.Three Steps Problem/README_EN.md
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Original file line number Diff line number Diff line change
Expand Up @@ -39,31 +39,223 @@
```python
class Solution:
def waysToStep(self, n: int) -> int:
if n < 3:
return n
a, b, c = 1, 2, 4
for _ in range(4, n + 1):
a, b, c = b, c, (a + b + c) % 1000000007
return c
mod = 10**9 + 7
for _ in range(n - 1):
a, b, c = b, c, (a + b + c) % mod
return a
```

```python
class Solution:
def waysToStep(self, n: int) -> int:
mod = 10**9 + 7

def mul(a: List[List[int]], b: List[List[int]]) -> List[List[int]]:
m, n = len(a), len(b[0])
c = [[0] * n for _ in range(m)]
for i in range(m):
for j in range(n):
for k in range(len(a[0])):
c[i][j] = (c[i][j] + a[i][k] * b[k][j] % mod) % mod
return c

def pow(a: List[List[int]], n: int) -> List[List[int]]:
res = [[4, 2, 1]]
while n:
if n & 1:
res = mul(res, a)
n >>= 1
a = mul(a, a)
return res

if n < 4:
return 2 ** (n - 1)
a = [[1, 1, 0], [1, 0, 1], [1, 0, 0]]
return sum(pow(a, n - 4)[0]) % mod
```

### **Java**

```java
class Solution {
public int waysToStep(int n) {
if (n < 3) {
return n;
final int mod = (int) 1e9 + 7;
int a = 1, b = 2, c = 4;
for (int i = 1; i < n; ++i) {
int t = a;
a = b;
b = c;
c = (((a + b) % mod) + t) % mod;
}
return a;
}
}
```

```java
class Solution {
private final int mod = (int) 1e9 + 7;

public int waysToStep(int n) {
if (n < 4) {
return (int) Math.pow(2, n - 1);
}
long[][] a = {{1, 1, 0}, {1, 0, 1}, {1, 0, 0}};
long[][] res = pow(a, n - 4);
long ans = 0;
for (long x : res[0]) {
ans = (ans + x) % mod;
}
return (int) ans;
}

private long[][] mul(long[][] a, long[][] b) {
int m = a.length, n = b[0].length;
long[][] c = new long[m][n];
for (int i = 0; i < m; ++i) {
for (int j = 0; j < n; ++j) {
for (int k = 0; k < b.length; ++k) {
c[i][j] = (c[i][j] + a[i][k] * b[k][j] % mod) % mod;
}
}
}
return c;
}

private long[][] pow(long[][] a, int n) {
long[][] res = {{4, 2, 1}};
while (n > 0) {
if ((n & 1) == 1) {
res = mul(res, a);
}
a = mul(a, a);
n >>= 1;
}
return res;
}
}
```

### **C++**

```cpp
class Solution {
public:
int waysToStep(int n) {
const int mod = 1e9 + 7;
int a = 1, b = 2, c = 4;
for (int i = 4; i <= n; ++i) {
for (int i = 1; i < n; ++i) {
int t = a;
a = b;
b = c;
c = ((a + b) % 1000000007 + t) % 1000000007;
c = (((a + b) % mod) + t) % mod;
}
return a;
}
};
```

```cpp
class Solution {
public:
int waysToStep(int n) {
if (n < 4) {
return pow(2, n - 1);
}
vector<vector<ll>> a = {{1, 1, 0}, {1, 0, 1}, {1, 0, 0}};
vector<vector<ll>> res = qpow(a, n - 4);
ll ans = 0;
for (ll x : res[0]) {
ans = (ans + x) % mod;
}
return ans;
}

private:
using ll = long long;
const int mod = 1e9 + 7;
vector<vector<ll>> mul(vector<vector<ll>>& a, vector<vector<ll>>& b) {
int m = a.size(), n = b[0].size();
vector<vector<ll>> c(m, vector<ll>(n));
for (int i = 0; i < m; ++i) {
for (int j = 0; j < n; ++j) {
for (int k = 0; k < b.size(); ++k) {
c[i][j] = (c[i][j] + a[i][k] * b[k][j] % mod) % mod;
}
}
}
return c;
}

vector<vector<ll>> qpow(vector<vector<ll>>& a, int n) {
vector<vector<ll>> res = {{4, 2, 1}};
while (n) {
if (n & 1) {
res = mul(res, a);
}
a = mul(a, a);
n >>= 1;
}
return res;
}
};
```

### **Go**

```go
func waysToStep(n int) int {
const mod int = 1e9 + 7
a, b, c := 1, 2, 4
for i := 1; i < n; i++ {
a, b, c = b, c, (a+b+c)%mod
}
return a
}
```

```go
const mod = 1e9 + 7

func waysToStep(n int) (ans int) {
if n < 4 {
return int(math.Pow(2, float64(n-1)))
}
a := [][]int{{1, 1, 0}, {1, 0, 1}, {1, 0, 0}}
res := pow(a, n-4)
for _, x := range res[0] {
ans = (ans + x) % mod
}
return
}

func mul(a, b [][]int) [][]int {
m, n := len(a), len(b[0])
c := make([][]int, m)
for i := range c {
c[i] = make([]int, n)
}
for i := 0; i < m; i++ {
for j := 0; j < n; j++ {
for k := 0; k < len(b); k++ {
c[i][j] = (c[i][j] + a[i][k]*b[k][j]%mod) % mod
}
}
}
return c
}

func pow(a [][]int, n int) [][]int {
res := [][]int{{4, 2, 1}}
for n > 0 {
if n&1 == 1 {
res = mul(res, a)
}
a = mul(a, a)
n >>= 1
}
return res
}
```

Expand All @@ -75,74 +267,97 @@ class Solution {
* @return {number}
*/
var waysToStep = function (n) {
if (n < 3) return n;
let a = 1,
b = 2,
c = 4;
for (let i = 3; i < n; i++) {
[a, b, c] = [b, c, (a + b + c) % 1000000007];
let [a, b, c] = [1, 2, 4];
const mod = 1e9 + 7;
for (let i = 1; i < n; ++i) {
[a, b, c] = [b, c, (a + b + c) % mod];
}
return c;
return a;
};
```

### **C**
```js
/**
* @param {number} n
* @return {number}
*/

```c
int waysToStep(int n) {
if (n < 3) {
return n;
const mod = 1e9 + 7;

var waysToStep = function (n) {
if (n < 4) {
return Math.pow(2, n - 1);
}
int a = 1, b = 2, c = 4, i = 4;
while (i++ <= n) {
int t = ((a + b) % 1000000007 + c) % 1000000007;
a = b;
b = c;
c = t;
const a = [
[1, 1, 0],
[1, 0, 1],
[1, 0, 0],
];
let ans = 0;
const res = pow(a, n - 4);
for (const x of res[0]) {
ans = (ans + x) % mod;
}
return ans;
};

function mul(a, b) {
const [m, n] = [a.length, b[0].length];
const c = Array.from({ length: m }, () => Array.from({ length: n }, () => 0));
for (let i = 0; i < m; ++i) {
for (let j = 0; j < n; ++j) {
for (let k = 0; k < b.length; ++k) {
c[i][j] =
(c[i][j] + Number((BigInt(a[i][k]) * BigInt(b[k][j])) % BigInt(mod))) % mod;
}
}
}
return c;
}

function pow(a, n) {
let res = [[4, 2, 1]];
while (n) {
if (n & 1) {
res = mul(res, a);
}
a = mul(a, a);
n >>= 1;
}
return res;
}
```

### **C++**
### **C**

```cpp
class Solution {
public:
int waysToStep(int n) {
if (n < 3) {
return n;
}
int a = 1, b = 2, c = 4, i = 4;
while (i++ <= n) {
int t = ((a + b) % 1000000007 + c) % 1000000007;
a = b;
b = c;
c = t;
}
return c;
```c
int waysToStep(int n) {
const int mod = 1e9 + 7;
int a = 1, b = 2, c = 4;
for (int i = 1; i < n; ++i) {
int t = a;
a = b;
b = c;
c = (((a + b) % mod) + t) % mod;
}
};
return a;
}
```

### **Rust**

```rust
impl Solution {
pub fn ways_to_step(n: i32) -> i32 {
let mut dp = [1, 2, 4];
let n = n as usize;
if n <= 3 {
return dp[n - 1];
}
for _ in 3..n {
dp = [
dp[1],
dp[2],
(((dp[0] + dp[1]) % 1000000007) + dp[2]) % 1000000007,
];
let (mut a, mut b, mut c) = (1, 2, 4);
let m = 1000000007;
for _ in 1..n {
let t = a;
a = b;
b = c;
c = ((a + b) % m + t) % m;
}
dp[2]
a
}
}
```
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