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feat: add solutions to lc problem: No.3283 #3496

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259 changes: 253 additions & 6 deletions solution/3200-3299/3283.Maximum Number of Moves to Kill All Pawns/README.md
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Expand Up @@ -99,32 +99,279 @@ edit_url: https://github.com/doocs/leetcode/edit/main/solution/3200-3299/3283.Ma

<!-- solution:start -->

### 方法一
### 方法一：BFS + 状态压缩 + 记忆化搜索

<!-- tabs:start -->
我们首先预处理出每个兵到棋盘上任意一个位置的马的最短距离，记录在数组 $\textit{dist}$ 中，即 $\textit{dist}[i][x][y]$ 表示第 $i$ 个兵到 $(x, y)$ 位置的马的最短距离。

接下来，我们设计一个函数 $\text{dfs}(\textit{last}, \textit{state}, \textit{k})$，其中 $\textit{last}$ 表示上一个吃掉的兵的编号，而 $\textit{state}$ 表示当前还剩下的兵的状态，而 $\textit{k}$ 表示当前是 Alice 还是 Bob 的回合。函数的返回值表示当前回合的最大移动次数。那么答案为 $\text{dfs}(n, 2^n-1, 1)$。这里我们初始时上一个吃掉的兵的编号记为 $n$，这也是马所在的位置。

函数 $\text{dfs}$ 的具体实现如下：

- 如果 $\textit{state} = 0$，表示没有兵了，返回 $0$；
- 如果 $\textit{k} = 1$，表示当前是 Alice 的回合，我们需要找到一个兵，使得吃掉这个兵后的移动次数最大，即 $\text{dfs}(i, \textit{state} \oplus 2^i, \textit{k} \oplus 1) + \textit{dist}[\textit{last}][x][y]$；
- 如果 $\textit{k} = 0$，表示当前是 Bob 的回合，我们需要找到一个兵，使得吃掉这个兵后的移动次数最小，即 $\text{dfs}(i, \textit{state} \oplus 2^i, \textit{k} \oplus 1) + \textit{dist}[\textit{last}][x][y]$。

为了避免重复计算，我们使用记忆化搜索，即使用哈希表记录已经计算过的状态。

时间复杂度 $O(n \times m^2 + 2^n \times n^2)$，空间复杂度 $O(n \times m^2 + 2^n \times n)$。其中 $n$ 和 $m$ 分别为兵的数量和棋盘的大小。

#### Python3

```python

class Solution:
def maxMoves(self, kx: int, ky: int, positions: List[List[int]]) -> int:
@cache
def dfs(last: int, state: int, k: int) -> int:
if state == 0:
return 0
if k:
res = 0
for i, (x, y) in enumerate(positions):
if state >> i & 1:
t = dfs(i, state ^ (1 << i), k ^ 1) + dist[last][x][y]
if res < t:
res = t
return res
else:
res = inf
for i, (x, y) in enumerate(positions):
if state >> i & 1:
t = dfs(i, state ^ (1 << i), k ^ 1) + dist[last][x][y]
if res > t:
res = t
return res

n = len(positions)
m = 50
dist = [[[-1] * m for _ in range(m)] for _ in range(n + 1)]
dx = [1, 1, 2, 2, -1, -1, -2, -2]
dy = [2, -2, 1, -1, 2, -2, 1, -1]
positions.append([kx, ky])
for i, (x, y) in enumerate(positions):
dist[i][x][y] = 0
q = deque([(x, y)])
step = 0
while q:
step += 1
for _ in range(len(q)):
x1, y1 = q.popleft()
for j in range(8):
x2, y2 = x1 + dx[j], y1 + dy[j]
if 0 <= x2 < m and 0 <= y2 < m and dist[i][x2][y2] == -1:
dist[i][x2][y2] = step
q.append((x2, y2))

ans = dfs(n, (1 << n) - 1, 1)
dfs.cache_clear()
return ans
```

#### Java

```java

class Solution {
private Integer[][][] f;
private Integer[][][] dist;
private int[][] positions;
private final int[] dx = {1, 1, 2, 2, -1, -1, -2, -2};
private final int[] dy = {2, -2, 1, -1, 2, -2, 1, -1};

public int maxMoves(int kx, int ky, int[][] positions) {
int n = positions.length;
final int m = 50;
dist = new Integer[n + 1][m][m];
this.positions = positions;
for (int i = 0; i <= n; ++i) {
int x = i < n ? positions[i][0] : kx;
int y = i < n ? positions[i][1] : ky;
Deque<int[]> q = new ArrayDeque<>();
q.offer(new int[] {x, y});
for (int step = 1; !q.isEmpty(); ++step) {
for (int k = q.size(); k > 0; --k) {
var p = q.poll();
int x1 = p[0], y1 = p[1];
for (int j = 0; j < 8; ++j) {
int x2 = x1 + dx[j], y2 = y1 + dy[j];
if (x2 >= 0 && x2 < m && y2 >= 0 && y2 < m && dist[i][x2][y2] == null) {
dist[i][x2][y2] = step;
q.offer(new int[] {x2, y2});
}
}
}
}
}
f = new Integer[n + 1][1 << n][2];
return dfs(n, (1 << n) - 1, 1);
}

private int dfs(int last, int state, int k) {
if (state == 0) {
return 0;
}
if (f[last][state][k] != null) {
return f[last][state][k];
}
int res = k == 1 ? 0 : Integer.MAX_VALUE;
for (int i = 0; i < positions.length; ++i) {
int x = positions[i][0], y = positions[i][1];
if ((state >> i & 1) == 1) {
int t = dfs(i, state ^ (1 << i), k ^ 1) + dist[last][x][y];
res = k == 1 ? Math.max(res, t) : Math.min(res, t);
}
}
return f[last][state][k] = res;
}
}
```

#### C++

```cpp

class Solution {
public:
int maxMoves(int kx, int ky, vector<vector<int>>& positions) {
int n = positions.size();
const int m = 50;
const int dx[8] = {1, 1, 2, 2, -1, -1, -2, -2};
const int dy[8] = {2, -2, 1, -1, 2, -2, 1, -1};
int dist[n + 1][m][m];
memset(dist, -1, sizeof(dist));
for (int i = 0; i <= n; ++i) {
int x = (i < n) ? positions[i][0] : kx;
int y = (i < n) ? positions[i][1] : ky;
queue<pair<int, int>> q;
q.push({x, y});
dist[i][x][y] = 0;
for (int step = 1; !q.empty(); ++step) {
for (int k = q.size(); k > 0; --k) {
auto [x1, y1] = q.front();
q.pop();
for (int j = 0; j < 8; ++j) {
int x2 = x1 + dx[j], y2 = y1 + dy[j];
if (x2 >= 0 && x2 < m && y2 >= 0 && y2 < m && dist[i][x2][y2] == -1) {
dist[i][x2][y2] = step;
q.push({x2, y2});
}
}
}
}
}

int f[n + 1][1 << n][2];
memset(f, -1, sizeof(f));
auto dfs = [&](auto&& dfs, int last, int state, int k) -> int {
if (state == 0) {
return 0;
}
if (f[last][state][k] != -1) {
return f[last][state][k];
}
int res = (k == 1) ? 0 : INT_MAX;

for (int i = 0; i < positions.size(); ++i) {
int x = positions[i][0], y = positions[i][1];
if ((state >> i) & 1) {
int t = dfs(dfs, i, state ^ (1 << i), k ^ 1) + dist[last][x][y];
if (k == 1) {
res = max(res, t);
} else {
res = min(res, t);
}
}
}
return f[last][state][k] = res;
};
return dfs(dfs, n, (1 << n) - 1, 1);
}
};
```

#### Go

```go

func maxMoves(kx int, ky int, positions [][]int) int {
n := len(positions)
const m = 50
dx := []int{1, 1, 2, 2, -1, -1, -2, -2}
dy := []int{2, -2, 1, -1, 2, -2, 1, -1}
dist := make([][][]int, n+1)
for i := range dist {
dist[i] = make([][]int, m)
for j := range dist[i] {
dist[i][j] = make([]int, m)
for k := range dist[i][j] {
dist[i][j][k] = -1
}
}
}

for i := 0; i <= n; i++ {
x := kx
y := ky
if i < n {
x = positions[i][0]
y = positions[i][1]
}
q := [][2]int{[2]int{x, y}}
dist[i][x][y] = 0

for step := 1; len(q) > 0; step++ {
for k := len(q); k > 0; k-- {
p := q[0]
q = q[1:]
x1, y1 := p[0], p[1]
for j := 0; j < 8; j++ {
x2 := x1 + dx[j]
y2 := y1 + dy[j]
if x2 >= 0 && x2 < m && y2 >= 0 && y2 < m && dist[i][x2][y2] == -1 {
dist[i][x2][y2] = step
q = append(q, [2]int{x2, y2})
}
}
}
}
}
f := make([][][]int, n+1)
for i := range f {
f[i] = make([][]int, 1<<n)
for j := range f[i] {
f[i][j] = make([]int, 2)
for k := range f[i][j] {
f[i][j][k] = -1
}
}
}
var dfs func(last, state, k int) int
dfs = func(last, state, k int) int {
if state == 0 {
return 0
}
if f[last][state][k] != -1 {
return f[last][state][k]
}

var res int
if k == 0 {
res = math.MaxInt32
}

for i, p := range positions {
x, y := p[0], p[1]
if (state>>i)&1 == 1 {
t := dfs(i, state^(1<<i), k^1) + dist[last][x][y]
if k == 1 {
res = max(res, t)
} else {
res = min(res, t)
}
}
}
f[last][state][k] = res
return res
}

return dfs(n, (1<<n)-1, 1)
}
```

<!-- tabs:end -->
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