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

No.3283.Maximum Number of Moves to Kill All Pawns

Author: yanglbme

solution/3200-3299/3283.Maximum Number of Moves to Kill All Pawns/README.md

@@ -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 -->