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Formatted question description: https://leetcode.ca/all/1263.html

# 1263. Minimum Moves to Move a Box to Their Target Location (Hard)

Storekeeper is a game in which the player pushes boxes around in a warehouse trying to get them to target locations.

The game is represented by a grid of size m x n, where each element is a wall, floor, or a box.

Your task is move the box 'B' to the target position 'T' under the following rules:

• Player is represented by character 'S' and can move up, down, left, right in the grid if it is a floor (empy cell).
• Floor is represented by character '.' that means free cell to walk.
• Wall is represented by character '#' that means obstacle  (impossible to walk there).
• There is only one box 'B' and one target cell 'T' in the grid.
• The box can be moved to an adjacent free cell by standing next to the box and then moving in the direction of the box. This is a push.
• The player cannot walk through the box.

Return the minimum number of pushes to move the box to the target. If there is no way to reach the target, return -1.

Example 1:

Input: grid = [["#","#","#","#","#","#"],
["#","T","#","#","#","#"],
["#",".",".","B",".","#"],
["#",".","#","#",".","#"],
["#",".",".",".","S","#"],
["#","#","#","#","#","#"]]
Output: 3
Explanation: We return only the number of times the box is pushed.

Example 2:

Input: grid = [["#","#","#","#","#","#"],
["#","T","#","#","#","#"],
["#",".",".","B",".","#"],
["#","#","#","#",".","#"],
["#",".",".",".","S","#"],
["#","#","#","#","#","#"]]
Output: -1


Example 3:

Input: grid = [["#","#","#","#","#","#"],
["#","T",".",".","#","#"],
["#",".","#","B",".","#"],
["#",".",".",".",".","#"],
["#",".",".",".","S","#"],
["#","#","#","#","#","#"]]
Output: 5
Explanation:  push the box down, left, left, up and up.


Example 4:

Input: grid = [["#","#","#","#","#","#","#"],
["#","S","#",".","B","T","#"],
["#","#","#","#","#","#","#"]]
Output: -1


Constraints:

• m == grid.length
• n == grid[i].length
• 1 <= m <= 20
• 1 <= n <= 20
• grid contains only characters '.', '#''S' , 'T', or 'B'.
• There is only one character 'S', 'B' and 'T' in the grid.

## Solution 1. BFS + DP

dp[sx][sy][bx][by] is the minimal number of pushes needed to make S at (sx, sy) and B at (bx, by).

// OJ: https://leetcode.com/problems/minimum-moves-to-move-a-box-to-their-target-location/
// Time: O((MN)^2)
// Space: O((MN)^2)
class Solution {
public:
int minPushBox(vector<vector<char>>& G) {
int dp[20][20][20][20] = {}, M = G.size(), N = G[0].size(), sx, sy, bx, by, tx, ty, dirs[4][2] = { {0,1},{0,-1},{1,0},{-1,0} }, ans = INT_MAX;
memset(dp, 0x3f, sizeof(dp));
for (int i = 0; i < M; ++i) {
for (int j = 0; j < N; ++j) {
if (G[i][j] == 'S') sx = i, sy = j;
else if (G[i][j] == 'B') bx = i, by = j;
else if (G[i][j] == 'T') tx = i, ty = j;
}
}
queue<vector<int>> q;
q.push({ sx, sy, bx, by, 0 });
dp[sx][sy][bx][by] = 0;
while (q.size()) {
auto v = q.front();
q.pop();
int sx = v[0], sy = v[1], bx = v[2], by = v[3], push = v[4];
if (bx == tx && by == ty) ans = min(ans, push);
for (auto [dx, dy] : dirs) {
int x = sx + dx, y = sy + dy;
if (x < 0 || y < 0 || x >= M || y >= N || G[x][y] == '#') continue;
int bx2 = bx, by2 = by, push2 = push;
if (x == bx && y == by) bx2 += dx, by2 += dy, ++push2;
if (bx2 < 0 || by2 < 0 || bx2 >= M || by2 >= N || G[bx2][by2] == '#' || push2 >= dp[x][y][bx2][by2]) continue;
dp[x][y][bx2][by2] = push2;
q.push({ x, y, bx2, by2, push2 });
}
}
return ans == INT_MAX ? -1 : ans;
}
};

• class Solution {
public int minPushBox(char[][] grid) {
final int BLOCK = -1;
final int WHITE = 0;
final int GRAY = 1;
final int BLACK = 2;
int rows = grid.length, columns = grid[0].length;
int[][][][] colors = new int[rows][columns][rows][columns];
int[][][][] distances = new int[rows][columns][rows][columns];
for (int bRow = 0; bRow < rows; bRow++) {
for (int bCol = 0; bCol < columns; bCol++) {
for (int pRow = 0; pRow < rows; pRow++) {
for (int pCol = 0; pCol < columns; pCol++)
distances[bRow][bCol][pRow][pCol] = Integer.MAX_VALUE;
}
}
}
for (int bRow = 0; bRow < rows; bRow++) {
for (int bCol = 0; bCol < columns; bCol++) {
for (int pRow = 0; pRow < rows; pRow++) {
for (int pCol = 0; pCol < columns; pCol++) {
if (grid[bRow][bCol] == '#' || grid[pRow][pCol] == '#') {
colors[bRow][bCol][pRow][pCol] = BLOCK;
distances[bRow][bCol][pRow][pCol] = -1;
}
}
}
}
}
int initialBoxRow = -1, initialBoxColumn = -1, initialPlayerRow = -1, initialPlayerColumn = -1, targetRow = -1, targetColumn = -1;
int count = 0;
outer:
for (int i = 0; i < rows; i++) {
for (int j = 0; j < columns; j++) {
char c = grid[i][j];
if (c == 'B') {
initialBoxRow = i;
initialBoxColumn = j;
count++;
} else if (c == 'S') {
initialPlayerRow = i;
initialPlayerColumn = j;
count++;
} else if (c == 'T') {
targetRow = i;
targetColumn = j;
count++;
}
if (count == 3)
break outer;
}
}
int[][] directions = { {-1, 0}, {0, 1}, {1, 0}, {0, -1} };
distances[initialBoxRow][initialBoxColumn][initialPlayerRow][initialPlayerColumn] = 0;
PriorityQueue<Status> queue = new PriorityQueue<Status>();
queue.offer(new Status(initialBoxRow, initialBoxColumn, initialPlayerRow, initialPlayerColumn, 0));
while (!queue.isEmpty()) {
Status status = queue.poll();
int boxRow = status.boxRow, boxColumn = status.boxColumn, playerRow = status.playerRow, playerColumn = status.playerColumn, distance = status.distance;
for (int[] direction : directions) {
int playerNewRow = playerRow + direction[0], playerNewColumn = playerColumn + direction[1];
if (playerNewRow < 0 || playerNewRow >= rows || playerNewColumn < 0 || playerNewColumn >= columns || grid[playerNewRow][playerNewColumn] == '#')
continue;
if (playerNewRow == boxRow && playerNewColumn == boxColumn) {
int boxNewRow = boxRow + direction[0], boxNewColumn = boxColumn + direction[1];
if (boxNewRow < 0 || boxNewRow >= rows || boxNewColumn < 0 || boxNewColumn >= columns || grid[boxNewRow][boxNewColumn] == '#')
continue;
if (boxNewRow == targetRow && boxNewColumn == targetColumn)
return distance + 1;
else if (colors[boxNewRow][boxNewColumn][playerNewRow][playerNewColumn] == WHITE) {
colors[boxNewRow][boxNewColumn][playerNewRow][playerNewColumn] = GRAY;
distances[boxNewRow][boxNewColumn][playerNewRow][playerNewColumn] = distance + 1;
queue.offer(new Status(boxNewRow, boxNewColumn, playerNewRow, playerNewColumn, distance + 1));
}
} else {
if (colors[boxRow][boxColumn][playerNewRow][playerNewColumn] == WHITE) {
colors[boxRow][boxColumn][playerNewRow][playerNewColumn] = GRAY;
distances[boxRow][boxColumn][playerNewRow][playerNewColumn] = distance;
queue.offer(new Status(boxRow, boxColumn, playerNewRow, playerNewColumn, distance));
}

}
}
colors[boxRow][boxColumn][playerRow][playerColumn] = BLACK;
}
int totalDistance = Integer.MAX_VALUE;
for (int i = 0; i < rows; i++) {
for (int j = 0; j < columns; j++) {
int distance = distances[targetRow][targetColumn][i][j];
totalDistance = Math.min(totalDistance, distance);
}
}
}
}

class Status implements Comparable<Status> {
int boxRow;
int boxColumn;
int playerRow;
int playerColumn;
int distance;

public Status() {

}

public Status(int boxRow, int boxColumn, int playerRow, int playerColumn, int distance) {
this.boxRow = boxRow;
this.boxColumn = boxColumn;
this.playerRow = playerRow;
this.playerColumn = playerColumn;
this.distance = distance;
}

public int compareTo(Status status2) {
return this.distance - status2.distance;
}
}

• // OJ: https://leetcode.com/problems/minimum-moves-to-move-a-box-to-their-target-location/
// Time: O((MN)^2)
// Space: O((MN)^2)
class Solution {
public:
int minPushBox(vector<vector<char>>& A) {
int M = A.size(), N = A[0].size(), tx, ty, sx, sy, bx, by, dirs[4][2] = { {0,1},{0,-1},{1,0},{-1,0} }, dp[20][20][20][20] = {}, ans = INT_MAX;
for (int i = 0; i < M; ++i) {
for (int j = 0; j < N; ++j) {
if (A[i][j] == 'S') sx = i, sy = j;
else if (A[i][j] == 'B') bx = i, by = j;
else if (A[i][j] == 'T') tx = i, ty = j;
}
}
memset(dp, 0x3f, sizeof(dp));
dp[sx][sy][bx][by] = 0;
queue<array<int, 4>> q{ { {sx, sy, bx, by} } };
while (q.size()) {
auto [sx, sy, bx, by] = q.front();
q.pop();
int step = dp[sx][sy][bx][by];
if (bx == tx && by == ty) ans = min(ans, step);
for (auto &[dx, dy] : dirs) {
int a = sx + dx, b = sy + dy, bx2 = bx, by2 = by, step2 = step;
if (a < 0 || a >= M || b < 0 || b >= N || A[a][b] == '#') continue;
if (a == bx && b == by) bx2 += dx, by2 += dy, step2++;
if (bx2 < 0 || bx2 >= M || by2 < 0 || by2 >= N || A[bx2][by2] == '#' || step2 >= dp[a][b][bx2][by2]) continue;
dp[a][b][bx2][by2] = step2;
q.push({a, b, bx2, by2});
}
}
return ans == INT_MAX ? -1 : ans;
}
};

• class Solution:
def minPushBox(self, grid: List[List[str]]) -> int:
def f(i: int, j: int) -> int:
return i * n + j

def check(i: int, j: int) -> bool:
return 0 <= i < m and 0 <= j < n and grid[i][j] != "#"

for i, row in enumerate(grid):
for j, c in enumerate(row):
if c == "S":
si, sj = i, j
elif c == "B":
bi, bj = i, j
m, n = len(grid), len(grid[0])
dirs = (-1, 0, 1, 0, -1)
q = deque([(f(si, sj), f(bi, bj), 0)])
vis = [[False] * (m * n) for _ in range(m * n)]
vis[f(si, sj)][f(bi, bj)] = True
while q:
s, b, d = q.popleft()
bi, bj = b // n, b % n
if grid[bi][bj] == "T":
return d
si, sj = s // n, s % n
for a, b in pairwise(dirs):
sx, sy = si + a, sj + b
if not check(sx, sy):
continue
if sx == bi and sy == bj:
bx, by = bi + a, bj + b
if not check(bx, by) or vis[f(sx, sy)][f(bx, by)]:
continue
vis[f(sx, sy)][f(bx, by)] = True
q.append((f(sx, sy), f(bx, by), d + 1))
elif not vis[f(sx, sy)][f(bi, bj)]:
vis[f(sx, sy)][f(bi, bj)] = True
q.appendleft((f(sx, sy), f(bi, bj), d))
return -1


• func minPushBox(grid [][]byte) int {
m, n := len(grid), len(grid[0])
var si, sj, bi, bj int
for i, row := range grid {
for j, c := range row {
if c == 'S' {
si, sj = i, j
} else if c == 'B' {
bi, bj = i, j
}
}
}
f := func(i, j int) int {
return i*n + j
}
check := func(i, j int) bool {
return i >= 0 && i < m && j >= 0 && j < n && grid[i][j] != '#'
}
q := [][]int{[]int{f(si, sj), f(bi, bj), 0} }
vis := make([][]bool, m*n)
for i := range vis {
vis[i] = make([]bool, m*n)
}
vis[f(si, sj)][f(bi, bj)] = true
dirs := [5]int{-1, 0, 1, 0, -1}
for len(q) > 0 {
p := q[0]
q = q[1:]
si, sj, bi, bj = p[0]/n, p[0]%n, p[1]/n, p[1]%n
d := p[2]
if grid[bi][bj] == 'T' {
return d
}
for k := 0; k < 4; k++ {
sx, sy := si+dirs[k], sj+dirs[k+1]
if !check(sx, sy) {
continue
}
if sx == bi && sy == bj {
bx, by := bi+dirs[k], bj+dirs[k+1]
if !check(bx, by) || vis[f(sx, sy)][f(bx, by)] {
continue
}
vis[f(sx, sy)][f(bx, by)] = true
q = append(q, []int{f(sx, sy), f(bx, by), d + 1})
} else if !vis[f(sx, sy)][f(bi, bj)] {
vis[f(sx, sy)][f(bi, bj)] = true
q = append([][]int{[]int{f(sx, sy), f(bi, bj), d} }, q...)
}
}
}
return -1
}

• function minPushBox(grid: string[][]): number {
const [m, n] = [grid.length, grid[0].length];
let [si, sj, bi, bj] = [0, 0, 0, 0];
for (let i = 0; i < m; ++i) {
for (let j = 0; j < n; ++j) {
if (grid[i][j] === 'S') {
[si, sj] = [i, j];
} else if (grid[i][j] === 'B') {
[bi, bj] = [i, j];
}
}
}
const f = (i: number, j: number): number => i * n + j;
const check = (i: number, j: number): boolean =>
i >= 0 && i < m && j >= 0 && j < n && grid[i][j] !== '#';

const q: Deque<[number, number, number]> = new Deque();
const vis: boolean[][] = new Array(m * n)
.fill(0)
.map(() => new Array(m * n).fill(false));
q.push([f(si, sj), f(bi, bj), 0]);
vis[f(si, sj)][f(bi, bj)] = true;
const dirs: number[] = [-1, 0, 1, 0, -1];
while (q.size() > 0) {
const [s, b, d] = q.shift()!;
const [si, sj] = [Math.floor(s / n), s % n];
const [bi, bj] = [Math.floor(b / n), b % n];
if (grid[bi][bj] === 'T') {
return d;
}
for (let k = 0; k < 4; ++k) {
const [sx, sy] = [si + dirs[k], sj + dirs[k + 1]];
if (!check(sx, sy)) {
continue;
}
if (sx === bi && sy === bj) {
const [bx, by] = [bi + dirs[k], bj + dirs[k + 1]];
if (!check(bx, by) || vis[f(sx, sy)][f(bx, by)]) {
continue;
}
vis[f(sx, sy)][f(bx, by)] = true;
q.push([f(sx, sy), f(bx, by), d + 1]);
} else if (!vis[f(sx, sy)][f(bi, bj)]) {
vis[f(sx, sy)][f(bi, bj)] = true;
q.unshift([f(sx, sy), f(bi, bj), d]);
}
}
}
return -1;
}

/* 以下是双向列队模板类 */
class CircularDeque<T> {
prev: CircularDeque<T> | null;
next: CircularDeque<T> | null;
begin: number;
end: number;
empty: boolean;
data: T[];
constructor(N: number) {
this.prev = this.next = null;
this.begin = this.end = 0;
this.empty = true;
this.data = Array(N);
}

isFull(): boolean {
return this.end === this.begin && !this.empty;
}

isEmpty(): boolean {
return this.empty;
}

push(val: T): boolean {
if (this.isFull()) return false;
this.empty = false;
this.data[this.end] = val;
this.end = (this.end + 1) % this.data.length;
return true;
}

front(): T | undefined {
return this.isEmpty() ? undefined : this.data[this.begin];
}

back(): T | undefined {
return this.isEmpty() ? undefined : this.data[this.end - 1];
}

pop(): T | undefined {
if (this.isEmpty()) return undefined;
const value = this.data[this.end - 1];
this.end = (this.end - 1) % this.data.length;
if (this.end < 0) this.end += this.data.length;
if (this.end === this.begin) this.empty = true;
return value;
}

unshift(val: T): boolean {
if (this.isFull()) return false;
this.empty = false;
this.begin = (this.begin - 1) % this.data.length;
if (this.begin < 0) this.begin += this.data.length;
this.data[this.begin] = val;
return true;
}

shift(): T | undefined {
if (this.isEmpty()) return undefined;
const value = this.data[this.begin];
this.begin = (this.begin + 1) % this.data.length;
if (this.end === this.begin) this.empty = true;
return value;
}

*values(): Generator<T, void, undefined> {
if (this.isEmpty()) return undefined;
let i = this.begin;
do {
yield this.data[i];
i = (i + 1) % this.data.length;
} while (i !== this.end);
}
}

class Deque<T> {
tail: CircularDeque<T>;
_size: number;
constructor(collection: T[] = []) {
this.tail = new CircularDeque<T>(128);
this._size = 0;
for (const item of collection) this.push(item);
}

size(): number {
return this._size;
}

push(val: T): void {
let last = this.tail.prev!;
if (last.isFull()) {
const inserted = new CircularDeque<T>(128);

this.tail.prev = inserted;
inserted.next = this.tail;

last.next = inserted;
inserted.prev = last;

last = inserted;
}
last.push(val);
this._size++;
}

back(): T | undefined {
if (this._size === 0) return;
return this.tail.prev!.back();
}

pop(): T | undefined {
if (this.head.next === this.tail) return undefined;
const last = this.tail.prev!;
const value = last.pop();
if (last.isEmpty()) {
this.tail.prev = last.prev;
last.prev!.next = this.tail;
}
this._size--;
return value;
}

unshift(val: T): void {
if (first.isFull()) {
const inserted = new CircularDeque<T>(128);

inserted.next = first;
first.prev = inserted;

first = inserted;
}
first.unshift(val);
this._size++;
}

shift(): T | undefined {
if (this.head.next === this.tail) return undefined;
const value = first.shift();
if (first.isEmpty()) {
}
this._size--;
return value;
}

front(): T | undefined {
if (this._size === 0) return undefined;
}

*values(): Generator<T, void, undefined> {