Shortest Path to Get All Keys
<p>You are given an <code>m x n</code> grid <code>grid</code> where:</p>

<ul>
	<li><code>&#39;.&#39;</code> is an empty cell.</li>
	<li><code>&#39;#&#39;</code> is a wall.</li>
	<li><code>&#39;@&#39;</code> is the starting point.</li>
	<li>Lowercase letters represent keys.</li>
	<li>Uppercase letters represent locks.</li>
</ul>

<p>You start at the starting point and one move consists of walking one space in one of the four cardinal directions. You cannot walk outside the grid, or walk into a wall.</p>

<p>If you walk over a key, you can pick it up and you cannot walk over a lock unless you have its corresponding key.</p>

<p>For some <code><font face="monospace">1 &lt;= k &lt;= 6</font></code>, there is exactly one lowercase and one uppercase letter of the first <code>k</code> letters of the English alphabet in the grid. This means that there is exactly one key for each lock, and one lock for each key; and also that the letters used to represent the keys and locks were chosen in the same order as the English alphabet.</p>

<p>Return <em>the lowest number of moves to acquire all keys</em>. If it is impossible, return <code>-1</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/0800-0899/0864.Shortest%20Path%20to%20Get%20All%20Keys/images/lc-keys2.jpg" style="width: 404px; height: 245px;" />
<pre>
<strong>Input:</strong> grid = [&quot;@.a..&quot;,&quot;###.#&quot;,&quot;b.A.B&quot;]
<strong>Output:</strong> 8
<strong>Explanation:</strong> Note that the goal is to obtain all the keys not to open all the locks.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/0800-0899/0864.Shortest%20Path%20to%20Get%20All%20Keys/images/lc-key2.jpg" style="width: 404px; height: 245px;" />
<pre>
<strong>Input:</strong> grid = [&quot;@..aA&quot;,&quot;..B#.&quot;,&quot;....b&quot;]
<strong>Output:</strong> 6
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/0800-0899/0864.Shortest%20Path%20to%20Get%20All%20Keys/images/lc-keys3.jpg" style="width: 244px; height: 85px;" />
<pre>
<strong>Input:</strong> grid = [&quot;@Aa&quot;]
<strong>Output:</strong> -1
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 30</code></li>
	<li><code>grid[i][j]</code> is either an English letter, <code>&#39;.&#39;</code>, <code>&#39;#&#39;</code>, or <code>&#39;@&#39;</code>.&nbsp;</li>
	<li>There is exactly one&nbsp;<code>&#39;@&#39;</code>&nbsp;in the grid.</li>
	<li>The number of keys in the grid is in the range <code>[1, 6]</code>.</li>
	<li>Each key in the grid is <strong>unique</strong>.</li>
	<li>Each key in the grid has a matching lock.</li>
</ul>
Bit Manipulation; Breadth-First Search; Array; Matrix
func shortestPathAllKeys(grid []string) int {
	m, n := len(grid), len(grid[0])
	var k, si, sj int
	for i, row := range grid {
		for j, c := range row {
			if c >= 'a' && c <= 'z' {
				// 累加钥匙数量
				k++
			} else if c == '@' {
				// 起点
				si, sj = i, j
			}
		}
	}
	type tuple struct{ i, j, state int }
	q := []tuple{tuple{si, sj, 0}}
	vis := map[tuple]bool{tuple{si, sj, 0}: true}
	dirs := []int{-1, 0, 1, 0, -1}
	ans := 0
	for len(q) > 0 {
		for t := len(q); t > 0; t-- {
			p := q[0]
			q = q[1:]
			i, j, state := p.i, p.j, p.state
			// 找到所有钥匙，返回当前步数
			if state == 1<<k-1 {
				return ans
			}
			// 往四个方向搜索
			for h := 0; h < 4; h++ {
				x, y := i+dirs[h], j+dirs[h+1]
				// 在边界范围内
				if x >= 0 && x < m && y >= 0 && y < n {
					c := grid[x][y]
					// 是墙，或者是锁，但此时没有对应的钥匙，无法通过
					if c == '#' || (c >= 'A' && c <= 'Z' && (state>>(c-'A')&1 == 0)) {
						continue
					}
					nxt := state
					// 是钥匙，更新状态
					if c >= 'a' && c <= 'z' {
						nxt |= 1 << (c - 'a')
					}
					// 此状态未访问过，入队
					if !vis[tuple{x, y, nxt}] {
						vis[tuple{x, y, nxt}] = true
						q = append(q, tuple{x, y, nxt})
					}
				}
			}
		}
		// 步数加一
		ans++
	}
	return -1
}