<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://assets.leetcode.com/uploads/2021/07/23/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://assets.leetcode.com/uploads/2021/07/23/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://assets.leetcode.com/uploads/2021/07/23/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>

#include <bits/stdc++.h>
using namespace std;

/* Macros {{{ */
/*  A lot of this is from some of Benq's submissions
    [https://codeforces.com/profile/Benq]
    Ugly af to the eyes, but with vim fold its barable
    Hopefully c++20 concepts can make all this stuff must cleaner */

/* Basics {{{ */
using ll = long long;
using ld = long double;
using str = string;

using pi = pair<int, int>;
using pll = pair<ll, ll>;
using pld = pair<ld, ld>;
#define mp make_pair
#define fi first
#define se second

#define arr array
#define ve vector
using vi = vector<int>;
using vll = vector<ll>;
using vld = vector<ld>;

using vpi = vector<pi>;
using vpll = vector<pll>;
using vpld = vector<pld>;

using vvi = vector<vi>;
using vvll = vector<vll>;
using vvld = vector<vld>;

using vvpi = vector<vpi>;
using vvpll = vector<vpll>;
using vvpld = vector<vpld>;

#define pb push_back
#define lb lower_bound
#define ub upper_bound
#define sz size()
#define rsz(a) resize(a)
#define all(x) x.begin(), x.end()
#define rall(x) x.rbegin(), x.rend()

#define For(i, a, b) for (int i = a; i < b; ++i)
#define Rof(i, a, b) for (int i = (b)-1; i >= (a); --i)
#define rep(a) For(_, 0, a)
#define each(a, x) for (auto &a : x)
#define reach(a, x) for (auto a = x.rbegin(); a != x.rend(); ++a)

template <typename T, typename U>
inline void cmin(T &x, U y) {
    if (y < x) x = y;
}
template <typename T, typename U>
inline void cmax(T &x, U y) {
    if (x < y) x = y;
}
/*}}}*/

/* IO {{{ */

/* Template Macros {{{ */
#define tcT template <class T
#define tcTU tcT, class U
#define tcTUU tcT, class... U
/*}}}*/

inline namespace Helpers { /*{{{*/
tcT, class = void > struct is_iterable : false_type {};
tcT > struct is_iterable<
          T, void_t<decltype(begin(declval<T>())), decltype(end(declval<T>()))>>
    : true_type {};
tcT > constexpr bool is_iterable_v = is_iterable<T>::value;

tcT, class = void > struct is_readable : false_type {};
tcT > struct is_readable<T, typename std::enable_if_t<is_same_v<
                                decltype(cin >> declval<T &>()), istream &>>>
    : true_type {};
tcT > constexpr bool is_readable_v = is_readable<T>::value;

tcT, class = void > struct is_printable : false_type {};
tcT > struct is_printable<T, typename std::enable_if_t<is_same_v<
                                 decltype(cout << declval<T>()), ostream &>>>
    : true_type {};
tcT > constexpr bool is_printable_v = is_printable<T>::value;
} /* namespace Helpers */
/*}}}*/

inline namespace Input { /*{{{*/
tcT > constexpr bool needs_input_v = !is_readable_v<T> && is_iterable_v<T>;
tcTUU > void re(T &t, U &...u);
tcTU > void re(pair<T, U> &p); /* pairs */

/* re: read{{{ */
tcT > typename enable_if<is_readable_v<T>, void>::type re(T &x) {
    cin >> x;
} /* default */
tcT > typename enable_if<needs_input_v<T>, void>::type re(
          T &i);                                   // vectors, arrays, etc...
tcTU > void re(pair<T, U> &p) { re(p.fi, p.se); }  // pairs
tcT > typename enable_if<needs_input_v<T>, void>::type re(T &i) {
    each(x, i) re(x);
}
tcTUU > void re(T &t, U &...u) {
    re(t);
    re(u...);
} /* read multiple}}} */

/* rv: resize and read vectors{{{ */
void rv(size_t) {}
tcTUU > void rv(size_t N, ve<T> &t, U &...u);
template <class... U>
void rv(size_t, size_t N2, U &...u);
tcTUU > void rv(size_t N, ve<T> &t, U &...u) {
    t.rsz(N);
    re(t);
    rv(N, u...);
}
template <class... U>
void rv(size_t, size_t N2, U &...u) {
    rv(N2, u...);
} /*}}}*/

/* dumb shortcuts to read in ints{{{ */
void decrement() {} /* subtract one from each */
tcTUU > void decrement(T &t, U &...u) {
    --t;
    decrement(u...);
}
#define ints(...)    \
    int __VA_ARGS__; \
    re(__VA_ARGS__);
#define int1(...)      \
    ints(__VA_ARGS__); \
    decrement(__VA_ARGS__); /*}}}*/
} /* namespace Input */
/*}}}*/

inline namespace ToString { /*{{{*/
tcT > constexpr bool needs_output_v = !is_printable_v<T> && is_iterable_v<T>;

/* ts: string representation to print */
tcT > typename enable_if<is_printable_v<T>, str>::type ts(T v) {
    stringstream ss;
    ss << fixed << setprecision(15) << v;
    return ss.str();
} /* default */
tcT > str bit_vec(T t) { /* bit vector to string */
    str res = "{";
    For(i, 0, t.sz) res += ts(t[i]);
    res += "}";
    return res;
}
str ts(ve<bool> v) { return bit_vec(v); }
template <size_t SZ>
str ts(bitset<SZ> b) {
    return bit_vec(b);
} /* bit vector */
tcTU > str ts(pair<T, U> p); /* pairs */
tcT > typename enable_if<needs_output_v<T>, str>::type ts(
          T v); /* vectors, arrays */
tcTU > str ts(pair<T, U> p) { return "(" + ts(p.fi) + ", " + ts(p.se) + ")"; }
tcT > typename enable_if<is_iterable_v<T>, str>::type ts_sep(T v, str sep) {
    /* convert container to string w/ separator sep */
    bool fst = 1;
    str res = "";
    for (const auto &x : v) {
        if (!fst) res += sep;
        fst = 0;
        res += ts(x);
    }
    return res;
}
tcT > typename enable_if<needs_output_v<T>, str>::type ts(T v) {
    return "{" + ts_sep(v, ", ") + "}";
}

/* for nested DS */
template <int, class T>
typename enable_if<!needs_output_v<T>, ve<str>>::type ts_lev(const T &v) {
    return {ts(v)};
}
template <int lev, class T>
typename enable_if<needs_output_v<T>, ve<str>>::type ts_lev(const T &v) {
    if (lev == 0 || !v.sz) return {ts(v)};
    ve<str> res;
    for (const auto &t : v) {
        if (res.sz) res.back() += ",";
        ve<str> tmp = ts_lev<lev - 1>(t);
        res.insert(end(res), all(tmp));
    }
    For(i, 0, res.sz) {
        str bef = " ";
        if (i == 0) bef = "{";
        res[i] = bef + res[i];
    }
    res.back() += "}";
    return res;
}
} /* namespace ToString */
/*}}}*/

inline namespace Output { /*{{{*/
template <class T>
void pr_sep(ostream &os, str, const T &t) {
    os << ts(t);
}
template <class T, class... U>
void pr_sep(ostream &os, str sep, const T &t, const U &...u) {
    pr_sep(os, sep, t);
    os << sep;
    pr_sep(os, sep, u...);
}
/* print w/ no spaces */
template <class... T>
void pr(const T &...t) {
    pr_sep(cout, "", t...);
}
/* print w/ spaces, end with newline */
void ps() { cout << "\n"; }
template <class... T>
void ps(const T &...t) {
    pr_sep(cout, " ", t...);
    ps();
}
/* debug to cerr */
template <class... T>
void dbg_out(const T &...t) {
    pr_sep(cerr, " | ", t...);
    cerr << endl;
}
void loc_info(int line, str names) {
    cerr << "Line(" << line << ") -> [" << names << "]: ";
}
template <int lev, class T>
void dbgl_out(const T &t) {
    cerr << "\n\n" << ts_sep(ts_lev<lev>(t), "\n") << "\n" << endl;
}
} /* namespace Output */
/*}}}}}}}}}*/

class Solution {
public:
    int shortestPathAllKeys(vector<string>& grid) {
        int n = grid.size(), m = grid[0].size();

        pi ipos;
        int k=0;
        vpi kpos('z');
        for(int r=0; r<n; ++r) for(int c=0; c<m; ++c) {
            char v = grid[r][c];
            if('a' <= v && v <= 'z') kpos[v - 'a'] = {r, c}, ++k;
            if(v == '@') ipos = {r, c};
        }
        kpos.rsz(k);

        queue<arr<int, 4>> q; // { r, c, msk, d }
        q.push({ ipos.fi, ipos.se, 0, 0 });
        ve<vvi> seen(n, vvi(m, vi(1<<k, 0)));

        while(q.size()) {
            auto [r, c, msk, d] = q.front(); q.pop();
            if(r < 0 || r >= n || c < 0 || c >= m) continue;
            if(grid[r][c] == '#') continue; // not wall
            if('A' <= grid[r][c] && grid[r][c] <= 'Z' && !( msk & (1<<(grid[r][c] - 'A')))) continue;

            if('a' <= grid[r][c] && grid[r][c] <= 'z') msk |= (1<<(grid[r][c] - 'a'));
            if(seen[r][c][msk]) continue;
            seen[r][c][msk] = 1;

            if(__builtin_popcount(msk) == k) return d;

            for(auto [dr, dc] : vpi({ {-1, 0}, {1, 0}, {0, -1}, {0, 1} }))
                q.push({ r+dr, c+dc, msk, d+1 });
        }

        return -1;
    }
};

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