Problem Description

In this problem, you are asked to find the shortest path to acquire all keys in a given grid. The grid representation is as follows:

• A '.' character represents an empty cell where you can walk.
• A '#' character represents a wall that you cannot pass through.
• An '@' character represents your starting point.
• Lowercase letters ('a' to 'f') represent keys.
• Uppercase letters ('A' to 'F') correspond to locks.

You are able to move one space at a time in one of the four cardinal directions (up, down, left, or right). You cannot exit the boundaries of the grid, and you cannot pass through walls. If you encounter a key, you can pick it up. You can only pass through a lock if you've already picked up the corresponding key. The keys and locks use the first k letters of the English alphabet, and there is exactly one key for each lock, and one lock for each key.

The task is to return the minimum number of moves required to pick up all keys. If it is not possible to obtain all the keys, the function should return -1.

Flowchart Walkthrough

To decide on an appropriate algorithm for solving LeetCode problem 864, Shortest Path to Get All Keys, we'll use the algorithm flowchart for guidance. You can follow along with the actual decision process visualized by visiting the Flowchart. Here's a systematic walkthrough:

Is it a graph?

• Yes: Although presented as a grid, the problem fundamentally involves movement through spaces (which are like nodes) with barriers or keys (acting like conditioned edges).

Is it a tree?

• No: The structure has multiple possible routes and cycles that depend on collecting various keys, hence it is not strictly hierarchical like a tree.

Is the problem related to directed acyclic graphs (DAGs)?

• No: We're dealing with a free-form grid where backtracking to previously visited cells may be necessary when new keys are obtained, thus the graph can have cycles.

Is the problem related to shortest paths?

• Yes: The goal is explicitly to find the shortest path that allows collection of all the keys.

Is the graph weighted?

• No: Each move from one cell to an adjacent one can be considered of uniform cost or weight.

Since we identified that the problem involves finding shortest paths in an unweighted graph, the flowchart's guidance would suggest using BFS for this problem. Breadth-First Search (BFS) is ideal for finding shortest paths in unweighted graphs, as it explores all nodes at the present depth level before moving on to nodes at the next depth level. Moreover, BFS can be adapted to handle states, such as which keys have been collected, which is crucial for this problem, where accessibility of parts of the grid depends on the keys possessed.

Conclusion: The flowchart indicates that BFS is the suitable algorithm for LeetCode problem 864 since it focuses on finding the shortest path in an unweighted graph where the state of traversal can change based on keys collected.

Intuition

The intuition behind the solution is to perform a breadth-first search (BFS) to explore the grid systematically. In a BFS, you visit each accessible cell in the grid by starting from your initial position and moving in all four cardinal directions. As you move, you'll track certain key aspects:

• The current position (i, j) on the grid.
• The set of keys you have collected so far (represented as a bit mask state).

The bit mask for the state allows tracking which keys have been picked up without needing a separate list or structure. By using bitwise operations, we convert the collection of keys to an integer, where the bits of the integer represent the presence or absence of the keys. For example, if there are two keys 'a' and 'b', and you've collected just 'a', the state might look like 01 (in binary).

As the BFS progresses, we aim to reach a state where state == (1 << k) - 1, which means all k keys have been collected, and the corresponding bits for all keys are set to 1. This check is efficient because (1 << k) - 1 produces a number with the first k bits set to 1.

A key aspect of this solution is the avoidance of duplicate states; we keep a set vis of visited (position, state) pairs to ensure we do not reprocess the same scenarios. Each time a key is picked up, the state changes, and the (position, state) pair is considered unique, allowing further exploration from this new scenario.

The BFS continues until either all keys are collected, or no further movement is possible. The variable ans is used to count the number of moves made, and if the state where all keys have been collected is reached, the corresponding value of ans is returned, indicating the shortest path. If the BFS concludes without collecting all keys, the function returns -1 to indicate that the task is impossible.

Learn more about Breadth-First Search patterns.

Solution Approach

The solution to this problem implements a BFS to explore the given grid efficiently. Here is a step-by-step explanation of the approach:

1. Identify Starting Point and Key Count:

   • Find the cell with the '@' character which is the starting position.
   • Calculate the number of keys k in the grid by counting the lowercase letters.

2. Initialize BFS Structures:

   • Use a queue q and initialize it with the tuple (si, sj, 0), where si and sj are starting cell coordinates, and 0 represents the initial state with no keys collected.
   • Create a set vis to keep track of the visited cells with particular states to avoid re-processing them. (si, sj, 0) is added to it as an initially visited state.

3. BFS Algorithm:

   • While the queue is not empty, process each element by popping from the left (dequeue).
   • If the current state reflects all keys have been collected (state == (1 << k) - 1), return the current number of moves ans.
   • Explore all possible movements (up, down, left, right) by adding or subtracting from the current cell coordinates.
   • For each move, check if the potential move is valid (i.e., within bounds, not walking into walls, having the key if it's a lock).
   • When encountering a key, update the state by setting the corresponding bit using a bitwise or operation nxt |= 1 << (ord(c) - ord('a')).
   • After processing all valid movements, increment the ans variable to reflect an increase in the number of moves.

4. State Management:

   • The state variable is a bitmask representing the keys collected. For example, if there are 6 possible keys, the state could be a number between 0 (no keys) and 63 (all keys, where binary 111111 is 63 in decimal).
   • If a key is found, it modifies the state by setting the corresponding bit to 1.

5. Avoid Duplicate States:

   • Before enqueueing the new state, check if (x, y, nxt) has not been visited to prevent duplicate processing.
   • If not, add (x, y, nxt) to the vis set and enqueue the new coordinates and state to the queue q.

6. Check for Solution:

   • The BFS will end in two cases:
     • All keys are collected, and the function returns the minimum number of steps ans.
     • The queue q is empty without all keys collected, indicating there is no solution, and the function returns -1.

The usage of BFS ensures that the shortest path to collect all keys is found because it explores all possible paths simultaneously, branching out in a breadth-first manner. By using a bitmask to represent the collected keys, the solution is both space-efficient and fast, as the check to see if we have all the keys becomes a simple bitwise comparison. The use of a set to track visited states prevents the algorithm from falling into infinite loops or reprocessing unnecessary states, thus ensuring the BFS terminates with the correct answer or -1 if a solution is not possible.

Example Walkthrough

Let's consider a simple example grid to understand how the solution works:

# Grid:
@.a
.#.
Ab.

Starting from the @ symbol, the task is to collect the keys represented by lowercase 'a' and 'b', passing through locks 'A' and 'B' if we have the corresponding keys, and determining the minimum number of moves to complete this task.

1. Identify Starting Point and Key Count:

   • The cell with the '@' character is at coordinates (0, 0).
   • There are two keys 'a' and 'b' in the grid.

2. Initialize BFS Structures:

   • Initialize the queue q with ((0, 0), 0) indicating that we start at the top-left corner with no keys collected.
   • The set vis starts with ((0, 0), 0) to represent that we've visited the starting position with no keys.

3. BFS Algorithm:

   • The initial state dequeued from q is ((0, 0), 0). We haven't collected any keys yet (state = 0).
   • From the start, we can go right to the cell (0, 1) with no change in state since it's an empty cell.
   • Next, we can go down to cell (1, 1), but it's a wall, so we ignore this move.

4. State Management:

   • Move to cell (0, 2), pick up key 'a'. Our state changes to (0, 2), with the state bitmask now being 01.

5. Avoid Duplicate States:

   • Add this new state to the vis set and push ((0, 2), 01) onto q.

6. BFS Continues:

   • Continue with BFS, dequeueing ((0, 2), 01) and we see the key 'a' is collected.
   • From this cell, we can move down to (1, 2) next to lock 'A'. Since we have key 'a', we can pass through.
   • From (1, 2), we move left to cell (1, 1) and then down to (2, 1) where key 'b' is located.
   • Now we collect key 'b', and our state bitmask becomes 11.

7. Check for Solution:

   • The condition (state == (1 << k) - 1) is met since we have two keys and our state bitmask is 11 (in binary), which equals 3 (in decimal) and is equal to (1 << 2) - 1.
   • Calculate the total number of moves: Move right to (0, 1), move right to (0, 2), collect key 'a', move down to (1, 2), move left to (1, 1), move down to (2, 1), and collect key 'b'. This is a total of 6 moves.

The result for this example grid is that the shortest path to acquire all keys is 6 moves. If this process had been continued until the queue was empty without collecting all keys, the result would be -1, indicating that acquiring all keys is not possible.

Solution Implementation

Time and Space Complexity

Time Complexity:

The time complexity of the algorithm is primarily determined by the breadth-first search (BFS) it performs to find the shortest path to collect all keys.

• The algorithm iterates through each cell in the grid, which gives us an initial O(m*n) complexity.
• In the worst case, every position on the grid could be visited with every possible key collection state, which is 2^k where k is the number of keys.
• The for _ in range(len(q)) loop iterates through each element in the queue, which in the worst case can contain all the cells times the number of states, resulting in m*n*2^k.
• Each cell expands to a maximum of 4 neighboring cells.

Combining these factors, the time complexity is O(m*n*4*2^k) simplified to O(m*n*2^k) as constants are generally omitted in big-O notation.

Space Complexity:

The space complexity involves the space used by the queue and the visited states:

• The queue could potentially store all cells with each possible key state, which would require O(m*n*2^k) space.
• The visited set also stores tuples of (row, column, key_state), which will contain at most m*n*2^k entries.

Hence, the space complexity is also O(m*n*2^k).

Learn more about how to find time and space complexity quickly using problem constraints.