Word Search

Background Knowledge

The "Word Search" problem falls under the category of Graphs & Search, which involves traversing through a data structure to find a specific pattern or solution. In this case, the data structure is a 2D grid of characters, and the goal is to search for a given word within this grid. The word can be constructed from sequentially adjacent cells, which means that each character in the word must be found in a cell that is directly horizontal or vertical to the previous character. This constraint implies that the search algorithm needs to explore the grid in a way that respects these adjacency relationships.

The concept of adjacency is crucial in this problem, as it defines how cells in the grid are connected to each other. In a 2D grid, each cell has four adjacent cells: one above, one below, one to the left, and one to the right. The search algorithm needs to consider these adjacency relationships when exploring the grid to find the word. Additionally, the problem statement mentions that each cell may only be used once, which means that the algorithm needs to keep track of visited cells to avoid revisiting them and entering an infinite loop.

The "Word Search" problem can be solved using various search algorithms, including Depth-First Search (DFS) and Breadth-First Search (BFS). However, due to the constraint that each cell may only be used once, DFS is a more natural fit for this problem. DFS involves exploring the grid by traversing as far as possible along each branch before backtracking, which allows the algorithm to efficiently search for the word while avoiding revisits to the same cell.

Algorithm/Approach

The general approach to solving the "Word Search" problem involves using a Depth-First Search (DFS) algorithm to traverse the 2D grid and search for the given word. The DFS algorithm will explore the grid by starting at each cell and checking if the current character matches the first character of the word. If it does, the algorithm will recursively explore the adjacent cells to find the remaining characters of the word.

Step-by-Step Strategy

To implement the solution, follow these steps:

• Initialize a 2D grid and a given word as input
• Define a DFS function that takes the current cell coordinates and the current index in the word as parameters
• Within the DFS function, check if the current character in the grid matches the current character in the word
• If it does, recursively call the DFS function on the adjacent cells (up, down, left, right) and increment the index in the word
• Keep track of visited cells to avoid revisiting them
• If the entire word is found, return True; otherwise, return False

Common Pitfalls

When implementing the solution, watch out for the following common pitfalls:

• Failing to keep track of visited cells, leading to infinite loops
• Not checking the boundaries of the grid when exploring adjacent cells
• Not handling the case where the word is not found in the grid

Time & Space Complexity

The expected time complexity of the solution is $O(N \cdot M \cdot 4^L)$, where $N$ and $M$ are the dimensions of the grid, and $L$ is the length of the word. The space complexity is $O(N \cdot M)$, which is used to store the visited cells. Note that the time complexity is exponential in the length of the word due to the recursive nature of the DFS algorithm.