Question

A number maze is an n×n grid of positive integers. A token starts in the upper left corner; your goal is to move the token to the lower-right corner. On each turn, you are allowed to move the token up, down, left, or right; the distance you may move the token is determined by the number on its current square. For example, if the token is on a square labeled 3, then you may move the token three steps up, three steps down, three steps left, or three steps right. However, you are never allowed to move the token off the edge of the board. In this problem, you will design and analyze an efficient algorithm that either returns the minimum number of moves required to solve a given number maze, or correctly reports that the maze has no solution. 1 A 5×5 maze that can be solved in eight moves Describe the solution to this problem at a high level, justify why it works, write down the pseudocode for your algorithm and analyze its running time. All four parts must be present to receive any credit! Hint: It is highly advisable that you read Section 5.7 in Jeff Erickson's online textbook before attempting to solve this problem.

Answer 1

An efficient algorithm for solving the number maze problem involves employing breadth-first search (BFS). By treating the maze as a graph where each cell is a node and valid moves are edges, BFS explores the grid level by level, guaranteeing the shortest path from the start to the end.

Step-by-step explanation:

Utilizing breadth-first search (BFS) proves effective due to its ability to systematically explore all possible moves from the current cell while ensuring the shortest path to the destination. Starting from the upper left corner and considering each cell as a node in a graph, BFS expands outward, allowing movement based on the number within each cell. This approach exhaustively explores all feasible paths, ultimately identifying the shortest one or determining if no solution exists.

To implement BFS for this problem, the pseudocode involves initializing a queue to track cell positions and distances, starting from the origin. The algorithm continually explores neighboring cells within the permitted distance until reaching the target cell or exhausting all possibilities. If the destination is reached, the algorithm returns the minimum number of moves required; otherwise, it reports the absence of a solution.

The algorithm's time complexity relies on the size of the maze, denoted as n×n. BFS examines each cell at most once, leading to a time complexity of O(
`n^2`) as it traverses through the entire grid. Since the algorithm explores only reachable cells, it efficiently identifies the shortest path or determines the lack of a viable solution within a reasonable time frame.