Rat in a Maze Problem in DSA Rust - Time & Space Complexity

Time Complexity: Rat in a Maze Problem

O(2^{2n})

Understanding Time Complexity

We want to understand how the time needed to solve the Rat in a Maze problem changes as the maze size grows.

Specifically, how does the search for a path through the maze take longer when the maze gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

fn solve_maze(maze: &Vec>, x: usize, y: usize, solution: &mut Vec>) -> bool {
    let n = maze.len();
    if x == n - 1 && y == n - 1 {
        solution[x][y] = 1;
        return true;
    }
    if x < n && y < n && maze[x][y] == 1 {
        solution[x][y] = 1;
        if solve_maze(maze, x + 1, y, solution) {
            return true;
        }
        if solve_maze(maze, x, y + 1, solution) {
            return true;
        }
        solution[x][y] = 0;
        return false;
    }
    false
}

This code tries to find a path from the top-left to the bottom-right of a maze using backtracking.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive calls exploring possible paths in the maze.
• How many times: Potentially, each cell is visited multiple times as the function tries different paths.

How Execution Grows With Input

As the maze size (n x n) grows, the number of possible paths grows very fast because the rat can move right or down at each step.

Input Size (n)	Approx. Operations
2	About 2 paths to check
4	About 20 paths to check
10	Thousands of paths to check

Pattern observation: The number of paths grows exponentially as the maze size increases.

Final Time Complexity

Time Complexity: O(2^{2n})

This means the time needed roughly quadruples with each increase in maze size n, making it very slow for large mazes.

Common Mistake

[X] Wrong: "The rat only moves right or down, so the time grows linearly with maze size."

[OK] Correct: The rat tries many possible paths, including backtracking, so the number of paths grows exponentially, not linearly.

Interview Connect

Understanding this problem helps you think about how recursive searches explore many possibilities and why some problems become very slow as input grows.

Self-Check

"What if we allowed the rat to move in all four directions instead of just right and down? How would the time complexity change?"

Practice

1. What is the main goal of the Rat in a Maze problem in data structures?

easy

A. Find a path from the start to the destination in a maze

B. Count the total number of cells in the maze

C. Calculate the shortest distance between two points in a graph

D. Sort the maze cells in ascending order

Solution

1. Step 1: Understand the problem goal

   The Rat in a Maze problem asks to find a path from the start cell (usually top-left) to the destination cell (usually bottom-right) in a maze represented by a grid.

2. Step 2: Identify the correct goal

   The goal is not counting cells, sorting, or shortest distance but finding a valid path through allowed cells.

3. Final Answer:

   Find a path from the start to the destination in a maze -> Option A

4. Quick Check:

   Goal = Find path [OK]

Hint: Focus on pathfinding, not counting or sorting [OK]

Common Mistakes:

• Confusing pathfinding with counting cells
• Thinking it finds shortest path always
• Mixing it with sorting or graph distance problems

2. Which of the following Rust function signatures is correct for a recursive function solving the Rat in a Maze problem?

easy

A. fn solve_maze(maze: Vec<i32>, x: usize, y: usize) -> Vec<(usize, usize)>

B. fn solve_maze(maze: Vec, x: i32, y: i32) -> i32

C. fn solve_maze(maze: &mut Vec<i32>, x: usize, y: usize) -> bool

D. fn solve_maze(maze: &Vec, x: usize, y: usize, path: &mut Vec<(usize, usize)>) -> bool

Solution

1. Step 1: Check parameter types for maze and coordinates

   The maze is a 2D grid, so it should be a reference to a vector of vectors of integers (&Vec). Coordinates are usize for indexing.

2. Step 2: Check return type and path parameter

   The function returns bool to indicate success and uses a mutable path vector to store the route.

3. Final Answer:

   fn solve_maze(maze: &Vec, x: usize, y: usize, path: &mut Vec<(usize, usize)>) -> bool -> Option D

4. Quick Check:

   Correct signature uses references and bool return [OK]

Hint: Use references for maze and bool return for success [OK]

Common Mistakes:

• Passing maze by value causing unnecessary copying
• Using wrong coordinate types (i32 instead of usize)
• Returning wrong types like i32 or Vec instead of bool

3. Given the following Rust code snippet for Rat in a Maze, what will be printed?

fn main() {
    let maze = vec![
        vec![1, 0, 0, 0],
        vec![1, 1, 0, 1],
        vec![0, 1, 0, 0],
        vec![1, 1, 1, 1],
    ];
    let mut path = Vec::new();
    if solve_maze(&maze, 0, 0, &mut path) {
        println!("Path found: {:?}", path);
    } else {
        println!("No path found");
    }
}

fn solve_maze(maze: &Vec, x: usize, y: usize, path: &mut Vec<(usize, usize)>) -> bool {
    let n = maze.len();
    if x == n - 1 && y == n - 1 {
        path.push((x, y));
        return true;
    }
    if x < n && y < n && maze[x][y] == 1 {
        path.push((x, y));
        if x + 1 < n && solve_maze(maze, x + 1, y, path) {
            return true;
        }
        if y + 1 < n && solve_maze(maze, x, y + 1, path) {
            return true;
        }
        path.pop();
    }
    false
}

medium

A. Path found: [(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (3, 3)]

B. Path found: [(0, 0), (0, 1), (1, 1), (2, 1), (3, 1), (3, 2), (3, 3)]

C. No path found

D. Path found: [(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (3, 2), (3, 3)]

Solution

1. Step 1: Trace the recursive calls from (0,0)

   Start at (0,0) which is 1 (allowed). Try moving down to (1,0) which is 1, then (2,0) is 0 (blocked), so backtrack. Then try right from (1,0) to (1,1) which is 1.

2. Step 2: Continue path to destination

   From (1,1), down to (2,1) is 1, then down to (3,1) is 1, right to (3,2) is 1, and right to (3,3) is 1 (destination). Path is [(0,0),(1,0),(1,1),(2,1),(3,1),(3,2),(3,3)].

3. Final Answer:

   Path found: [(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (3, 3)] -> Option A

4. Quick Check:

   Recursive path = Path found: [(0, 0), (1, 0), (1, 1), (2, 1), (3, 1), (3, 2), (3, 3)] [OK]

Hint: Follow down then right moves recursively [OK]

Common Mistakes:

• Assuming path goes right first instead of down
• Not backtracking correctly on blocked cells
• Confusing coordinates order in path

4. Identify the bug in this Rust code snippet for Rat in a Maze and select the fix:

fn solve_maze(maze: &Vec, x: usize, y: usize, path: &mut Vec<(usize, usize)>) -> bool {
    let n = maze.len();
    if x == n - 1 && y == n - 1 {
        path.push((x, y));
        return true;
    }
    if x < n && y < n && maze[x][y] == 1 {
        path.push((x, y));
        if solve_maze(maze, x + 1, y, path) {
            return true;
        }
        if solve_maze(maze, x, y + 1, path) {
            return true;
        }
        path.pop();
    }
    false
}

medium

A. Change maze parameter to mutable reference

B. Remove path.pop() to keep all visited cells

C. Add boundary checks before recursive calls to avoid out-of-bounds

D. Return i32 instead of bool for success

Solution

1. Step 1: Analyze recursive calls for boundary safety

   The recursive calls use x+1 and y+1 without checking if they exceed maze size, risking out-of-bounds panic.

2. Step 2: Fix by adding boundary checks before recursion

   Adding checks like x + 1 < n and y + 1 < n before calling solve_maze prevents invalid indexing.

3. Final Answer:

   Add boundary checks before recursive calls to avoid out-of-bounds -> Option C

4. Quick Check:

   Boundary checks prevent panic [OK]

Hint: Always check indices before recursive calls [OK]

Common Mistakes:

• Ignoring index bounds causing runtime panic
• Changing maze to mutable unnecessarily
• Removing path.pop() causing wrong paths

5. You want to modify the Rat in a Maze solution in Rust to find all possible paths from start to finish instead of just one. Which approach is best?

hard

A. Use a queue to perform breadth-first search and return the shortest path only

B. Return a vector of paths and explore all possible moves recursively without stopping at first success

C. Stop recursion as soon as one path is found to save time

D. Modify the maze to mark visited cells permanently to avoid revisiting

Solution

1. Step 1: Understand requirement to find all paths

   Finding all paths means exploring every possible route without stopping at the first success.

2. Step 2: Use recursion to explore all moves and collect paths

   Return a vector of paths and do not return early. Backtrack properly to explore all options.

3. Final Answer:

   Return a vector of paths and explore all possible moves recursively without stopping at first success -> Option B

4. Quick Check:

   Explore all paths recursively [OK]

Hint: Collect all paths, don't stop at first success [OK]

Common Mistakes:

• Stopping recursion early losing other paths
• Using BFS only for shortest path, not all paths
• Marking visited permanently blocking other paths