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

Time Complexity: Rat in a Maze Problem

O(2^{n^2})

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.

function solveMaze(maze: number[][], x: number, y: number, solution: number[][]): boolean {
  const n = maze.length;
  if (x === n - 1 && y === n - 1) {
    solution[x][y] = 1;
    return true;
  }
  if (x >= 0 && y >= 0 && x < n && y < n && maze[x][y] === 1) {
    solution[x][y] = 1;
    if (solveMaze(maze, x + 1, y, solution)) return true;
    if (solveMaze(maze, x, y + 1, solution)) return true;
    solution[x][y] = 0;
    return false;
  }
  return 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 each cell in the maze.
• How many times: Potentially, each cell is visited multiple times during backtracking.

How Execution Grows With Input

As the maze size grows, the number of possible paths grows very fast because the rat can move in multiple directions at each step.

Input Size (n)	Approx. Operations
2	Few recursive calls, easy to check all paths.
5	Many more paths to explore, operations increase quickly.
10	Huge number of possible paths, operations grow very fast.

Pattern observation: The time grows very fast, much faster than just doubling or squaring; it grows exponentially with maze size.

Final Time Complexity

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

This means the time needed can grow extremely fast as the maze size increases, because the rat tries many possible paths.

Common Mistake

[X] Wrong: "The rat only moves forward, so time grows linearly with maze size."

[OK] Correct: The rat can move in multiple directions and backtrack, so many paths are checked, making the time grow much faster than linear.

Interview Connect

Understanding this problem helps you think about how exploring many options affects time, a skill useful for many search and path problems.

Self-Check

"What if we added a rule that the rat can only move right or down? How would the time complexity change?"

Practice

1. What is the main goal of the Rat in a Maze Problem in data structures and algorithms?

easy

A. Sort the cells of the maze in ascending order

B. Count the number of walls in the maze

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

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

Solution

1. Step 1: Understand the problem context

   The Rat in a Maze problem involves navigating through a maze represented as a grid to find a path from the start cell to the destination cell.

2. Step 2: Identify the goal

   The goal is to find a valid path that the rat can follow to reach the destination without crossing walls or blocked cells.

3. Final Answer:

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

4. Quick Check:

   Goal = Find path [OK]

Hint: Remember: Rat must find a path, not count or sort cells [OK]

Common Mistakes:

• Confusing path finding with counting walls
• Thinking the problem is about sorting cells
• Assuming shortest path is always required

2. Which of the following is the correct TypeScript syntax to declare a 2D maze array of size 4x4 filled with zeros?

easy

A. const maze: number[][] = Array.from({length: 4}, () => Array(4).fill(0));

B. const maze: number[][] = [[0]*4]*4;

C. const maze: number[][] = new Array(4, 4).fill(0);

D. const maze: number[][] = Array(4).fill(Array(4).fill(0));

Solution

1. Step 1: Understand array initialization in TypeScript

   Using Array.from with a mapping function creates independent inner arrays, avoiding reference issues.

2. Step 2: Analyze each option

   const maze: number[][] = Array(4).fill(Array(4).fill(0)); creates inner arrays by reference, causing all rows to point to the same array. const maze: number[][] = Array.from({length: 4}, () => Array(4).fill(0)); correctly creates separate arrays for each row. Options B and C use invalid syntax for 2D arrays.

3. Final Answer:

   const maze: number[][] = Array.from({length: 4}, () => Array(4).fill(0)); -> Option A

4. Quick Check:

   Correct 2D array init = const maze: number[][] = Array.from({length: 4}, () => Array(4).fill(0)); [OK]

Hint: Use Array.from with a function to avoid shared references [OK]

Common Mistakes:

• Using fill with nested arrays causing shared references
• Incorrect syntax for 2D array creation
• Confusing array size parameters

3. Given the following maze and code snippet, what will be the output?

const maze = [
  [1, 0, 0, 0],
  [1, 1, 0, 1],
  [0, 1, 0, 0],
  [1, 1, 1, 1]
];

function isSafe(x: number, y: number): boolean {
  return x >= 0 && y >= 0 && x < 4 && y < 4 && maze[x][y] === 1;
}

function solveMaze(x: number, y: number, path: string[]): boolean {
  if (x === 3 && y === 3) {
    path.push(`(${x},${y})`);
    return true;
  }
  if (isSafe(x, y)) {
    path.push(`(${x},${y})`);
    if (solveMaze(x + 1, y, path)) return true;
    if (solveMaze(x, y + 1, path)) return true;
    path.pop();
  }
  return false;
}

const path: string[] = [];
if (solveMaze(0, 0, path)) {
  console.log(path.join(' -> '));
} else {
  console.log('No path found');
}

medium

A. (0,0) -> (0,1) -> (1,1) -> (2,1) -> (3,1) -> (3,2) -> (3,3)

B. (0,0) -> (1,0) -> (1,1) -> (2,1) -> (3,1) -> (3,2) -> (3,3)

C. No path found

D. (0,0) -> (1,0) -> (2,0) -> (3,0) -> (3,1) -> (3,2) -> (3,3)

Solution

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

   Starting at (0,0), the rat moves down to (1,0) since maze[1][0] = 1. Then right to (1,1), down to (2,1), down to (3,1), right to (3,2), and right to (3,3).

2. Step 2: Verify path validity and output

   All these cells are safe (value 1). The path is recorded and printed as shown in (0,0) -> (1,0) -> (1,1) -> (2,1) -> (3,1) -> (3,2) -> (3,3).

3. Final Answer:

   (0,0) -> (1,0) -> (1,1) -> (2,1) -> (3,1) -> (3,2) -> (3,3) -> Option B

4. Quick Check:

   Path found = (0,0) -> (1,0) -> (1,1) -> (2,1) -> (3,1) -> (3,2) -> (3,3) [OK]

Hint: Follow down then right moves recursively to find path [OK]

Common Mistakes:

• Assuming movement to blocked cells (0 values)
• Confusing row and column indices
• Ignoring backtracking when path fails

4. Identify the bug in the following TypeScript code snippet for the Rat in a Maze problem:

function solveMaze(x: number, y: number, path: string[]): boolean {
  if (x === 3 && y === 3) {
    path.push(`(${x},${y})`);
    return true;
  }
  if (isSafe(x, y)) {
    path.push(`(${x},${y})`);
    if (solveMaze(x + 1, y, path)) return true;
    if (solveMaze(x, y + 1, path)) return true;
    path.pop();
  }
  return false;
}

medium

A. isSafe function is not called before pushing to path

B. Incorrect base case coordinates for destination

C. Missing backtracking step to remove current cell from path when dead end is reached

D. The function should move left and up instead of right and down

Solution

1. Step 1: Analyze the recursion and path tracking

   The function adds the current cell to path but does not remove it if both recursive calls fail, causing incorrect path accumulation.

2. Step 2: Identify missing backtracking

   Backtracking requires removing the current cell from path when no path forward is found, which is missing here.

3. Final Answer:

   Missing backtracking step to remove current cell from path when dead end is reached -> Option C

4. Quick Check:

   Backtracking needed = Missing backtracking step to remove current cell from path when dead end is reached [OK]

Hint: Always pop path on dead ends to backtrack [OK]

Common Mistakes:

• Forgetting to remove cell from path on failure
• Incorrect base case coordinates
• Ignoring isSafe checks before path push

5. You want to modify the Rat in a Maze algorithm to find all possible paths from start (0,0) to destination (N-1,N-1) in an NxN maze. Which approach should you use?

hard

A. Use backtracking to explore all safe moves, storing each complete path separately

B. Stop at the first found path and return immediately

C. Use a greedy approach to always move right or down only

D. Convert the maze to a graph and use Dijkstra's algorithm to find shortest path

Solution

1. Step 1: Understand the requirement to find all paths

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

2. Step 2: Choose backtracking to explore all safe moves

   Backtracking allows exploring all safe moves recursively, storing each complete path separately before backtracking to try other routes.

3. Final Answer:

   Use backtracking to explore all safe moves, storing each complete path separately -> Option A

4. Quick Check:

   All paths = Backtracking exploration [OK]

Hint: Backtrack fully to find all paths, not just one [OK]

Common Mistakes:

• Stopping after first path found
• Using greedy moves that miss paths
• Using shortest path algorithms instead of all paths