Count Paths With the Given XOR Value

This problem requires calculating the number of paths from the top-left to bottom-right corner of a grid where the XOR of all numbers along the path equals a given value. The dynamic programming approach efficiently handles this using state transitions. Focus on keeping track of the XOR results at each step using a DP table.

Problem Statement

You are given a 2D integer array grid of size m x n. You must calculate the number of distinct paths from the top-left cell (0, 0) to the bottom-right cell (m - 1, n - 1). Each path must satisfy the constraint that the XOR of all values along the path equals a given integer k.

The grid values and k range from 0 to 15. The goal is to compute how many paths exist that satisfy this condition, using dynamic programming techniques to track possible XOR results along each path.

Examples

Example 1

Input: grid = [[2, 1, 5], [7, 10, 0], [12, 6, 4]], k = 11

Output: 3

The 3 paths are:

Example 2

Input: grid = [[1, 3, 3, 3], [0, 3, 3, 2], [3, 0, 1, 1]], k = 2

Output: 5

The 5 paths are:

Example 3

Input: grid = [[1, 1, 1, 2], [3, 0, 3, 2], [3, 0, 2, 2]], k = 10

Output: 0

Example details omitted.

Constraints

• 1 <= m == grid.length <= 300
• 1 <= n == grid[r].length <= 300
• 0 <= grid[r][c] < 16
• 0 <= k < 16

Solution Approach

Dynamic Programming Table Setup

Use a 3D DP table where dp[i][j][x] represents the number of ways to reach cell (i, j) with an XOR value of x. The goal is to fill the table by considering valid transitions from adjacent cells (top and left), and updating the XOR value at each step.

State Transitions

At each cell (i, j), calculate the new XOR by including the current cell's value. For each cell, consider the two possible directions (from above and from the left) and update the DP table accordingly. This way, we build up the paths and track the XOR value for each path dynamically.

Final Calculation

Once the DP table is filled, the answer will be found in dp[m-1][n-1][k], which gives the number of paths to the bottom-right corner with the exact XOR value of k.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity depends on the number of cells m * n and the number of possible XOR values (16), resulting in a time complexity of O(m * n * 16). Space complexity is similarly O(m * n * 16) due to the DP table size.

What Interviewers Usually Probe

• Look for a candidate's ability to apply dynamic programming to state transitions effectively.
• Assess whether the candidate understands the necessity of tracking XOR states in paths.
• Check if the candidate handles the grid traversal and transition properly, avoiding unnecessary recomputation.

Common Pitfalls or Variants

Common pitfalls

• Candidates may forget to handle boundary conditions when transitioning from cells that are not top or left.
• Some may struggle with the memory management of the 3D DP table, especially when the grid size is large.
• Not considering the XOR of previous steps correctly may lead to incorrect path calculations.

Follow-up variants

• Variant where grid size increases but the XOR value remains small.
• Grid with higher values of k where optimization strategies like bitmasking could help.
• Variant involving negative or different range of values in the grid.

FAQ

What is the primary approach to solve the 'Count Paths With the Given XOR Value' problem?

The problem is solved using dynamic programming with a 3D DP table to track the number of paths to each grid cell with a specific XOR value.

What is the time complexity of solving the 'Count Paths With the Given XOR Value' problem?

The time complexity is O(m * n * 16), where m and n are the dimensions of the grid, and 16 represents the number of possible XOR values.

Can you explain the state transition for this problem?

State transitions occur by considering paths from adjacent cells (top and left) and updating the XOR value at each step, using a dynamic programming approach.

How can I handle large grids efficiently in the 'Count Paths With the Given XOR Value' problem?

By using a 3D DP table, the problem remains manageable even for larger grids, but optimizations like rolling arrays or space compression can help reduce memory usage.

What are the common pitfalls when solving this problem?

Common mistakes include failing to manage boundary conditions, not properly tracking the XOR state, and inefficient space usage in the DP table.