Count Paths With the Given XOR Value - Solution & Explanation

Problem Overview: Given a matrix of integers, count how many paths from the top-left cell to the bottom-right cell produce a XOR value equal to k. You can only move right or down. Each path accumulates XOR from all visited cells.

Approach 1: Brute Force DFS Enumeration (Exponential Time)

Explore every possible path from (0,0) to (m-1,n-1) using recursion. At each step, move either right or down and maintain the running XOR of visited cells. When you reach the bottom-right cell, check if the accumulated XOR equals k. This approach performs a full traversal of the path space, which grows exponentially with grid size. Time complexity is O(2^(m+n)) and space complexity is O(m+n) due to recursion depth. It works only for very small matrices.

Approach 2: DFS with Memoization (DP on State) (O(m * n * X))

Instead of recomputing paths repeatedly, cache results based on the state (row, col, currentXor). From a cell, recursively explore the right and down neighbors while updating the XOR using nextXor = currentXor ^ grid[r][c]. If the same state appears again, return the stored result immediately. The key insight: once you reach a cell with a specific XOR value, the number of ways to finish the path will always be the same. Time complexity becomes O(m * n * X), where X is the number of possible XOR states. Space complexity is also O(m * n * X) for the memo table. This approach uses ideas from dynamic programming and bit manipulation.

Approach 3: Bottom-Up Dynamic Programming (State Transition DP) (O(m * n * X))

Define a DP state dp[r][c][x] representing the number of ways to reach cell (r,c) with XOR value x. Initialize the starting state using the value of grid[0][0]. For each cell, propagate counts to the right and down neighbors while updating the XOR with the destination cell value. This effectively builds the solution row by row across the matrix. The final answer is dp[m-1][n-1][k]. Time complexity remains O(m * n * X) and space complexity is O(m * n * X). Using rolling arrays can reduce memory usage.

Recommended for interviews: The dynamic programming approach with XOR state tracking. Interviewers expect you to recognize that brute force enumerates too many paths, then introduce a DP state that includes the current XOR. Showing the brute force idea demonstrates understanding of the path structure, while the optimized DP solution shows strong knowledge of array traversal and state-based dynamic programming.