Find Kth Largest XOR Coordinate Value

Find Kth Largest XOR Coordinate Value

What is this problem about?

The Find Kth Largest XOR Coordinate Value coding problem gives you a 2D matrix. You define a new matrix value where value[i][j] is the XOR sum of all matrix[r][c] where $0 \le r \le i$ and $0 \le c \le j$. You need to find the $k$-th largest value in this derived value matrix.

Why is this asked in interviews?

Google uses this to test your proficiency with Bit Manipulation and 2D Prefix Sums. It evaluations whether you can apply the prefix sum pattern to XOR operations (which are their own inverse). It also tests your ability to find the $k$-th largest element in a large set ($O(NM)$ total values) using efficient selection algorithms like Quickselect or a Priority Queue.

Algorithmic pattern used

The problem uses 2D Prefix XOR and Selection.

1. Calculate the cumulative XOR matrix: val[i][j] = matrix[i][j] ^ val[i-1][j] ^ val[i][j-1] ^ val[i-1][j-1].
2. Collect all $N imes M$ values into a list.
3. Find the $k$-th largest element:
   • Max-Heap: $O(NM \log k)$.
   • Quickselect: $O(NM)$ average time.

Example explanation

Matrix:

5 2
1 6

1. (0,0): 5.
2. (0,1): $5 \oplus 2 = 7$.
3. (1,0): $5 \oplus 1 = 4$.
4. (1,1): $5 \oplus 2 \oplus 1 \oplus 6 = 0$. Values: [5, 7, 4, 0]. Sorted: [7, 5, 4, 0]. $k=2$: Result is 5.

Common mistakes candidates make

• Incorrect XOR formula: Forgetting to XOR the val[i-1][j-1] term back in (since it was included twice by the row and column XORs).
• Inefficient Sorting: Sorting the entire list ($O(NM \log NM)$) when $k$ is small or when Quickselect could be used.
• Memory Management: Creating too many copies of the matrix.

Interview preparation tip

XOR behaves mathematically exactly like addition in prefix sum formulas. If you know how to do a 2D prefix sum for additions, you know how to do it for XOR. The only difference is that $A \oplus A = 0$, so the "subtraction" step is also just XOR.