1738. Find Kth Largest XOR Coordinate Value
Description
You are given a 2D matrix
of size m x n
, consisting of non-negative integers. You are also given an integer k
.
The value of coordinate (a, b)
of the matrix is the XOR of all matrix[i][j]
where 0 <= i <= a < m
and 0 <= j <= b < n
(0-indexed).
Find the kth
largest value (1-indexed) of all the coordinates of matrix
.
Example 1:
Input: matrix = [[5,2],[1,6]], k = 1 Output: 7 Explanation: The value of coordinate (0,1) is 5 XOR 2 = 7, which is the largest value.
Example 2:
Input: matrix = [[5,2],[1,6]], k = 2 Output: 5 Explanation: The value of coordinate (0,0) is 5 = 5, which is the 2nd largest value.
Example 3:
Input: matrix = [[5,2],[1,6]], k = 3 Output: 4 Explanation: The value of coordinate (1,0) is 5 XOR 1 = 4, which is the 3rd largest value.
Constraints:
m == matrix.length
n == matrix[i].length
1 <= m, n <= 1000
0 <= matrix[i][j] <= 106
1 <= k <= m * n
Solutions
Solution 1: Two-dimensional Prefix XOR + Sorting or Quick Selection
We define a two-dimensional prefix XOR array $s$, where $s[i][j]$ represents the XOR result of the elements in the first $i$ rows and the first $j$ columns of the matrix, i.e.,
$$ s[i][j] = \bigoplus_{0 \leq x \leq i, 0 \leq y \leq j} matrix[x][y] $$
And $s[i][j]$ can be calculated from the three elements $s[i - 1][j]$, $s[i][j - 1]$ and $s[i - 1][j - 1]$, i.e.,
$$ s[i][j] = s[i - 1][j] \oplus s[i][j - 1] \oplus s[i - 1][j - 1] \oplus matrix[i - 1][j - 1] $$
We traverse the matrix, calculate all $s[i][j]$, then sort them, and finally return the $k$th largest element. If you don't want to use sorting, you can also use the quick selection algorithm, which can optimize the time complexity.
The time complexity is $O(m \times n \times \log (m \times n))$ or $O(m \times n)$, and the space complexity is $O(m \times n)$. Here, $m$ and $n$ are the number of rows and columns of the matrix, respectively.
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