Find in Shifted Sorted Matrix

Understanding the Matrix Shift Problem

Suppose you have a matrix where each row and column is sorted in ascending order. Your task is to apply two specific types of shifts, described below, and locate a given element x in the transformed matrix.

Shift Types

Left Circular Shift (l): - Definition: Moves the first column to the last position, shifting all other columns one step left. - Example:

 Before:
 ```py
 [
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9]
 ]
 ```
 After:
 ```py
 [
    [2, 3, 1],
    [5, 6, 4],
    [8, 9, 7]
 ]
 ```

• Visual Explanation: Imagine the columns on a circular conveyor belt. The first column wraps to the end, while others shift left.

Top Circular Shift (u): - Definition: Moves the first row to the last position, shifting all other rows one step up. - Example:

 Before:
 ```py
 [
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9]
 ]
 ```
 After:
 ```py
 [
    [4, 5, 6],
    [7, 8, 9],
    [1, 2, 3]
 ]
 ```

• Visual Explanation: Picture the rows on a circular escalator. The first row descends to the bottom, while others shift up.

Applying the Shifts to a Specific Example

Let's consider a sequence of shifts like [l,u,l,u,u] and apply them to the given matrix:

matrix = [
    [ 1, 2, 3, 4, 5],
    [ 6, 7, 8, 9,10],
    [11,12,13,14,15],
    [16,17,18,19,20]
]

After applying these rotations, the matrix transforms to:

matrix = [
    [9 ,10, 6, 7, 8],
    [14,15,11,12,13],
    [19,20,16,17,18],
    [ 4, 5, 1, 2, 3]
]

Here's an illustration to help visualize the problem:

Your Task

Given this final matrix and a specific value x, how can you efficiently find the location of this element in the array and return it?

Constraints

1. Matrix Size: Up to 10^5 x 10^5
2. Time Complexity: O(nlogn)
3. Memory Complexity: O(1)
4. Element Existence: The value x is in the matrix.
5. Minimum Elements: At least 1 element in the matrix.
6. Sorting Order: Always sorted in all rows and columns.
7. Uniqueness: Contains unique elements.

Considering these constraints and the above example, can you develop a method to locate the specified element in the transformed matrix?

```py
def findElementInMatrix(matrix, x):
    # Fill in