Problem Description

In this problem, we are provided with a 2D grid that represents a matrix of size m x n and an integer k. The task is to perform a 'shift' operation on the grid exactly k times, where each shift operation moves elements in the grid to their adjacent right position with wrapping around at the edges. This means that:

• An element at grid[i][j] moves to grid[i][j + 1].
• An element at the end of a row (grid[i][n - 1]) moves to the starting of the next row (grid[i + 1][0]).
• The bottom-right element (grid[m - 1][n - 1]) moves to the very first position of the grid (grid[0][0]).

We need to return the grid as it looks after performing all k shifts.

Intuition

To solve this problem, we aim to simulate the process of shifting elements. However, directly shifting elements in the grid k times would be inefficient, especially for large values of k. Instead, we can take advantage of the observation that after m * n shifts, the grid will look exactly as it started (here, m * n represents the total number of elements in the grid).

By using this fact, we can calculate the effective number of shifts needed by taking the remainder of k divided by the total number of elements m * n (k % (m * n)). This way, we minimize the number of shifts to no more than m * n - 1.

To implement the shifts, the solution takes the following steps:

1. Flatten the grid into a one-dimensional list t, thus simplifying the problem to shifting elements in an array.

2. Since we are shifting to the right, we take the last k elements of the array t[-k:] and move them to the front, effectively performing a rotation. The remaining elements t[:-k] are then appended to the result of the previously rotated elements to form the new array after the shift.

3. Finally, we map the elements back to the original grid shape (m x n) by iterating through the matrix indices and assigning the values from the shifted list back into the grid.

The given solution is efficient and avoids unnecessary computations by leveraging modular arithmetic and array slicing to achieve the desired shifts before reconstructing the final matrix.

Solution Approach

The solution implementation uses a straightforward approach to solve the problem by manipulating the grid as a one-dimensional list and then reshaping it back to the 2D grid. Here's the breakdown of the implementation:

1. First, we determine the size of the grid with m, n = len(grid), len(grid[0]). This gives us the number of rows (m) and columns (n) in the grid.

2. Since the grid is effectively a circular data structure after m * n shifts, we compute the effective shifts needed: k %= m * n. This step uses the modulus operator to ensure we only shift the minimum required number of times as shifting m * n times would return the grid to its original configuration.

3. We then flatten the grid into a one-dimensional list t: t = [grid[i][j] for i in range(m) for j in range(n)]. This step uses list comprehension to traverse all elements in a row-major order.

4. The next step involves rotating the list (shifting the elements to the right by k places). This is accomplished with list slicing: t = t[-k:] + t[:-k]. Here, t[-k:] obtains the last k elements which will move to the front of the list, and t[:-k] gets the elements before the last k, which will now follow the shifted k elements.

5. Finally, we map the elements from the rotated list back to the 2D grid form. This is done using nested loops where we traverse the rows and columns of the grid and set each element to its corresponding value from the rotated list:

   for i in range(m):
       for j in range(n):
           grid[i][j] = t[i * n + j]

   • For each index (i, j) in the grid, we calculate the one-dimensional index using i * n + j, which is then used to access the correct value from the list t.

6. The modified grid after all k shifts is then returned.

The algorithm makes use of simple list manipulations, modular arithmetic, and the efficiency of Python's list slicing operations to provide a swift resolution to the grid shifting problem. It avoids the more computationally expensive direct simulation of the shifts and elegantly manages the wrapping around behavior by treating the 2D grid as a circular list.

Example Walkthrough

Let's illustrate the solution approach using a small example grid and a number of shifts k. Consider the following grid grid with dimensions 2 x 3 and an integer k = 4.

grid = [
    [1, 2, 3],
    [4, 5, 6]
]

1. Determine Grid Size: First, we recognize the grid has m = 2 rows and n = 3 columns.

2. Compute Effective Shifts: Since the total number of elements is m * n = 6, we compute the effective number of shifts k %= 6, which is k = 4 % 6 = 4. It means we only need to shift elements four positions to the right.

3. Flatten the Grid: Next, we flatten the grid into a one-dimensional list.

   t = [1, 2, 3, 4, 5, 6]

4. Rotate the List: To simulate the shift operation, we perform a rotation on the list. Given k = 4, we slice and re-arrange the list:

   t[-k:] = [3, 4, 5, 6]
   t[:-k] = [1, 2]

   After rotation, the list t will look like:

   t = [3, 4, 5, 6] + [1, 2] = [3, 4, 5, 6, 1, 2]

5. Map Rotated List Back to Grid: Lastly, we convert the rotated list back into a 2D grid form using the formulas from our approach. This will restructure the list into rows and columns based on the grid's dimensions.

   grid[0][0] = t[0 * 3 + 0] = t[0] = 3
   grid[0][1] = t[0 * 3 + 1] = t[1] = 4
   grid[0][2] = t[0 * 3 + 2] = t[2] = 5
   grid[1][0] = t[1 * 3 + 0] = t[3] = 6
   grid[1][1] = t[1 * 3 + 1] = t[4] = 1
   grid[1][2] = t[1 * 3 + 2] = t[5] = 2

   The final grid after 4 shifts will be:

   grid = [
       [3, 4, 5],
       [6, 1, 2]
   ]

And that's how you apply the given solution to perform shift operations on a grid. The grid grid after four shifts turns out to be [ [3, 4, 5], [6, 1, 2] ], which demonstrates the effectiveness of this method for moving elements to their right positions while wrapping around the edges.

Solution Implementation

Time and Space Complexity

The given Python code performs a shift operation on a 2D grid by k places.

Time Complexity

The time complexity of the code is determined by several factors:

1. Calculating the total number of cells in the grid, which is m * n. This is constant-time operation O(1).

2. Flattening the grid into a 1D list, which involves iterating through all the cells. This takes O(m * n) time.

3. Slicing the list to perform the shift operation. The list slicing operation in Python takes O(k) time in the worst case, where k is the number of elements being sliced. In this case, it's also O(m * n) because k is at most m * n after the modulo operation.

4. Iterating through the grid again to place the shifted values back into the 2D grid. This is another O(m * n) operation.

Combining these steps, the overall time complexity is O(m * n) because all other operations are constant time or also O(m * n).

Space Complexity

The space complexity is determined by the additional space used by the algorithm, not including the space used by the input and output.

1. Storing the flattened grid as a 1D list t requires O(m * n) space.

2. The space needed for storing shifted list slices before concatenation is also O(m * n).

Overall, the space complexity is O(m * n) due to the additional list created to store the grid elements.

Learn more about how to find time and space complexity quickly using problem constraints.