Problem Description

In this problem, we are working with a matrix grid with dimensions m x n (where m is the number of rows and n is the number of columns), filled with positive integers. The primary objective is to continually perform a specific operation on the grid until it becomes empty. The operations can be detailed as follows:

• Firstly, we delete the elements with the greatest value from each row. This operation may remove more than one element per row if multiple elements share the greatest value.
• Secondly, we identify the maximum of the values deleted in this round and add it to the answer. This step essentially means that after each round of deletions, only the single highest value from among all the rows is added to the running total.

It is crucial to note that eliminating elements results in the grid having one less column after each operation. Consequently, with each iteration, the grid's dimensions change, shrinking in the column direction. The task is to return the cumulative total of the maximum values added after each operation has been completed and the grid is empty.

Intuition

To arrive at a solution for this problem, we should recognize that we are not required to modify the original grid structure. Instead, we need to repeatedly find and sum the largest elements removed based on the current iteration.

As we are required to sum the maximum elements of each row, and then take the maximum of these per operation, the key insight is that sorting each row will facilitate easy access to the maximum values. By sorting, we always have the largest value at the end of each row, and this enables quick removal and processing.

To sum the largest elements removed from each row, we utilize Python's zip function combined with an unpacking operator *, which effectively transposes the sorted grid. Transposing the grid after sorting allows us to use the intuitive max function on the columns, which now represent the original rows' maximum elements.

This leads to a concise and efficient process:

1. Sort each row of grid so that the element with the greatest value is at the end of each row (making it easy to identify).
2. Transpose the grid, turning rows into columns, effectively grouping the elements to be deleted together.
3. Sum the maximum elements across what are now the columns, which correspond to the greatest elements of the original rows.

The final sum is the answer we return. This solution avoids manipulating the matrix extensively. Instead, it leverages Python's built-in functions for sorting, transposing, and finding the max to perform the necessary computations efficiently.

Learn more about Sorting and Heap (Priority Queue) patterns.

Solution Approach

The implementation of the solution follows a straightforward approach that heavily utilizes built-in Python functions to handle the primary operations. Here is a step by step walk-through of the algorithm:

1. Sorting Rows: The initial step involves sorting each row in the grid. This is important because, according to the problem description, we need to delete the greatest value from each row. Sorting each row guarantees that the largest element is at the end of the row, which makes identifying these elements straightforward.

   Algorithmically, the sort() method in Python rearranges the elements in each list (which corresponds to a row in the grid) in ascending order in-place, meaning no additional data structures are necessary for this step.

   for row in grid:
       row.sort()

2. Transposing the Matrix: After sorting, the solution transposes the grid using the zip(*grid) function. In effect, this creates an iterator of tuples where each tuple contains elements from the rows of the grid that are in the same column position. Since the grid rows are sorted, the last elements (which are now at the first position in each tuple) are the largest elements from each row.

   Transposing is a pattern that allows us to treat each tuple as a representation of a column, making it easy to find the maximum element across these columns. By following this pattern, we avoid having to manually loop through each row to find and delete the maximum element, which can be computationally expensive.

3. Summing the Maximum Elements: The final step involves summing the maximum elements of the previously transposed 'columns' (which, in reality, are representing the maximum elements from each row). This is done using a generator expression within the sum function. The max function is called on each column, and the return values, which are the largest elements from each original row, are summed up to form the answer.

   return sum(max(col) for col in zip(*grid))

It's important to note that this approach doesn't change the original grid size since we don't need to keep track of the number of columns decreased after each operation. The zip() function naturally ends when the shortest row (or column after transposition) is exhausted, which aligns with the original grid shrinking in dimensions as the problem statement suggests.

The algorithm makes a single pass through each row to sort it and a single pass to sum the maxima, resulting in a time complexity that is efficient given the constraints of the problem. The spatial complexity is also minimal as no additional data structures are required; the sorting and transposition operations are performed in-place (except for the space utilized by zip, which is negligible as it returns an iterator).

By carefully using these built-in functions and patterns, we obtain an elegant solution that requires only a few lines of code but still adheres to the requirements of the problem and achieves optimal performance.

Example Walkthrough

Let's take a small example to illustrate the solution approach with a grid that looks as follows:

grid = [
  [3, 1, 2],
  [4, 5, 6],
  [9, 8, 7]
]

Step-by-step, we would perform the following actions:

1. Sorting Rows: We start by sorting each row of our grid. After sorting, the grid becomes:

   [
     [1, 2, 3],
     [4, 5, 6],
     [7, 8, 9]
   ]

   This is achieved using the sort() function in Python on all the rows:

   for row in grid:
       row.sort()

2. Transposing the Matrix: After sorting, we transpose the grid to work with the maximum elements more conveniently. The transposed grid (zip(*grid)) turns our rows into tuples that act like columns:

   [
     (1, 4, 7),
     (2, 5, 8),
     (3, 6, 9)
   ]

   Each tuple contains elements that were previously in the same column. For example, 1, 4, 7 were the first elements of their respective rows but are now grouped together after transposition.

3. Summing the Maximum Elements: We find the maximum value in each of these tuples (which are our transposed 'columns') and sum them together. These maximum values are 3, 6, and 9 from the 'columns':

   max(1, 4, 7) = 7
   max(2, 5, 8) = 8
   max(3, 6, 9) = 9

   The sum of these max values gives us the total: 7 + 8 + 9 = 24. This is the final answer to our problem.

   In Python, this can be done with the following code:

   return sum(max(col) for col in zip(*grid))

And that's our expected result, a total of 24. This walkthrough demonstrates the usage of sorting rows, transposing the matrix, and summing the maximum elements, thus achieving the goal of efficiently computing the cumulative total after performing the operations dictated by the problem.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code consists of two parts:

1. Sorting each row in the grid list.
2. Finding the maximum value of each column after the rows have been sorted.

The grid size is defined by m x n, where m is the number of rows and n is the number of columns.

For the first part, we sort each row individually. The sort operation on a row of length n has a time complexity of O(n log n). Since there are m such rows, this part of the process has a total time complexity of O(m * n log n).

For the second part, after sorting, we find the maximum element in each column using a generator expression that zips the sorted rows together. This operation is O(m) for each column since it involves iterating through every row for the given column to find the max. There are n columns, so the total time complexity for this part is O(n * m).

Therefore, the total time complexity of the entire code is O(m * n log n + n * m). Since m * n log n would likely dominate for larger values of n, we might consider this the more significant term, and the overall time complexity can be approximated by O(m * n log n).

Space Complexity

For space complexity, we need to consider the additional space used by the algorithm excluding the input itself:

1. Sorting is done in-place in the provided grid list, not requiring additional space that depends on the size of the input. However, Python's sort method under the hood may use O(n) space for each row due to the Timsort algorithm it implements.

2. The generator expression used to calculate the sum of maximum values in each column does not require significant additional space; it will take O(1) space as it evaluates each value on-the-fly and does not store the entire column.

Thus, the additional space is O(n), needed for the in-place sorting of one row at a time, which is the most significant additional space used by the algorithm.

Considering the constraints above, the overall space complexity of the algorithm is O(n).

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