Problem Description

Given a 2-dimensional grid nums, where each cell contains an integer, our task is to return a list of all the elements of nums but in a specific order—diagonal order. In this ordering, elements from the same diagonal (elements with the same sum of their row and column indices) should be grouped together. Additionally, within each diagonal group, elements should be sorted based on their column indices. This creates a zigzag pattern if we visualize it, as we traverse diagonals starting from the top row, moving down-right on the grid, and then up to the next diagonal start point.

The challenge is to implement this in a way that respects the diagonal grouping and within-group ordering, and to do so as efficiently as possible.

Intuition

The intuition behind the solution lies in recognizing that elements that are on the same diagonal share a certain property: the sum of their row and column indices is constant. With this realization, we can iterate through every element in the nums grid, calculate the sum of its indices, and use this sum as a way to group elements into their respective diagonals.

To maintain the within-diagonal order (where elements with lower column indices come first), we keep track of the column index alongside the diagonal index sum. This means that for each element, we store a triple containing the sum (which represents the diagonal), the column index, and the value itself.

Once we have gone through all the elements and collected this information, we sort this list of triples. The sort will naturally group elements from the same diagonal together (because of the sums), and within each group, it will order elements by their column indices.

Finally, we extract the third element of each triple (which is the value from the original grid) to get our result in the correct diagonal order. The fact that the Python sort function sorts tuples lexicographically, first by the first element, then by the second, etc., makes it particularly well-suited for this task.

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

Solution Approach

The implementation of the solution makes efficient use of Python's list and sorting capabilities.

Here's a step-by-step explanation of the approach:

1. Initialize an empty list arr: This list is used to store the triples (sum of indices, column index, value) for each element in the nums array.

2. Iterate over the grid: We use a nested loop to go through each element of the nums grid. The outer loop (for i, row in enumerate(nums)) iterates over the rows, and the inner loop (for j, v in enumerate(row)) goes over each element in a row. Here, i is the row index, j is the column index, and v is the value at the nums[i][j] position.

3. Compute the diagonal index and store triples: For each grid element nums[i][j], we compute the sum of the indices (i + j) which uniquely identifies the diagonal to which the element belongs. We then create a triple (i + j, j, v) and append it to our arr list. The triple contains the diagonal index, the column index, and the value itself, respectively.

4. Sort the list of triples: We call arr.sort() to sort the list of triples. The sort method uses the natural lexicographical order for tuples, so it will first sort them by the diagonal index, and then by their column index within the same diagonal. This operation arranges the elements in exactly the order required for the final result.

5. Extract the results: The last line is a list comprehension [v[2] for v in arr] that goes through the sorted list of triples and extracts the third element from each triple (the value). This creates a new list of integers which is our final, diagonally ordered list of elements.

The key algorithms and data structures used in this approach are:

• Nested Loops: To iterate over a 2D array.
• List Comprehension: A concise way to construct lists.
• Enumerate Function: Provides a neat way to get both the index and the value when iterating over a list.
• Triple (Tuple): An immutable data structure to store the related elements - sum of indices, column index, and value.
• Sorting: A built-in Python method to sort lists in ascending order based on the first element of the tuple, and if they are the same, based on the second element, and so on.

This approach has a time complexity of O(N log N), where N is the total number of elements in the nums array, because the most expensive operation is sorting the list of triples. The spatial complexity is O(N) since we need to create a list arr that holds as many triples as there are elements in the original array.

Example Walkthrough

Let's use a small 3x3 grid example to illustrate the solution approach:

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

Following the steps in the solution approach:

1. Initialize an empty list arr:

   At the beginning, arr is an empty list: arr = []

2. Iterate over the grid:

   Now we start the nested loop over the grid:

   • For i = 0, row = [1, 2, 3]:
     • For j = 0, v = 1: Compute diagonal index i + j = 0, append (0, 0, 1) to arr.
     • For j = 1, v = 2: Compute diagonal index i + j = 1, append (1, 1, 2) to arr.
     • For j = 2, v = 3: Compute diagonal index i + j = 2, append (2, 2, 3) to arr.
   • For i = 1, row = [4, 5, 6]:
     • For j = 0, v = 4: Compute diagonal index i + j = 1, append (1, 0, 4) to arr.
     • For j = 1, v = 5: Compute diagonal index i + j = 2, append (2, 1, 5) to arr.
     • For j = 2, v = 6: Compute diagonal index i + j = 3, append (3, 2, 6) to arr.
   • For i = 2, row = [7, 8, 9]:
     • For j = 0, v = 7: Compute diagonal index i + j = 2, append (2, 0, 7) to arr.
     • For j = 1, v = 8: Compute diagonal index i + j = 3, append (3, 1, 8) to arr.
     • For j = 2, v = 9: Compute diagonal index i + j = 4, append (4, 2, 9) to arr.

   arr now looks like this: [(0, 0, 1), (1, 1, 2), (2, 2, 3), (1, 0, 4), (2, 1, 5), (3, 2, 6), (2, 0, 7), (3, 1, 8), (4, 2, 9)]

3. Sort the list of triples:

   After sorting, arr should look like this: [(0, 0, 1), (1, 0, 4), (1, 1, 2), (2, 0, 7), (2, 1, 5), (2, 2, 3), (3, 1, 8), (3, 2, 6), (4, 2, 9)]

   Notice the elements are grouped by the sum of their indices (the diagonals) and sorted by their column indices within each group.

4. Extract the results:

   By extracting the third element of each tuple, we get the diagonally ordered list of elements: [1, 4, 2, 7, 5, 3, 8, 6, 9]

That's the diagonal order traversal of the 2D nums grid using the proposed solution approach.

Solution Implementation

Time and Space Complexity

The time complexity of the code is determined by the number of operations it performs. The given Python code iterates over all the elements in the list of lists to create a flat list of tuples (arr). The iteration has a complexity of O(N), where N is the total number of elements in the nums list of lists. Additionally, the code sorts the arr list, which has O(N log N) complexity for the Timsort algorithm used in Python's sort() method. Since the sort is the most expensive operation here, the overall time complexity is O(N log N).

The space complexity of the code considers the additional space used by the algorithm outside the input and output data. The main extra space used is for the arr list of tuples. Since each element in the input is transformed into a tuple and stored in arr, the space complexity is O(N), where N is again the total number of elements in nums.

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