Problem Description

In this problem, we are asked to implement a class called SubrectangleQueries which encapsulates a 2D rectangle grid of integers provided during instantiation. The class needs to support two operations:

1. updateSubrectangle(int row1, int col1, int row2, int col2, int newValue): This method allows updating all values within a specified subrectangle to a new given value. The subrectangle to be updated is defined by its upper left coordinate (row1, col1) and its bottom right coordinate (row2, col2).

2. getValue(int row, int col): This method retrieves the current value at a specific coordinate (row, col) in the rectangle.

These methods must efficiently reflect any updates made by updateSubrectangle when getValue is called.

Intuition

The naive approach to solve the updateSubrectangle operation would be to iterate over every cell in the subrectangle and update it to newValue. However, this could become inefficient when there are many updates before a call to getValue, especially if the subrectangle being updated is large.

To optimize this, we can use an approach where we track only the updates made rather than applying them immediately to the entire subrectangle. Whenever an update operation is performed, we record the details of the update—the coordinates of the subrectangle and the new value—in a list of operations, ops.

Then, when getValue is invoked for a specific cell, we iterate through the list of updates in reverse chronological order (latest operation first) because the most recent value is what we’re interested in. We check if the queried cell falls within the subrectangle of an update. If it does, we return the newValue from the first update operation that includes the cell. Otherwise, if no update operations include the cell, we return the original value of the cell from the initial rectangle grid.

This approach is more efficient in scenarios where there are multiple updates and fewer getValue calls, as it avoids unnecessary updates to the entire subrectangle when the value of only a few cells might be retrieved later.

Solution Approach

The solution for the SubrectangleQueries class leverages a key concept in programming known as lazy updating combined with the use of a history list to save update operations. Let's break down the two primary methods provided by the solution.

When the class is initialized with a 2D array representing the rectangle, we store this array and initialize an empty list self.ops to record update operations:

def __init__(self, rectangle: List[List[int]]):
    self.g = rectangle
    self.ops = []

The updateSubrectangle method doesn't modify the original grid immediately. Instead, it appends the update information as a tuple (row1, col1, row2, col2, newValue) to self.ops:

def updateSubrectangle(self, row1: int, col1: int, row2: int, col2: int, newValue: int) -> None:
    self.ops.append((row1, col1, row2, col2, newValue))

During the getValue method, we iterate backward through self.ops to check if the given row and col coordinates fall within any of the recorded subrectangles. If they do, it means that this was the last update that touched the cell before the getValue request, and the newValue from that update is returned immediately without checking earlier updates:

def getValue(self, row: int, col: int) -> int:
    for r1, c1, r2, c2, v in self.ops[::-1]:
        if r1 <= row <= r2 and c1 <= col <= c2:
            return v
    return self.g[row][col]

This approach is an application of the lazy evaluation pattern, as the updates to the grid are deferred and only evaluated when needed. By applying this strategy, the algorithm ensures that unnecessary cell updates are avoided, reducing the number of operations to be O(1) for each updateSubrectangle call, and O(k) for each getValue call, where k is the number of update operations.

In conclusion, the solution approach utilizes a history list mechanism to deftly manage multiple updates and retrieve operations without redundantly modifying the entire rectangle upon each update. This way, the process becomes markedly more efficient for scenarios involving many update operations and relatively few reads.

Example Walkthrough

Let's use a small example to illustrate the solution approach for the SubrectangleQueries class. Suppose we initialize our SubrectangleQueries class with the following 2D rectangle grid:

1 2 3
4 5 6
7 8 9

The grid is instantiated as:

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

1. We perform an update on the subrectangle from (0, 0) to (1, 1) with a new value of 10. Our update call will be like this:

subrectangleQueries.updateSubrectangle(0, 0, 1, 1, 10)

This will not change the grid immediately but will record this operation in self.ops.

2. If we now call the getValue method to retrieve the value at (0, 1), which is part of the recently updated subrectangle, the method will do the following:

value = subrectangleQueries.getValue(0, 1)

Since the most recent update included this cell with coordinates (0, 1), the method will return 10, which was the new value set for that subrectangle.

3. Let's add another update, changing the value of the subrectangle from (1, 1) to (2, 2) to 20:

subrectangleQueries.updateSubrectangle(1, 1, 2, 2, 20)

Again, this updates the operation list but leaves the grid unchanged.

4. Another call to getValue for the coordinate (1, 1) would now return 20, as this is the newest value for that location due to the latest update.

value = subrectangleQueries.getValue(1, 1)

5. If we ask for the value at (2, 0), which has not been touched by any update operations, the getValue method finds that there are no updates affecting it and thus returns the original value 7 from the original grid:

value = subrectangleQueries.getValue(2, 0)

Throughout the example, we see that the updateSubrectangle method appends update operation details to the self.ops list but doesn't alter the original grid itself. When retrieving a value with getValue, the method checks the updates in reverse chronological order to see if they affect the cell in question. If they do, the latest value is returned. If not, the original grid value is returned. This optimization allows efficient handling of updates and retrievals by deferring actual updates until needed.

Solution Implementation

Time and Space Complexity

Time Complexity

• __init__(self, rectangle: List[List[int]]): This method initializes the object with the given rectangle. The time complexity is O(1) since it's simply storing the reference to rectangle and initializing an empty list ops.

• updateSubrectangle(self, row1: int, col1: int, row2: int, col2: int, newValue: int) -> None: This method records an update operation by appending a tuple to the ops list representing the subrectangle update parameters. The time complexity for each update is O(1) because appending to a list in Python is an amortized constant time operation.

• getValue(self, row: int, col: int) -> int: This method retrieves the value of the cell at the specified row and column. It iterates over the ops list in reverse to find the most recent update that covers the cell in question. If k is the number of updates, the worst time complexity is O(k) because it might need to inspect every update in the worst case.

Space Complexity

• The space complexity for maintaining the rectangle is O(m * n), where m is the number of rows and n is the number of columns in the given rectangle, since it stores the entire grid.

• The space complexity for maintaining the ops list is O(u), where u is the number of update operations made. Each operation is stored as a tuple with five integers, so the total space taken by ops is proportional to the number of updates.

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