Efficient Cumulative Sum: Optimize Your Python Code

Tired of algorithms that make you wait longer than the coffee line on a Monday morning? ☕ We have the solution.

🔮 Problem Statement

Develop a function max_after_operations_master(n, queries) that optimizes array manipulation and maximum search after multiple updates. 🤖

The problem consists of initializing an array of zeros with n positions. You also receive a matrix queries of m rows and 3 columns. Each row of the matrix represents a sum operation that must be applied to the array.

n (int): Number of positions in the array that will be initialized with zeros.
queries (int[m][3]): Matrix of integers with m rows and 3 columns. Each row contains:
- a: Initial index (inclusive) of the range to sum.
- b: Final index (inclusive) of the range to sum.
- k: Value to be added to each element of the range a to b.

The function must perform the sum operations described in the queries matrix, applying the value k to the range of indices specified by a and b (both inclusive). After completing all operations, the function must return the maximum value found in the array.

IMPORTANT NOTE: The function must be optimized to run in less than one second.

Example:

>>> max_after_operations_master(10, [[1, 3, 5], [2, 5, 1]])
6

Visual Procedure:

Considering the queries matrix:

a   b   k
1   3   5
2   5   1

Array initialized to 0 (size 10): [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
Applying the first operation (row 1): [5, 5, 5, 0, 0, 0, 0, 0, 0, 0]
Applying the second operation (row 2): [5, 6, 6, 1, 1, 0, 0, 0, 0, 0]

The maximum value in the final array is 6.

🧩 Step-by-Step Solution

The solution is based on the “cumulative differences” technique to achieve high efficiency. Instead of iterating over each range and adding the value k, we modify the array only at the beginning and end of the range. Subsequently, we perform a final pass to calculate the cumulative sums and find the maximum.

operations = [0] * n

We initialize a list operations of size n with all elements set to zero. This list will act as our canvas where we will paint the effects of each operation.

for query in queries:
    a, b, k = query
    operations[a - 1] += k
    operations[b] -= k

We iterate through the queries matrix. For each row (operation), we extract the values a, b, and k. We increment the value at position a - 1 (initial index) by k and decrement the value at position b (just after the final index) by k. This is the heart of the optimization: instead of adding k to the entire range, we mark the start and end of the increment.

maxval = 0
itt = 0
for num in operations:
    itt += num
    if itt > maxval:
        maxval = itt

Finally, we iterate over the operations list, calculating the cumulative sum at each position. We keep track of the maximum value found so far. The magic lies in the fact that the cumulative sum “propagates” the increments and decrements, simulating the effect of adding k to each range.

Complete Solution:

def max_after_operations_master(n, queries):
	"level: master; points: 10; time: 1"
	operations = [0] * n
	for query in queries:
		a, b, k = query
		operations[a - 1] += k
		operations[b] -= k
	maxval = 0
	itt = 0
	for num in operations:
		itt += num
		if itt > maxval:
			maxval = itt
	return maxval

🧠 Key Concepts

The central concept is the cumulative differences technique. This technique transforms a range update problem into a problem of updating individual points, followed by a prefix sum calculation. The key is to understand that adding k to a range [a, b] is equivalent to adding k at position a and subtracting k at position b+1. This avoids iterating over the entire range for each operation, drastically reducing the time complexity.

The time complexity of the resulting algorithm is O(m + n), where m is the number of queries and n is the size of the array. This is significantly more efficient than a naive solution that would iterate over each range for each query, which would have a complexity of O(m * n * (average range size)).

Did you know…? The cumulative differences technique is not only used in array manipulation problems, but also in image and signal processing to perform convolution operations efficiently.

💫 Final Thoughts

This solution demonstrates the power of algorithmic optimization. The cumulative differences technique transforms a problem that could be prohibitively slow into an efficient task.

Possible improvements could include parallelizing the cumulative sum calculation, especially on architectures with multiple cores. However, for typical test cases, the current solution is already fast enough.

If you want to master these techniques and become a master of optimization, I invite you to explore more articles on this blog. The path of efficient code awaits you! 😉