docs: explain Strassen's algorithm complexity in docstrings (#14925)

* docs: explain Strassen's algorithm complexity in docstrings

Expand actual_strassen() docstring to describe the divide-and-conquer
approach (7 recursive multiplications instead of 8) and note time
complexity O(n^log2(7)) ~= O(n^2.807) vs O(n^3) for naive matrix
multiplication, plus space complexity O(n^2). Also expand strassen()
docstring to explain the padding/trimming wrapper logic. No behavior
changes; all existing doctests pass.

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* Fix Ruff 0.16 lint failures

* Fix grammar in docstrings and comments

Corrected minor grammatical issues in docstrings and comments.

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---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>

Author: 3 people

divide_and_conquer/strassen_matrix_multiplication.py

@@ -1,3 +1,7 @@
+"""
+https://en.wikipedia.org/wiki/Strassen_algorithm
+"""
+
 from __future__ import annotations
 
 import math
@@ -32,7 +36,7 @@ def matrix_subtraction(matrix_a: list, matrix_b: list):
 
 def split_matrix(a: list) -> tuple[list, list, list, list]:
     """
-    Given an even length matrix, returns the top_left, top_right, bot_left, bot_right
+    Given an even-length matrix, returns the top_left, top_right, bot_left, bot_right
     quadrant.
 
     >>> split_matrix([[4,3,2,4],[2,3,1,1],[6,5,4,3],[8,4,1,6]])
@@ -75,6 +79,19 @@ def actual_strassen(matrix_a: list, matrix_b: list) -> list:
     """
     Recursive function to calculate the product of two matrices, using the Strassen
     Algorithm. It only supports square matrices of any size that is a power of 2.
+
+    Strassen's algorithm reduces the number of recursive multiplications needed to
+    multiply two n x n matrices from the 8 required by the naive divide-and-conquer
+    approach down to 7, at the cost of a few extra matrix additions/subtractions
+    (which are cheaper, O(n^2), operations). Each matrix is split into four
+    (n/2) x (n/2) quadrants; 7 products of quadrant combinations are computed
+    recursively, and those products are combined with additions/subtractions to
+    form the four quadrants of the result.
+
+    Time complexity: O(n^log2(7)) ~= O(n^2.807), an improvement over the O(n^3) of
+    the standard/naive matrix multiplication algorithm.
+    Space complexity: O(n^2) for storing the intermediate quadrant matrices, plus
+    O(log n) recursion stack depth.
     """
     if matrix_dimensions(matrix_a) == (2, 2):
         return default_matrix_multiplication(matrix_a, matrix_b)
@@ -106,6 +123,13 @@ def actual_strassen(matrix_a: list, matrix_b: list) -> list:
 
 def strassen(matrix1: list, matrix2: list) -> list:
     """
+    Multiplies two matrices using Strassen's algorithm, which runs in
+    O(n^log2(7)) ~= O(n^2.807) time, compared to O(n^3) for naive matrix
+    multiplication. This implementation pads both input matrices with zeros
+    until they are square matrices whose dimension is a power of 2 (required
+    by the divide-and-conquer recursion in actual_strassen), performs the
+    multiplication, then trims the padding back off the result.
+
     >>> strassen([[2,1,3],[3,4,6],[1,4,2],[7,6,7]], [[4,2,3,4],[2,1,1,1],[8,6,4,2]])
     [[34, 23, 19, 15], [68, 46, 37, 28], [28, 18, 15, 12], [96, 62, 55, 48]]
     >>> strassen([[3,7,5,6,9],[1,5,3,7,8],[1,4,4,5,7]], [[2,4],[5,2],[1,7],[5,5],[7,8]])