Merge a3cb0ee8a3 into 9a572dec2b

* feat: add Matrix Exponentiation method
docs: updated the header documentation and added new documentation for
the new function.

* feat: added new function matrix exponetiation method

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* feat: This function uses the tail-recursive form of the Euclidean algorithm to calculate

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* reduced the number of characters per line in the comments

* removed unwanted code

* feat: Implemented a new function to swaap numbers without dummy variable

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* removed previos code

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* Update maths/fibonacci.py

Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>

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---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>

graphs/kahns_algorithm_topo.py

@@ -1,36 +1,61 @@
-def topological_sort(graph):
+def topological_sort(graph: dict[int, list[int]]) -> list[int] | None:
     """
-    Kahn's Algorithm is used to find Topological ordering of Directed Acyclic Graph
+    Perform topological sorting of a Directed Acyclic Graph (DAG)
-    using BFS
+    using Kahn's Algorithm via Breadth-First Search (BFS).
+
+    Topological sorting is a linear ordering of vertices in a graph such that for
+    every directed edge u → v, vertex u comes before vertex v in the ordering.
+
+    Parameters:
+    graph: Adjacency list representing the directed graph where keys are
+           vertices, and values are lists of adjacent vertices.
+
+    Returns:
+    The topologically sorted order of vertices if the graph is a DAG.
+    Returns None if the graph contains a cycle.
+
+    Example:
+    >>> graph = {0: [1, 2], 1: [3], 2: [3], 3: [4, 5], 4: [], 5: []}
+    >>> topological_sort(graph)
+    [0, 1, 2, 3, 4, 5]
+
+    >>> graph_with_cycle = {0: [1], 1: [2], 2: [0]}
+    >>> topological_sort(graph_with_cycle)
     """
+
     indegree = [0] * len(graph)
     queue = []
-    topo = []
+    topo_order = []
-    cnt = 0
+    processed_vertices_count = 0
 
+    # Calculate the indegree of each vertex
     for values in graph.values():
         for i in values:
             indegree[i] += 1
 
+    # Add all vertices with 0 indegree to the queue
     for i in range(len(indegree)):
         if indegree[i] == 0:
             queue.append(i)
 
+    # Perform BFS
     while queue:
         vertex = queue.pop(0)
-        cnt += 1
+        processed_vertices_count += 1
-        topo.append(vertex)
+        topo_order.append(vertex)
-        for x in graph[vertex]:
-            indegree[x] -= 1
-            if indegree[x] == 0:
-                queue.append(x)
 
-    if cnt != len(graph):
+        # Traverse neighbors
-        print("Cycle exists")
+        for neighbor in graph[vertex]:
-    else:
+            indegree[neighbor] -= 1
-        print(topo)
+            if indegree[neighbor] == 0:
+                queue.append(neighbor)
+
+    if processed_vertices_count != len(graph):
+        return None  # no topological ordering exists due to cycle
+    return topo_order  # valid topological ordering
 
 
-# Adjacency List of Graph
+if __name__ == "__main__":
-graph = {0: [1, 2], 1: [3], 2: [3], 3: [4, 5], 4: [], 5: []}
+    import doctest
-topological_sort(graph)
+
+    doctest.testmod()

maths/fibonacci.py

@@ -7,6 +7,8 @@ the Binet's formula function because the Binet formula function  uses floats
 
 NOTE 2: the Binet's formula function is much more limited in the size of inputs
 that it can handle due to the size limitations of Python floats
+NOTE 3: the matrix function is the fastest and most memory efficient for large n
+
 
 See benchmark numbers in __main__ for performance comparisons/
 https://en.wikipedia.org/wiki/Fibonacci_number for more information
@@ -17,6 +19,9 @@ from collections.abc import Iterator
 from math import sqrt
 from time import time
 
+import numpy as np
+from numpy import ndarray
+
 
 def time_func(func, *args, **kwargs):
     """
@@ -230,6 +235,88 @@ def fib_binet(n: int) -> list[int]:
     return [round(phi**i / sqrt_5) for i in range(n + 1)]
 
 
+def matrix_pow_np(m: ndarray, power: int) -> ndarray:
+    """
+    Raises a matrix to the power of 'power' using binary exponentiation.
+
+    Args:
+        m: Matrix as a numpy array.
+        power: The power to which the matrix is to be raised.
+
+    Returns:
+        The matrix raised to the power.
+
+    Raises:
+        ValueError: If power is negative.
+
+    >>> m = np.array([[1, 1], [1, 0]], dtype=int)
+    >>> matrix_pow_np(m, 0)  # Identity matrix when raised to the power of 0
+    array([[1, 0],
+           [0, 1]])
+
+    >>> matrix_pow_np(m, 1)  # Same matrix when raised to the power of 1
+    array([[1, 1],
+           [1, 0]])
+
+    >>> matrix_pow_np(m, 5)
+    array([[8, 5],
+           [5, 3]])
+
+    >>> matrix_pow_np(m, -1)
+    Traceback (most recent call last):
+        ...
+    ValueError: power is negative
+    """
+    result = np.array([[1, 0], [0, 1]], dtype=int)  # Identity Matrix
+    base = m
+    if power < 0:  # Negative power is not allowed
+        raise ValueError("power is negative")
+    while power:
+        if power % 2 == 1:
+            result = np.dot(result, base)
+        base = np.dot(base, base)
+        power //= 2
+    return result
+
+
+def fib_matrix_np(n: int) -> int:
+    """
+    Calculates the n-th Fibonacci number using matrix exponentiation.
+    https://www.nayuki.io/page/fast-fibonacci-algorithms#:~:text=
+    Summary:%20The%20two%20fast%20Fibonacci%20algorithms%20are%20matrix
+
+    Args:
+        n: Fibonacci sequence index
+
+    Returns:
+        The n-th Fibonacci number.
+
+    Raises:
+        ValueError: If n is negative.
+
+    >>> fib_matrix_np(0)
+    0
+    >>> fib_matrix_np(1)
+    1
+    >>> fib_matrix_np(5)
+    5
+    >>> fib_matrix_np(10)
+    55
+    >>> fib_matrix_np(-1)
+    Traceback (most recent call last):
+        ...
+    ValueError: n is negative
+    """
+    if n < 0:
+        raise ValueError("n is negative")
+    if n == 0:
+        return 0
+
+    m = np.array([[1, 1], [1, 0]], dtype=int)
+    result = matrix_pow_np(m, n - 1)
+    return int(result[0, 0])
+
+
 if __name__ == "__main__":
     from doctest import testmod
 
@@ -242,3 +329,4 @@ if __name__ == "__main__":
     time_func(fib_memoization, num)  # 0.0100 ms
     time_func(fib_recursive_cached, num)  # 0.0153 ms
     time_func(fib_recursive, num)  # 257.0910 ms
+    time_func(fib_matrix_np, num)  # 0.0000 ms

strings/min_cost_string_conversion.py

@@ -17,11 +17,27 @@ def compute_transform_tables(
     delete_cost: int,
     insert_cost: int,
 ) -> tuple[list[list[int]], list[list[str]]]:
+    """
+    Finds the most cost efficient sequence
+    for converting one string into another.
+
+    >>> costs, operations = compute_transform_tables("cat", "cut", 1, 2, 3, 3)
+    >>> costs[0][:4]
+    [0, 3, 6, 9]
+    >>> costs[2][:4]
+    [6, 4, 3, 6]
+    >>> operations[0][:4]
+    ['0', 'Ic', 'Iu', 'It']
+    >>> operations[3][:4]
+    ['Dt', 'Dt', 'Rtu', 'Ct']
+
+    >>> compute_transform_tables("", "", 1, 2, 3, 3)
+    ([[0]], [['0']])
+    """
     source_seq = list(source_string)
     destination_seq = list(destination_string)
     len_source_seq = len(source_seq)
     len_destination_seq = len(destination_seq)
-
     costs = [
         [0 for _ in range(len_destination_seq + 1)] for _ in range(len_source_seq + 1)
     ]
@@ -31,33 +47,51 @@ def compute_transform_tables(
 
     for i in range(1, len_source_seq + 1):
         costs[i][0] = i * delete_cost
-        ops[i][0] = f"D{source_seq[i - 1]:c}"
+        ops[i][0] = f"D{source_seq[i - 1]}"
 
     for i in range(1, len_destination_seq + 1):
         costs[0][i] = i * insert_cost
-        ops[0][i] = f"I{destination_seq[i - 1]:c}"
+        ops[0][i] = f"I{destination_seq[i - 1]}"
 
     for i in range(1, len_source_seq + 1):
         for j in range(1, len_destination_seq + 1):
             if source_seq[i - 1] == destination_seq[j - 1]:
                 costs[i][j] = costs[i - 1][j - 1] + copy_cost
-                ops[i][j] = f"C{source_seq[i - 1]:c}"
+                ops[i][j] = f"C{source_seq[i - 1]}"
             else:
                 costs[i][j] = costs[i - 1][j - 1] + replace_cost
-                ops[i][j] = f"R{source_seq[i - 1]:c}" + str(destination_seq[j - 1])
+                ops[i][j] = f"R{source_seq[i - 1]}" + str(destination_seq[j - 1])
 
             if costs[i - 1][j] + delete_cost < costs[i][j]:
                 costs[i][j] = costs[i - 1][j] + delete_cost
-                ops[i][j] = f"D{source_seq[i - 1]:c}"
+                ops[i][j] = f"D{source_seq[i - 1]}"
 
             if costs[i][j - 1] + insert_cost < costs[i][j]:
                 costs[i][j] = costs[i][j - 1] + insert_cost
-                ops[i][j] = f"I{destination_seq[j - 1]:c}"
+                ops[i][j] = f"I{destination_seq[j - 1]}"
 
     return costs, ops
 
 
 def assemble_transformation(ops: list[list[str]], i: int, j: int) -> list[str]:
+    """
+    Assembles the transformations based on the ops table.
+
+    >>> ops = [['0', 'Ic', 'Iu', 'It'],
+    ...        ['Dc', 'Cc', 'Iu', 'It'],
+    ...        ['Da', 'Da', 'Rau', 'Rat'],
+    ...        ['Dt', 'Dt', 'Rtu', 'Ct']]
+    >>> x = len(ops) - 1
+    >>> y = len(ops[0]) - 1
+    >>> assemble_transformation(ops, x, y)
+    ['Cc', 'Rau', 'Ct']
+
+    >>> ops1 = [['0']]
+    >>> x1 = len(ops1) - 1
+    >>> y1 = len(ops1[0]) - 1
+    >>> assemble_transformation(ops1, x1, y1)
+    []
+    """
     if i == 0 and j == 0:
         return []
     elif ops[i][j][0] in {"C", "R"}: