Improve comments, add doctests for kahns_algorithm_topo.py (TheAlgori…

…thms#11668)

* Improve comments, add doctests for kahns_algorithm_topo.py

* Improve function docstring

* [pre-commit.ci] auto fixes from pre-commit.com hooks

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* Rename variables, remove print

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Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.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
-    using BFS
+    Perform topological sorting of a Directed Acyclic Graph (DAG)
+    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 = []
-    cnt = 0
+    topo_order = []
+    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
-        topo.append(vertex)
-        for x in graph[vertex]:
-            indegree[x] -= 1
-            if indegree[x] == 0:
-                queue.append(x)
-
-    if cnt != len(graph):
-        print("Cycle exists")
-    else:
-        print(topo)
-
-
-# Adjacency List of Graph
-graph = {0: [1, 2], 1: [3], 2: [3], 3: [4, 5], 4: [], 5: []}
-topological_sort(graph)
+        processed_vertices_count += 1
+        topo_order.append(vertex)
+
+        # Traverse neighbors
+        for neighbor in graph[vertex]:
+            indegree[neighbor] -= 1
+            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
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()