Add function docstrings, comments and type hints (#10893)

* Add function docstrings, comments and type hints

* Fix type mismatch

* Fix type hint error

* Fix float to int error

* Update ford_fulkerson.py

* Update ford_fulkerson.py

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

for more information, see https://pre-commit.ci

* Update ford_fulkerson.py

---------

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

networking_flow/ford_fulkerson.py

@@ -1,39 +1,95 @@
-# Ford-Fulkerson Algorithm for Maximum Flow Problem
 """
+Ford-Fulkerson Algorithm for Maximum Flow Problem
+* https://en.wikipedia.org/wiki/Ford%E2%80%93Fulkerson_algorithm
+
 Description:
-    (1) Start with initial flow as 0;
-    (2) Choose augmenting path from source to sink and add path to flow;
+    (1) Start with initial flow as 0
+    (2) Choose the augmenting path from source to sink and add the path to flow
 """
+graph = [
+    [0, 16, 13, 0, 0, 0],
+    [0, 0, 10, 12, 0, 0],
+    [0, 4, 0, 0, 14, 0],
+    [0, 0, 9, 0, 0, 20],
+    [0, 0, 0, 7, 0, 4],
+    [0, 0, 0, 0, 0, 0],
+]
 
 
-def bfs(graph, s, t, parent):
-    # Return True if there is node that has not iterated.
-    visited = [False] * len(graph)
-    queue = []
-    queue.append(s)
-    visited[s] = True
+def breadth_first_search(graph: list, source: int, sink: int, parents: list) -> bool:
+    """
+    This function returns True if there is a node that has not iterated.
+
+    Args:
+        graph: Adjacency matrix of graph
+        source: Source
+        sink: Sink
+        parents: Parent list
+
+    Returns:
+        True if there is a node that has not iterated.
+
+    >>> breadth_first_search(graph, 0, 5, [-1, -1, -1, -1, -1, -1])
+    True
+    >>> breadth_first_search(graph, 0, 6, [-1, -1, -1, -1, -1, -1])
+    Traceback (most recent call last):
+        ...
+    IndexError: list index out of range
+    """
+    visited = [False] * len(graph)  # Mark all nodes as not visited
+    queue = []  # breadth-first search queue
+
+    # Source node
+    queue.append(source)
+    visited[source] = True
 
     while queue:
-        u = queue.pop(0)
-        for ind in range(len(graph[u])):
-            if visited[ind] is False and graph[u][ind] > 0:
+        u = queue.pop(0)  # Pop the front node
+        # Traverse all adjacent nodes of u
+        for ind, node in enumerate(graph[u]):
+            if visited[ind] is False and node > 0:
                 queue.append(ind)
                 visited[ind] = True
-                parent[ind] = u
-
-    return visited[t]
+                parents[ind] = u
+    return visited[sink]
 
 
-def ford_fulkerson(graph, source, sink):
-    # This array is filled by BFS and to store path
+def ford_fulkerson(graph: list, source: int, sink: int) -> int:
+    """
+    This function returns the maximum flow from source to sink in the given graph.
+
+    CAUTION: This function changes the given graph.
+
+    Args:
+        graph: Adjacency matrix of graph
+        source: Source
+        sink: Sink
+
+    Returns:
+        Maximum flow
+
+    >>> test_graph = [
+    ...     [0, 16, 13, 0, 0, 0],
+    ...     [0, 0, 10, 12, 0, 0],
+    ...     [0, 4, 0, 0, 14, 0],
+    ...     [0, 0, 9, 0, 0, 20],
+    ...     [0, 0, 0, 7, 0, 4],
+    ...     [0, 0, 0, 0, 0, 0],
+    ... ]
+    >>> ford_fulkerson(test_graph, 0, 5)
+    23
+    """
+    # This array is filled by breadth-first search and to store path
     parent = [-1] * (len(graph))
     max_flow = 0
-    while bfs(graph, source, sink, parent):
-        path_flow = float("Inf")
+
+    # While there is a path from source to sink
+    while breadth_first_search(graph, source, sink, parent):
+        path_flow = int(1e9)  # Infinite value
         s = sink
 
         while s != source:
-            # Find the minimum value in select path
+            # Find the minimum value in the selected path
             path_flow = min(path_flow, graph[parent[s]][s])
             s = parent[s]
 
@@ -45,17 +101,12 @@ def ford_fulkerson(graph, source, sink):
             graph[u][v] -= path_flow
             graph[v][u] += path_flow
             v = parent[v]
+
     return max_flow
 
 
-graph = [
-    [0, 16, 13, 0, 0, 0],
-    [0, 0, 10, 12, 0, 0],
-    [0, 4, 0, 0, 14, 0],
-    [0, 0, 9, 0, 0, 20],
-    [0, 0, 0, 7, 0, 4],
-    [0, 0, 0, 0, 0, 0],
-]
+if __name__ == "__main__":
+    from doctest import testmod
 
-source, sink = 0, 5
-print(ford_fulkerson(graph, source, sink))
+    testmod()
+    print(f"{ford_fulkerson(graph, source=0, sink=5) = }")