Add shortest path by BFS (#1870)

* Create breadth_first_search_shortest_path.py

* updating DIRECTORY.md

* Reduce side effect of `shortest_path`

For the sake of future testing and documentation -

* fixup! Format Python code with psf/black push

* Fix typo `separately`

* Change to get() from dictionary

Co-Authored-By: Christian Clauss <cclauss@me.com>

* Move graph to the top

* fixup! Format Python code with psf/black push

* Add doctest for shortest path

* Add doctest for BFS

* fixup! Format Python code with psf/black push

* Add typings for breadth_first_search_shortest_path

* fixup! Format Python code with psf/black push

* Remove assert from doctests

* Add blank line to doctest

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
Co-authored-by: John Law <johnlaw.po@gmail.com>
Co-authored-by: Christian Clauss <cclauss@me.com>
Co-authored-by: John Law <johnlaw@linux.com>

DIRECTORY.md

@@ -213,6 +213,7 @@
   * [Bfs](https://github.com/TheAlgorithms/Python/blob/master/graphs/bfs.py)
   * [Bfs Shortest Path](https://github.com/TheAlgorithms/Python/blob/master/graphs/bfs_shortest_path.py)
   * [Breadth First Search](https://github.com/TheAlgorithms/Python/blob/master/graphs/breadth_first_search.py)
+  * [Breadth First Search Shortest Path](https://github.com/TheAlgorithms/Python/blob/master/graphs/breadth_first_search_shortest_path.py)
   * [Check Bipartite Graph Bfs](https://github.com/TheAlgorithms/Python/blob/master/graphs/check_bipartite_graph_bfs.py)
   * [Check Bipartite Graph Dfs](https://github.com/TheAlgorithms/Python/blob/master/graphs/check_bipartite_graph_dfs.py)
   * [Depth First Search](https://github.com/TheAlgorithms/Python/blob/master/graphs/depth_first_search.py)

graphs/breadth_first_search_shortest_path.py

@@ -0,0 +1,81 @@
+"""Breath First Search (BFS) can be used when finding the shortest path 
+from a given source node to a target node in an unweighted graph.
+"""
+graph = {
+    "A": ["B", "C", "E"],
+    "B": ["A", "D", "E"],
+    "C": ["A", "F", "G"],
+    "D": ["B"],
+    "E": ["A", "B", "D"],
+    "F": ["C"],
+    "G": ["C"],
+}
+
+from typing import Dict
+
+
+class Graph:
+    def __init__(self, graph: Dict[str, str], source_vertex: str) -> None:
+        """Graph is implemented as dictionary of adjancency lists. Also,
+        Source vertex have to be defined upon initialization.
+        """
+        self.graph = graph
+        # mapping node to its parent in resulting breadth first tree
+        self.parent = {}
+        self.source_vertex = source_vertex
+
+    def breath_first_search(self) -> None:
+        """This function is a helper for running breath first search on this graph.
+        >>> g = Graph(graph, "G")
+        >>> g.breath_first_search()
+        >>> g.parent
+        {'G': None, 'C': 'G', 'A': 'C', 'F': 'C', 'B': 'A', 'E': 'A', 'D': 'B'}
+        """
+        visited = {self.source_vertex}
+        self.parent[self.source_vertex] = None
+        queue = [self.source_vertex]  # first in first out queue
+
+        while queue:
+            vertex = queue.pop(0)
+            for adjancent_vertex in self.graph[vertex]:
+                if adjancent_vertex not in visited:
+                    visited.add(adjancent_vertex)
+                    self.parent[adjancent_vertex] = vertex
+                    queue.append(adjancent_vertex)
+
+    def shortest_path(self, target_vertex: str) -> str:
+        """This shortest path function returns a string, describing the result:
+        1.) No path is found. The string is a human readable message to indicate this.
+        2.) The shortest path is found. The string is in the form `v1(->v2->v3->...->vn)`,
+            where v1 is the source vertex and vn is the target vertex, if it exists separately.
+
+        >>> g = Graph(graph, "G")
+        >>> g.breath_first_search()
+
+        Case 1 - No path is found.
+        >>> g.shortest_path("Foo")
+        'No path from vertex:G to vertex:Foo'
+
+        Case 2 - The path is found.
+        >>> g.shortest_path("D")
+        'G->C->A->B->D'
+        >>> g.shortest_path("G")
+        'G'
+        """
+        if target_vertex == self.source_vertex:
+            return f"{self.source_vertex}"
+        elif not self.parent.get(target_vertex):
+            return f"No path from vertex:{self.source_vertex} to vertex:{target_vertex}"
+        else:
+            return self.shortest_path(self.parent[target_vertex]) + f"->{target_vertex}"
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+    g = Graph(graph, "G")
+    g.breath_first_search()
+    print(g.shortest_path("D"))
+    print(g.shortest_path("G"))
+    print(g.shortest_path("Foo"))