Added graphs/travelling_salesman_problem.py

Author: XenoBytesX

graphs/travelling_salesman_problem.py

@@ -0,0 +1,96 @@
+"""
+Author:- Sanjay Muthu <https://github.com/XenoBytesX>
+
+The Problem:
+    In the travelling salesman problem you have to find the path
+    in which a salesman can start from a source node and visit all the other nodes
+    ONCE and return to the source node in the shortest possible way
+
+Complexity:
+    There are n! permutations of all the nodes in the graph and
+    for each permutation we have to go through all the nodes
+    which will take O(n) time per permutation.
+    So the total time complexity will be O(n * n!).
+
+    Time Complexity:- O(n * n!)
+
+    It takes O(n) space to store the list of nodes.
+
+    Space Complexity:- O(n)
+
+Wiki page:- <https://en.wikipedia.org/wiki/Travelling_salesman_problem>
+"""
+
+from itertools import permutations
+
+
+def floyd_warshall(graph, v):
+    dist = graph
+    for k in range(v):
+        for i in range(v):
+            for j in range(v):
+                dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j])
+    return dist
+
+
+def travelling_salesman_problem(graph, v):
+    """
+    Returns the shortest path that a salesman can take to visit all the nodes
+    in the graph and return to the source node
+
+    The graph in the below testcase looks something like this:-
+                                    2
+                        _________________________
+                        |                       |
+              2         ^                 6     ^
+    (0) <------------> (1)          (2)<------>(3)
+     ^                               ^
+     |              3                |
+     |_______________________________|
+
+    >>> graph = []
+    >>> graph.append([0, 2, 3, float('inf')])
+    >>> graph.append([2, 0, float('inf'), 2])
+    >>> graph.append([3, float('inf'), 0, 6])
+    >>> graph.append([float('inf'), 2, 6, 0])
+    >>> travelling_salesman_problem(graph, 4)
+    13
+    """
+
+    # See graphs_floyd_warshall.py for implementation
+    shortest_distance_matrix = floyd_warshall(graph, v)
+    nodes = list(range(v))
+
+    min_distance = float("inf")
+
+    # Go through all the permutations of the nodes to find out
+    # which of the order of nodes lead to the shortest distance
+    # Permutations(nodes) returns a list of all permutations of the list
+    # Permutations are all the possible ways to arrange the elements of the list
+    for permutation in permutations(nodes):
+        current_dist = 0
+        current_node = 0
+
+        # Find the distance of the current permutation by going through all the nodes
+        # in order and add the distance between the nodes to the current_dist
+        for node in permutation:
+            current_dist += shortest_distance_matrix[current_node][node]
+            current_node = node
+
+        # Add the distance to come back to the source node
+        current_dist += shortest_distance_matrix[current_node][0]
+
+        # If the current distance is less than the minimum distance found so far,
+        # update the minimum distance
+        # print(permutation, current_dist)
+        min_distance = min(min_distance, current_dist)
+
+    # NOTE: We are assuming 0 is the source node here.
+
+    return min_distance
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()