Merge pull request TheAlgorithms#239 from damelLP/graph_algos

Added Tarjan's algorithm for finding strongly connected components

Graphs/tarjans_scc.py

@@ -0,0 +1,78 @@
+from collections import deque
+
+
+def tarjan(g):
+    """
+    Tarjan's algo for finding strongly connected components in a directed graph
+
+    Uses two main attributes of each node to track reachability, the index of that node within a component(index),
+    and the lowest index reachable from that node(lowlink).
+
+    We then perform a dfs of the each component making sure to update these parameters for each node and saving the
+    nodes we visit on the way.
+
+    If ever we find that the lowest reachable node from a current node is equal to the index of the current node then it
+    must be the root of a strongly connected component and so we save it and it's equireachable vertices as a strongly
+    connected component.
+
+    Complexity: strong_connect() is called at most once for each node and has a complexity of O(|E|) as it is DFS.
+    Therefore this has complexity O(|V| + |E|) for a graph G = (V, E)
+
+    """
+
+    n = len(g)
+    stack = deque()
+    on_stack = [False for _ in range(n)]
+    index_of = [-1 for _ in range(n)]
+    lowlink_of = index_of[:]
+
+    def strong_connect(v, index, components):
+        index_of[v] = index  # the number when this node is seen
+        lowlink_of[v] = index  # lowest rank node reachable from here
+        index += 1
+        stack.append(v)
+        on_stack[v] = True
+
+        for w in g[v]:
+            if index_of[w] == -1:
+                index = strong_connect(w, index, components)
+                lowlink_of[v] = lowlink_of[w] if lowlink_of[w] < lowlink_of[v] else lowlink_of[v]
+            elif on_stack[w]:
+                lowlink_of[v] = lowlink_of[w] if lowlink_of[w] < lowlink_of[v] else lowlink_of[v]
+
+        if lowlink_of[v] == index_of[v]:
+            component = []
+            w = stack.pop()
+            on_stack[w] = False
+            component.append(w)
+            while w != v:
+                w = stack.pop()
+                on_stack[w] = False
+                component.append(w)
+            components.append(component)
+        return index
+
+    components = []
+    for v in range(n):
+        if index_of[v] == -1:
+            strong_connect(v, 0, components)
+
+    return components
+
+
+def create_graph(n, edges):
+    g = [[] for _ in range(n)]
+    for u, v in edges:
+        g[u].append(v)
+    return g
+
+
+if __name__ == '__main__':
+    # Test
+    n_vertices = 7
+    source = [0, 0, 1, 2, 3, 3, 4, 4, 6]
+    target = [1, 3, 2, 0, 1, 4, 5, 6, 5]
+    edges = [(u, v) for u, v in zip(source, target)]
+    g = create_graph(n_vertices, edges)
+
+    assert [[5], [6], [4], [3, 2, 1, 0]] == tarjan(g)