feat(algs/aps): add backedge method of finding articulation points

Author: jessicayung

algorithms/articulation_points.py

@@ -16,10 +16,13 @@ class find_articulation_points(Graph):
     def __init__(self, adjacency_matrix, num_vertices):
         Graph.__init__(self, adjacency_matrix, num_vertices)
         self.saved_adj_mat = self.adj_mat
+        self.backedges = np.zeros(num_vertices)
+        self.potential_articulation_points = [[] for i in range(num_vertices)]
+        self.articulation_points = []
+        # self.min_discovery_time = np.full(num_vertices, num_vertices)
 
     def brute_force(self):
         """Alternative brute force approach: find islands using a DFS for each vertex."""
-        self.articulation_points = []
         original_islands = self.find_num_islands()
         print("Original number of islands: ", original_islands)
         for vertex in range(self.v):
@@ -32,10 +35,45 @@ def brute_force(self):
             self.adj_mat = self.saved_adj_mat
         return self.articulation_points
 
-adjmat = np.zeros((6,6))
-edges = [[0,1],[0,5],[1,2],[1,3],[2,3],[2,4],[3,4]]
-test1 = find_articulation_points(adjmat, 6)
-test1.add_edges(edges)
-test1.brute_force()
-print(test1.articulation_points)
+    def back_edge(self, start):
+        """ Back edge method of finding articulation points.
+
+        Back edge: an edge that connects a node to an ancestor of its parent.
+
+        Claim: If a vertex u has a child v and none of the vertices in the v-subtree have a back edge to ANY vertex discovered before u (this works bc data structure is a tree), u is an articulation point.
+        - i.e. if u is removed, then the v-subtree will be disconnected from the graph.
+        """
+        # print("at vertex", start)
+        if self.visited[start]:
+            # print("Already visited")
+            return
+
+        # print("mark as visited")
+        # Mark current cell as visited
+        self.visited[start] = True
+        self.steps_taken[start] = self.steps
+        self.steps += 1
+        # print(self.visited)
+
+        # Visit every neighbouring cell
+        for cell in self.neighbour(start):
+            if self.steps_taken[start] > self.steps_taken[cell]:
+                self.backedges[start] += 1
+                self.potential_articulation_points[start].append(cell)
+            self.back_edge(cell)
+        if self.backedges[start] == 1:
+            ap = self.potential_articulation_points[start][0]
+            if ap not in self.articulation_points:
+                self.articulation_points.append(ap)
+
+
+def test1():
+    adjmat = np.zeros((6,6))
+    edges = [[0,1],[0,5],[1,2],[1,3],[2,3],[2,4],[3,4]]
+    test1 = find_articulation_points(adjmat, 6)
+    test1.add_edges(edges)
+    test1.back_edge(1)
+    # test1.brute_force()
+    print("Articulation points:", test1.articulation_points)
 
+test1()