Added BCC and binary lifting

Author: bjorn-martinsson

pyrival/graphs/bcc.py

@@ -0,0 +1,52 @@
+"""
+Given a directed graph, find_SCC returns a list of lists containing 
+the strongly connected components in topological order.
+
+Note that this implementation can be also be used to check if a directed graph is a
+DAG, and in that case it can be used to find the topological ordering of the nodes.
+"""
+
+def find_SCC(graph):
+    SCC, S, P = [], [], []
+    depth = [0] * len(graph)
+ 
+    stack = list(range(len(graph)))
+    while stack:
+        node = stack.pop()
+        if node < 0:
+            d = depth[~node] - 1
+            if P[-1] > d:
+                SCC.append(S[d:])
+                del S[d:], P[-1]
+                for node in SCC[-1]:
+                    depth[node] = -1
+        elif depth[node] > 0:
+            while P[-1] > depth[node]:
+                P.pop()
+        elif depth[node] == 0:
+            S.append(node)
+            P.append(len(S))
+            depth[node] = len(S)
+            stack.append(~node)
+            stack += graph[node]
+    return SCC[::-1]
+
+"""
+Given an undirected simple graph, find_BCC returns a list of lists 
+containing the biconnected components of the graph. Runs in O(n + m) time.
+"""
+def find_BCC(graph):
+    d = 0
+    depth = [0] * len(graph)
+    stack = list(range(len(graph)))
+    while stack:
+        node = stack.pop()
+        if node < 0:
+            d -= 1
+        elif not depth[node]:
+            d = depth[node] = d + 1
+            stack.append(~node)
+            stack += graph[node]
+    
+    graph2 = [[v for v in g if d + 2 > depth[v] != d - 1] for g,d in zip(graph, depth)]
+    return find_SCC(graph2)

pyrival/graphs/binary_lifting_on_tree.py

@@ -0,0 +1,106 @@
+"""
+Binary lifting technique applied to a tree.
+There are three different uses of this implementation
+
+1. Computing LCA in O(log n) time.
+   
+   Example:
+           0
+           |
+           1
+          / \
+         2   3
+
+       graph = [[1], [0, 2, 3], [1], [1]]
+       BL = binary_lift(graph, root=0)
+       print(BL.lca(2, 3)) # prints 1
+
+2. Compute the distance between two nodes in O(log n) time.
+
+   Example:
+       graph = [[1], [0, 2, 3], [1], [1]]
+       BL = binary_lift(graph)
+       print(BL.distance(2, 3)) # prints 2
+
+3. Compute the sum/min/max/... of the weight 
+   of a path between a pair of nodes in O(log n) time.
+
+   res = Path[0]
+   for node in Path[1:]:
+     res = f(res, node)
+   return res
+   
+   Example:
+       
+       graph = [[1], [0, 2, 3], [1], [1]]
+       data = [1, 10, 20, 5]
+       BL = binary_lift(graph, data, f = lambda a,b: a + b)
+       print(BL(2, 3)) # prints 35
+"""
+
+class binary_lift:
+    def __init__(self, graph, data=(), f=min, root=0):
+        n = len(graph)
+
+        parent = [-1] * (n + 1)
+        depth = self.depth = [-1] * n
+        bfs = [root]
+        depth[root] = 0
+        for node in bfs:
+            for nei in graph[node]:
+                if depth[nei] == -1:
+                    parent[nei] = node
+                    depth[nei] = depth[node] + 1
+                    bfs.append(nei)
+
+        data = self.data = [data]
+        parent = self.parent = [parent]
+        self.f = f
+
+        for _ in range(max(depth).bit_length()):
+            old_data = data[-1]
+            old_parent = parent[-1]
+            
+            data.append([f(val, old_data[p]) for val,p in zip(old_data, old_parent)])
+            parent.append([old_parent[p] for p in old_parent])
+    
+    def lca(self, a, b):
+        depth = self.depth
+        parent = self.parent
+
+        if depth[a] < depth[b]:
+            a,b = b,a
+        
+        d = depth[a] - depth[b]
+        for i in range(d.bit_length()):
+            if (d >> i) & 1:
+                a = parent[i][a]
+
+        for i in range(depth[a].bit_length())[::-1]:
+            if parent[i][a] != parent[i][b]:
+                a = parent[i][a]
+                b = parent[i][b]
+
+        if a != b:
+            return parent[0][a]
+        else:
+            return a
+    
+    def distance(self, a, b):
+        return self.depth[a] + self.depth[b] - 2 * self.depth[self.lca(a,b)] 
+
+    def __call__(self, a, b):
+        depth = self.depth
+        parent = self.parent
+        data = self.data
+        f = self.f
+        
+        c = self.lca(a, b)
+        val = data[0][c]
+        for x,d in (a, depth[a] - depth[c]), (b, depth[b] - depth[c]):
+            for i in range(d.bit_length()):
+                if (d >> i) & 1:
+                    val = f(val, data[i][x])
+                    x = parent[i][x]
+
+        return val