1. Articulation Points 2 of 2 (Algorithm)
COMP261 Lecture 9
Articulation Points 2 of 2 (Algorithm)

2. Outline Recursive Articulation Points Finding
Iterative Articulation Points Finding

3. Articulation Points Key ideas of algorithm:
Record count number of nodes as you search
From each recursive search of a subtree, return the minimum count number that the subtree nodes can "reach back" to.
Compare the "reach back" of each subtree to count number of this node, if smaller then there is an alternative path

4. Articulation points: Recursive
Initialise: for each node: node.count ← , articulationPoints ← { } start.count ← 0, numSubtrees ← 0 for each neighbour of start if neighbour.count =  then recArtPts(neighbour, 1, start) numSubtrees ++ if numSubtrees > 1 then add start to articulationPoints recArtPts(node, count, fromNode): node.count ← count, reachBack ← count, for each neighbour of node other than fromNode if neighbour.count <  then reachBack = min(neighbour.count, reachBack) else childReach = recArtPts(neighbour, count +1, node) if childReach >= count then add node to articulationPoints reachBack = min(childReach, reachBack ) return reachBack B C D A E M P L J
Back edge
No back edge from the child Q

5. Recursive algorithm
count: 2
parent: F
reachBack: 2
Recursion may be annoying Can we do it iteratively? G 2
count: 3
parent: G
reachBack: 3 J H 3
count: 4
parent: J
reachBack: 4
\ 3 L 4 3
count: 5
parent: L
reachBack: 5 N
\ 3 M M 5 6 3 P
count: 7
parent: M
reachBack: 7
count: 6
parent: M
reachBack: 6 Q 7 6
\ 4
\ 3

6. Removing Recursion The process is essentially a DFS
Use stack for DFS (last-in-first-out)
iterDFS:
initialise an empty stack
Pick a root, push root onto stack
repeat until stack is empty:
pop node from stack
process node
for each unvisited neighbour of node
push neighbour onto stack

7. Articulation points: Iterative
Initialise: for each node: node.count ←  articulationPoints ← { } start.count ← 0, numSubtrees ← 0 for each neighbour of start if neighbour.count =  then recArtPts(neighbour, 1, start) numSubtrees ++ if numSubtrees > 1 then add start to articulationPoints
iterArtPts(neighbour, 1, start)

8. Articulation points: Iterative
iterArtPts(node, count, root): stack ← an empty stack, push <node, count, root> onto stack repeat until stack is empty: pop <node, count, fromNode> from stack node.count ← count, node.reachBack ← count updateReachBack(node) for each neighbour of node other than fromNode push <neighbour, count+1, node> onto stack Require calculation of neighbour.reachBack
Process node
How to update?

9. Articulation points: Iterative
iterArtPts(node, count, root): stack ← an empty stack, push <node, count, root> onto stack repeat until stack is empty: pop <node, count, fromNode> from stack node.count ← count, node.reachBack ← count for each neighbour of node other than fromNode if neighbour.count <  then node.reachBack = min(neighbour.count, node.reachBack) else childReach = recArtPts(neighbour, count+1, node) if childReach >= count then add node to articulationPoints node.reachBack = min(childReach, node.reachBack) push <neighbour, count+1, node> onto stack
Process node
Still need recursion for childReach

10. Articulation points: Iterative
Without recursion, a node needs to be processed
When first reached: initialise count, reachBack, …
After processing each neighbour: update reachBack
After the last neighbour: transfer reachBack to fromNode
However, the simple stack only processes a node when first reached
Need more sophisticated stack

11. Articulation points: Iterative
A more sophisticated stack
while stack is not empty
peek at element on top of stack
perform next action of element
Initialise count/reachBack
Process neighbour (may push onto the stack)
Transfer reachBack to fromNode
if all actions have been done then
remove element from stack

12. Stack for Iterative Articulation Points
Stack elements contain
node: graph node to be processed
count
reachBack
children: unvisited neighbours
fromNode
When peek at an element
First time: initialise count, reachBack and children
Process a child: poll a child; if visited, update reachBack; otherwise push onto stack
Last time: determine if fromNode is articulation point, update fromNode’s reachBack

13. Articulation points: Iterative
Initialise: for each node: node.count ←  articulationPoints ← { } start.count ← 0, numSubtrees ← 0 for each neighbour of start if neighbour.count =  then iterArtPts(neighbour, 1, start) numSubtrees ++ if numSubtrees > 1 then add start to articulationPoints

14. Articulation Points with Stack
iterArtPts (firstNode, count, root): push <firstNode, count, root> onto stack repeat until stack is empty: peek at <node, count, fromNode> from stack if node.count =  then node.count ← count, node.reachBack ← count, node.children ← new queue for each neighbour of node if neighbour ≠ fromNode then add neighbour to node.children else if node.children not empty then child ← node.children.dequeue() if child.count < ∞ then node.reachBack ← min(node.reachBack, child.count) else push 〈child, count+1, node〉 onto stack else if node ≠ firstNode then if node.reachBack ≥ fromNode.count then add fromNode to articulationPoints fromNode.reachBack = min (fromNode.reachBack, node.reachBack) remove <node, count, fromNode> from stack
First time
Children to
process
Last time