1

PQ Trees, PC Trees, and Planar Graphs

Hsu & McConnell

Presented by Roi Barkan

2

Outline

Planar Graphs Definition Some thoughts Basic Strategy

PC Tree Algorithm Review of PC Trees The Algorithm Complexity Analysis

3

Planar Graphs

Graphs that can be drawn on a plane, without intersecting edges.

Examples: Borders Between Countries Trees, Forests Simple Cycles

Counter-Examples: K5

K3,3

4

Basic Non-Planar Graphs

5

Some Thoughts

Every Non-Planar Has a K5 or K3,3 Subgraph (Kuratowski, 1930)

Articulation Vertices – Divide and Conquer.

The Problem – Cycles Everything else is either inside or outside. Cut-Set Cycle in a Graph– A cycle that

breaks the graph’s connectivity.

6

More Thoughts

Small Variations on Trees are allowed Connect one vertex to another. Connect all leaves to the root.

“Minimal” Biconnected Tree Biconnected: No Articulation Vertices Root Must Have a Single Child. Connect Root to a Descendant of its Child. Connect All Leaves to Ancestors.

7

Existing Solutions

Hopcroft & Tarjan – 1974 First Linear Time Solution

PQ-Tree related solutions Early 90’s Fairly complicated

PC-Tree Solution – 1999 Presented here

8

Our Strategy

Find a Cut-Set Cycle Replace it With a Simple Place-Holder Recursively Work on the Inside and

the Outside of the Graph Put Everything Back Together On Error – Find a Kuratowski Sub-

Graph.

9

Some Preprocessing

Split by Articulation Vertices DFS Scan the Graph

Reminder: Back-Edges connect vertices with their ancestors.

Label the Vertices According to a Post-Order Traversal of the DFS-Tree

Keep a List of Back-Vertices, Sorted in Ascending (Post-)Order

10

Data Representation

Same building blocks as in PC-Trees. P-nodes represent regular vertices in the

graph (store their label) C-nodes represent “place-holders” for

cycles Very intuitive: reserves cyclic order of edges.

Each P-node Knows its DFS-Parent, Each Tree-Edge Knows Who’s the Parent

11

Flow of The Algorithm

On each step we will work on a single back-vertex, according to their order in the list.

Recursion Ends when the list only holds the root. The root has to be a back-vertex When it’s the only back-vertex we know

how to embed the graph

12

Finding a Cut-Set Cycle

Let i be the current back-vertex. r is i’s son on the path to the back edge.

Let Tr be the DFS-subtree who’s root is r.

We will view Tr as a PC-tree No internal back edges (i is minimal) Back edges to i will be considered full Back edges to ancestors of i will be

considered empty

13

Full-Partial Labeling

X is a Full Vertex When (deg(x)-1) of its Neighbors are Full

X is an Empty Vertex When (deg(x)-1) of its Neighbors are Empty

X is Partial if it is not Full and not Empty A Terminal Edge Connects Partial

Vertices. Algorithm: Start with Full Leaves, and

Scan Up the PC-Graph

14

Terminal Path

All Terminal Edges Are Connected Claim: If they do not form a Path

The Graph isn’t Planar. (Proved Later) Claim: If we can’t Flip Full and Empty

Vertices to Opposite side of the Path The Graph isn’t Planar. (Proved Later)

15

Tr As a PC-Tree

16

The Actual Cut-Set Cycle

17

Pitfall

A Vertex can have a degree of 2. It can Full and Empty Ambiguous. Easily detected:

X’s Full Neighbor is the Only One Left on the Update List.

Deg(X) = 2 In That Case – Only X is a Terminal

Node.

18

Complete Labeling Algorithm

L List of Full Leaves While L is not Empty

X = pop(L) If L is Empty, and X has a neighbor of

degree 2 output X as the Only Terminal Node.

For each Neighbor Y of X Increment Y’s counter. If the counter reached deg(Y)-1 Add Y to L.

19

Finding The Terminal Path

Climb Up the DFS Tree From All Partial Vertices, in an Interlaced Manner.

Trim a Possible Apex. If We End Up with a Path Output it. If We End Up with a Tree Non-

Planar.

20

Our Strategy - Reminder

Find a Cut-Set Cycle Replace it With a Simple Place-Holder Recursively Work on the Inside and

the Outside of the Graph Put Everything Back Together On Error – Find a Kuratowski Sub-

Graph.

21

Place-Holder For A Cycle

22

Fine Details

Resulting Graph has the Same Number of Edges, But One Less Cycle.

A Vertex of Degree 2 on the Cycle Becomes an Articulation Vertex Avoid it by Contracting The Vertex.

We Want to Avoid Having Neighboring C Nodes Contract them as well

23

Fine Details (2)

Parent-bits of the edges need to be kept consistent. Actually not very hard. The new C-node is a Child of i. Vertices in the terminal path that lost their

parents are “adopted” by the new C-node Edge contraction is easy to fix.

24

How It Is Done

25

How It Is Done (2)

26

How It Is Done (3)

27

Another Example

28

Another Example (2)

29

Another Example (3)

30

Another Example (4)

31

Our Strategy - Reminder

Find a Cut-Set Cycle Replace it With a Simple Place-Holder Recursively Work on the Inside and

the Outside of the Graph Put Everything Back Together On Error – Find a Kuratowski Sub-

Graph.

32

Recursive Work

The Inner Side of the Cycle is Easy A tree (Tr) where all the leaves, and the

root are connected to a single node – i. The Outer Side is Done Recursively

The new graph has fewer back-edges, and thus is simpler.

33

Putting It Together

Each Phase Remembers: Contracted Edges The C-node it created

Connecting Inner and Outer Parts is Easy The C-node preserves cyclic order.

34

Handling Errors

Two Possible Cases: Terminal edges form a tree. Terminal path can’t be flipped correctly.

Basic Idea Walk from i down the tree to problematic

leaves Move up through back edges Down the tree back to i K5 or K3,3

35

Terminal Edges Form a Tree

36

Points to Remember

Full Leaves have Back-Edges to i. Empty Leaves have Back-Edges to

ancestors of i. Every C-node in the Graph originated

from a cycle Every path through a C-node, represents

two paths in the original graph.

37

Paths Through C-Nodes

38

When W is a P-Node

39

When W is a C-Node (1)

40

When W is a C-Node (2)

41

When W is a C-Node (2.1)

42

When W is a C-Node (2.2)

43

A Path that won’t Flip

The Problematic Node is a C-Node Has an Empty and a Full Sub-Tree on

the Same Side of the Path Terminal Edges Lead to Empty/Full

Sub-Trees As Well

44

A Path That Won’t Flip (2)

45

Complexity Analysis

Preprocessing DFS Scan Post-order scan of the DFS tree

Actual Processing Finding Terminal Path Splitting Graph, Putting Back Together Ordering Internal Sub-Tree Recursive Call

46

Complexity Analysis (2)

Preprocessing – Linear Finding Terminal Path

Every Node scanned not on the Terminal Path will be removed from the graph.

Only need to worry about total lengths of terminal paths

Splitting and Joining O(Terminal Path) Sub-Trees Once for each node.

47

How Long Are The Paths

Vertex is on the Terminal Path Only two are at the edges Linear cost Others lose at least one edge (to full

subtree) O(|E|) total amount

48

Some Minor Details

Terminal Path Flipping is Easy P-nodes sorted through the labeling

process Flipping done in O(1) due to cyclic lists.

Terminal Path Splitting is Easy Simply use some pointer tricks

49

Summary

Intuitive, Linear Algorithm to a Non-Trivial Problem

Data Structure Actually Represents the Problem Domain

A Great Way to End A Great Seminar

50

Questions ?

51

Complexity Analysis

Use Amortized Complexity If we work hard in phase i, we make the job

easier for the next phases. Potential Function:

|Gi| – Number of Vertices and Edges in Gi

|Ci| - Number of C-Nodes in Gi

|Pi| - Number of P-Nodes in Gi

, 2 deg 1i

i ix P

M i G C x