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K5  K 3,3

4 Basic Non-Planar Graphs

5 Some Thoughts Every Non-Planar Has a K 5 or K 3,3 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 T r be the DFS-subtree who’s root is r. We will view T r 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 T r 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 (T r ) 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  K 5 or K 3,3

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: |G i | – Number of Vertices and Edges in G i |C i | - Number of C-Nodes in G i |P i | - Number of P-Nodes in G i