PQ Trees, PC Trees, and Planar Graphs

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PQ Trees, PC Trees, and Planar Graphs

• Hsu McConnell
• Presented by Roi Barkan

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Outline

• Planar Graphs
• Definition
• Some thoughts
• Basic Strategy
• PC Tree Algorithm
• Review of PC Trees
• The Algorithm
• Complexity Analysis

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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

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Basic Non-Planar Graphs
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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
  graphs connectivity.

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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.

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Existing Solutions

• Hopcroft Tarjan 1974
• First Linear Time Solution
• PQ-Tree related solutions
• Early 90s
• Fairly complicated
• PC-Tree Solution 1999
• Presented here

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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.

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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

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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 Whos the Parent

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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 its the only back-vertex we know how to
  embed the graph

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Finding a Cut-Set Cycle

• Let i be the current back-vertex.
• r is is son on the path to the back edge.
• Let Tr be the DFS-subtree whos 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

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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

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Terminal Path

• All Terminal Edges Are Connected
• Claim If they do not form a Path ? The Graph
  isnt Planar. (Proved Later)
• Claim If we cant Flip Full and Empty Vertices
  to Opposite side of the Path ? The Graph isnt
  Planar. (Proved Later)

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Tr As a PC-Tree
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The Actual Cut-Set Cycle
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Pitfall

• A Vertex can have a degree of 2.
• It can Full and Empty ? Ambiguous.
• Easily detected
• Xs Full Neighbor is the Only One Left on the
  Update List.
• Deg(X) 2
• In That Case Only X is a Terminal Node.

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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 Ys counter.
• If the counter reached deg(Y)-1 ? Add Y to L.

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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.

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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.

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Place-Holder For A Cycle
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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

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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.

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How It Is Done
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How It Is Done (2)
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How It Is Done (3)
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Another Example
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Another Example (2)
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Another Example (3)
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Another Example (4)
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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.

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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.

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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.

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Handling Errors

• Two Possible Cases
• Terminal edges form a tree.
• Terminal path cant 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

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Terminal Edges Form a Tree
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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.

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Paths Through C-Nodes
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When W is a P-Node
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When W is a C-Node (1)
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When W is a C-Node (2)
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When W is a C-Node (2.1)
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When W is a C-Node (2.2)
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A Path that wont 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

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A Path That Wont Flip (2)
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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

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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.

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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

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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

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Summary

• Intuitive, Linear Algorithm to a Non-Trivial
  Problem
• Data Structure Actually Represents the Problem
  Domain
• A Great Way to End A Great Seminar

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Questions ?
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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