Anchored Drawings of Planar Graphs Angelini, Da Lozzo, Di - - PowerPoint PPT Presentation

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Anchored Drawings of Planar Graphs Angelini, Da Lozzo, Di - - PowerPoint PPT Presentation

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22nd International Symposium on Graph Drawing 24-26 September 2014, Wrzburg, Germany Anchored Drawings of Planar Graphs Angelini, Da Lozzo, Di Bartolomeo, Di Battista, Hong, Patrignani, Roselli Applicative Context Drawing a graph on a

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Anchored Drawings

  • f Planar Graphs

Angelini, Da Lozzo, Di Bartolomeo, Di Battista, Hong, Patrignani, Roselli

22nd International Symposium on Graph Drawing 24-26 September 2014, Würzburg, Germany

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Applicative Context

  • Drawing a graph on a geographical map
  • Vertices have fixed positions
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Drawing Nicely

  • Our idea:

○ Let vertices move “a bit” around their positions ○ Check if this allows a planar drawing of the graph

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Anchored Graph Drawing Problem

  • Instance

○ Planar graph G ○ Initial vertex positions α(v) ○ Maximum distance δ

  • Question

○ Does G admits a planar drawing ○ ...such that vertices move by distance at most δ ○ ...from their initial positions α?

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SLIDE 5

Considered Settings

Distance Function Drawing Style

“Euclidean” “Manhattan” “Uniform” d = (dx2 + dy2)1/2 d = dx + dy d = max(dx,dy) Straight-line Rectilinear

Vertex Region

dy dx dy dx dy dx

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Previous work

  • NP-hard: straight-line and disks of different size

○

  • Godau. On the difficulty of embedding planar graphs with inaccuracies. 1995
  • NP-hard: rectilinear and δ = inf

○ Garg, Tamassia. On the comp. compl. of upward and rectilinear planarity test. 2001

  • Application of force-directed algorithms

○ Abellanas et. al. Network drawing with geographical constraints on vertices. 2005

  • Iterative adjustments that preserve mental map

○ Lyons et. al. Algorithms for cluster busting in anchored graph drawing. 1998

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SLIDE 7

Assumption

  • No overlap between vertex regions

○ Or two vertices may invert their positions

■ Very confusing for a user

○ Relationship with Clustered Planarity with drawn clusters

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Our Results

Metric Straight-line Rectilinear Manhattan NP-hard NP-hard Euclidean NP-hard NP-hard Uniform NP-hard Polynomial

dx

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Our Results

Metric Straight-line Rectilinear Manhattan NP-hard NP-hard Euclidean NP-hard NP-hard Uniform NP-hard Polynomial

dx

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Polynomial Case

  • Connected graph
  • Uniform distance ( regions)
  • Rectilinear drawing
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Edge Pipes

  • We call pipe the convex hull of two regions

○ Minus the regions

  • An edge can be drawn only inside a pipe
  • In this setting pipes “get rectilinear” too
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Rectilinear Edges

  • An edge is either horizontal or vertical
  • Can be deduced by the region positions
  • Visibility is required between two endpoints
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Trimming

  • Regions and pipes can trim each other
  • A trimmed area cannot be used
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General Strategy

  • 1. Start from the initial region/pipe configuration
  • 2. While (a trim is possible):
  • a. Trim unusable parts of pipes and regions
  • b. Check if a negative configuration is obtained
  • 3. Flag the instance as positive
  • 4. Draw edges according to the current pipes
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Trimming Pipes

  • VP-overlaps can trim a pipe
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Trimming Regions

  • VP-overlaps can trim a region
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Negative Instances

PP-overlap (Unavoidable crossing) No visibility

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An Example of Execution

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An Example of Execution

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An Example of Execution

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SLIDE 21

An Example of Execution

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An Example of Execution

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SLIDE 23

NP-hard Case

  • Euclidean distance ( regions)
  • Straight-line drawing
  • Reduction from Planar 3-SAT
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SLIDE 24

Planar 3-SAT

(x1 ˅ ¬x2 ˅ x5) ˄ (x2 ˅ x3 ˅ ¬x4) ˄ (x1 ˅ ¬x3 ˅ x5) ˄ (x3 ˅ x4 ˅ x5) C1 C2 C3 C4 x1 x2 x3 x4 x5 c1 c2 c3 c4

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Planar 3-SAT - Gadgets

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Planar 3-SAT - Variable Gadget

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SLIDE 27

Planar 3-SAT - Clause Gadget

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Planar 3-SAT - Truth Propagation

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Planar 3-SAT - Not Gadget

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Planar 3-SAT - Turn Gadget

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Planar 3-SAT - Split Gadget

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Variable Gadget

True configuration False configuration

T F F T

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Truth Propagation

F T F T

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SLIDE 34

Not Gadget

F T F T

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Turn Gadget

F T F T

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Split Gadget

F T T F F T

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Clause Gadget

F T F T F T x y b a

F-T-T case The gadget is planar

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SLIDE 38

Clause Gadget

F T F T F T

F-F-F case The gadget is NOT planar

a b x y

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SLIDE 39

Clause Gadget

F T F T F T

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SLIDE 40

Clause Gadget

F T F T F T

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SLIDE 41

Clause Gadget

F T F T F T

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Clause Gadget

F T F T F T

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SLIDE 43

Clause Gadget

F T F T F T

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SLIDE 44

Clause Gadget

F T F T F T

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Open Problems

  • Do the hard problems belong to NP?
  • Still hard with biconnected gadgets. What if

triconnected?

  • What if we allow regions to partially overlap?
  • What if we allow some crossings?
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SLIDE 46

Applicative Context

  • Drawing a graph on a geographical map
  • Vertices have fixed positions
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SLIDE 47

Clause Gadget (master slide)

F T F T F T

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Challenges

  • Vertex cluttering, edge crossings
  • Techniques exist to mitigate cluttering
  • However, crossings are still an issue