advanced seminar on graph drawing – planar orientations olga maksin victor makarenkov department...
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![Page 1: Advanced Seminar on Graph Drawing – Planar Orientations Olga Maksin Victor Makarenkov Department of Computer Science. Ben-Gurion University of the Negev](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649ec65503460f94bd1e26/html5/thumbnails/1.jpg)
Advanced Seminar on Advanced Seminar on Graph Drawing – Planar Graph Drawing – Planar OrientationsOrientationsOlga MaksinVictor Makarenkov
Department of Computer Science.
Ben-Gurion University of the Negev.
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ContentsContentsDominance DrawingsUndirected Planar GraphsPlanar Orthogonal DrawingsPlanar Straight Line DrawingsReal-World Example
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Dominance Drawings - Dominance Drawings - MotivationMotivationDraw a nice planar st-graphs
with:◦Linear time complexity◦Small number of bends◦Small area◦Presentation of symmetries
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Dominance drawing A dominance drawing of a
digraph G is a drawing Γ of G such that:
Dominance drawing is upward.
u
v
iffX(u) <= X(v)
Y(u) <= Y(v)
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LemmaLemmaAny straight line dominance
drawing Γ of a reduced planar st-graph G is planar.
Proof (sketched) :◦Assume the negation -> K3,3 ->
contradiction.
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Dominance straight line Dominance straight line Input : Reduced planar st-graph
G=(V,E)Output : Straight line dominance
drawing Γ of G
3 Stages:◦Preprocessing – Data structures◦Preliminary Layout◦Compaction
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PreprocessingPreprocessing
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• For each v • list of outgoing edges (clockwise)
•Next (e), Pred (e)
• firstout(v)• lastout(v)• firstin(v)• lastin(v)
• For each e = <u,v>
• head(e) = v
firstout(s) = <s,w>Lastin(t) = <q,t>Next(<s,w>)=<s,k>
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Preliminary LayoutPreliminary Layout
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Preliminary Layout contPreliminary Layout cont..
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CompactionCompaction
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If Y(u) > Y(v) or(Firstout(u) = lastout(u) and firstin(v) = lastin(v))Then x(v) = x(u) + 1Else x(v) = x(u)
Two ordered lists by X and Y coordinate
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Compaction contCompaction cont..
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If X(u) > X(v) or(Firstout(u) = lastout(u) and firstin(v) = lastin(v))Then y(v) = y(u) + 1Else y(v) = y(u)
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Compaction conclusionCompaction conclusion..
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ExampleExample
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Algorithm AnalysisAlgorithm AnalysisTheorem 4.9. Let G be a
reduced planar st-graph with n vertices. Algorithm Dominance-Straight-Line constructs in O(n) time a planar straight line dominance grid drawing Γ of G with O(n2) area.
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Display of symmetriesDisplay of symmetriesTheorem 4.10. Let G be a reduced
planar st-graph. And Γ be the corresponding straight line drawing constructed by algorithm dominance-straight-line. We have :◦Simply isomorphic components of G
have drawings in Γ that are congruent up to a translation.
◦Axially isomorphic components of G have drawings in Γ that are congruent up to a translation and reflection.
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Display of symmetries Display of symmetries cont.cont.Rotationally isomorphic components
of G have drawings in Γ that are congruent up to a translation and 180o rotation.
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Minimum area dominance Minimum area dominance drawingsdrawings
EL set of edges {<u,v>} such that <u,v> is the rightmost incoming edge of v and leftmost outgoing of u.
ER – vice versa.EH – {<u,v>} the only outgoing edge of
u and the only incoming of v. EH is intersection of EL and ER
Area = (n -1 – (mL - mH)) * (n -1-(mR - mH))Minimum area : EH is empty
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Minimum area dominance Minimum area dominance drawingsdrawingsCompute mL and mR in
preprocessing phaseIf Y(u) > Y(v) or(Firstout(u) = lastout(u) and firstin(v) =
lastin(v) and mL <= mR (mL > mR for x))Then x(v) = x(u) + 1Else x(v) = x(u)
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Minimum area contMinimum area cont..Theorem 4.12. Let G be a
reduced planar st-graph with n vertices. A minimum area Dominance-Straight-Line grid drawing of G can be constructed in O(n) time.
Note: Symmetry not guaranteed.
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General planar st-graphsGeneral planar st-graphsInput : Planar st-graph G=(V,E)Output : Polyline dominance
drawing Γ of G◦Construct reduced G’◦Run Algorithm Dominance-straight-
line◦Remove dummies
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General planar st-graphs General planar st-graphs contcont..
Adding a dummy vertex:
At least n-1 edges are not transitive => at most 2n – 5 dummies => 2n-5 bends.
Recall at most 3n-6 edges in planar graph.
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Dominance Drawings - Dominance Drawings - SummarySummaryΓ is planar , upward, grid,
dominance, polyline.Γ has O(n2) area.Γ has at most 2n-5 bends. Every
edge at most one bend.Γ presents symmetric and
isomorphic components.
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ContentsContentsDominance DrawingsUndirected Planar GraphsPlanar Orthogonal DrawingsPlanar Straight Line DrawingsReal-World Example
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Extension - Undirected Planar Extension - Undirected Planar GraphsGraphs
TessellationVisibilityUpward polyline drawings
O(n) time , O(n2) area , 2n-5 bends for planar polyline.
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The extensionThe extensionConstruct a planar embedding of G.Make it biconnected (dummy edges).Let s and t be 2 vertices on external face.St-numberingOrient edges according to st-numbering.
s t1
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ContentsContentsDominance DrawingsUndirected Planar GraphsPlanar Orthogonal DrawingsPlanar Straight Line DrawingsReal-World Example
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Planar Orthogonal DrawingsGraphs with degree less than or
equal to four.Uses visibility representations as
an intermediate construction.At most 2 bends for each edge
(except two edges for s and t with four bends each). Total: 2n+4
O(n) time , O(n2) area
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Orthogonal-from-VisibilityOrthogonal-from-VisibilityInput : biconnected planar graph
G=(V,E) with n vertices of degree at most 4.
Output : planar orthogonal grid drawing of G.◦Construct planar embedding -> planar
st -graph◦Create paths◦Run Constrained-Visibility ◦Construct a planar orthogonal grid
drawing
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Create paths◦n – 2 directed paths associated to
vertices distinct from s, t
◦Unify paths sharing edges
Orthogonal-from-VisibilityOrthogonal-from-Visibilitycontcont..
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Orthogonal-from-VisibilityOrthogonal-from-Visibilitycontcont..Run Constrained –Visibility with
respect to ∏ nonintersecting paths.
Prespecified edges vertically alligned.
For 2 edges on same path, the edge segments have same x coordinate.
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Orthogonal-from-VisibilityOrthogonal-from-Visibilitycont.cont.
Construct a planar orthogonal grid drawing:◦For each v: draw at the intersection of vertex
segment with the edge segments of its path◦For s (t): intersection of its segment with
edge segment of median outgoing (incoming).
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Orthogonal-from-VisibilityOrthogonal-from-Visibilitycont.cont.For each e=(u,v): orthogonal chain through
the following points :Placement of u Intersection of Γ(u) and Γ(e) Intersection Γ(e) and Γ(v)Placement of v
Γ(u)
Γ(v)
Γ(e)
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ExampleExample
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ContentsContentsDominance DrawingsUndirected Planar GraphsPlanar Orthogonal DrawingsPlanar Straight Line DrawingsReal-World Example
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Planar straight linePlanar straight lineEvery planar graph admits a
straight line drawing.Existence is not the same as nice
one. Until 1988 vertices exponentially close together (in minimum unit of distance).
So no visualization can be made.
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Planar straight linePlanar straight lineTheorem 4.17. Every n-vertex
planar graph has a planar straight line grid drawing with O(n2) area.
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Planar straight linePlanar straight lineHow ?
◦Orientation◦One vertex at a time◦Partial order on vertices, edges and
faces.
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ContentsContentsDominance DrawingsUndirected Planar GraphsPlanar Orthogonal DrawingsPlanar Straight Line DrawingsReal-World Example
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UMLUML
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USE exampleUSE exampleForce directed methods