Graph Drawing by Force-directed Placement Jana Ludolph.
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Transcript of Graph Drawing by Force-directed Placement Jana Ludolph.
![Page 1: Graph Drawing by Force-directed Placement Jana Ludolph.](https://reader036.fdocuments.us/reader036/viewer/2022062404/55204d6649795902118bc1a6/html5/thumbnails/1.jpg)
Graph Drawing by Force-directed Placement
Jana Ludolph
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Graph Drawing by Force-Directed Placement2
Übersicht
Problemstellung Eades Kamada und Kawai Davidson und Harel Fruchtermann und Reingold
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Graph Drawing by Force-Directed Placement3
Problemstellung
Gegeben: Graph G=(V,E)– Ungerichtet– Gerade Kanten
Gesucht: ästhetisch ansprechende Darstellung des Graphen
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Problemstellung
1) Knoten gleichmäßig im Raum verteilen
2) Kantenkreuzungen vermeiden
3) Kanten gleich lang zeichnen
4) Symmetrien hervorheben
5) Nicht über den gegebenen Rahmen hinauszeichnen
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Die Federanalogie [Eades '84]
Knoten entsprechen Stahlringen Kanten entsprechen Zugfedern Ziel: Energielevel im System minimieren
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Die Federanalogie
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Eades '84
Hookes Gesetz
F = D(x – x0)
wird nicht genau berechnet Für die anziehenden Kräfte nur
Nachbarknoten betrachten: Θ(|E|) Für die abstoßenden Kräfte alle Knoten
betrachten: Θ(|V|2)
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Kamada und Kawai '89
x1
x2
x3
x4
x5
x6
l13 = 1
l15 = 3
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Kamada und Kawai '89
Anziehende Kräfte zwischen allen Knoten, die verbunden sind
Optimierungsfunktion:
Σ D(|xi - xj| - lij)2
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Simulated Annealing
Randomisierte Startpositionen Hohe Anfangstemperatur Cooling- Funktion Evtl. Übergang zu einer minderwertigen
Anordung Hill climbing
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Davidson und Harel '91
Simulated Annealing Optimierungsfunktion mit gewichteten
Termen:– Knotenverteilung– Nähe zu Grenzen– Kantenlängen– Kantenkreuzung
Sehr langsam
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Fruchtermann und Reingold '91
Zwei Bedingungen:1. Nachbarknoten sollen nah beieinander
gezeichnet werden
2. Nicht benachbarte Knoten sollen nicht zu nah beieinander gezeichnet werden
Betrachtung der anziehenden und abstoßenden Kräfte wie Eades
Hauptsache schnell
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Fruchtermann und Reingold
Berechnung der optimalen Distanz k:
Bestimmung der anziehenden (attractive)
und abstoßenden (repulsive) Kräfte durch:
||V
Flächek
k
ddfa
2
)( d
kdfr
2
)(
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Fruchtermann und Reingold
1) Berechne für jeden Knoten v aus V:Alle abstoßenden Kräfte zu allen anderen Knoten mittels fr. → Θ(|V|2)
2) Berechne zu jeder Kante e aus E:Für beide Knoten jeweils die anziehende Kraft mittels fa. → Θ(|E|)
3) Gesamtkraft auf Knoten berechnen; Geschwindigkeit abhängig von Temperatur
4) Knoten müssen im Rahmen bleiben
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Rahmenproblem
Sticky vertex
Weiterrutschen
Elastische Kollision
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Die Rastervariante
x1 x2
x3
x4
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Die Rastervariante
k x1 x2
x3
x4
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Die Rastervariante
k x1 x2
x3
x4
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Die Rastervariante
kk x1 x2
x3
x4
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Die Rastervariante
Abstoßende Kräfte nur zwischen Knoten „in der Nähe“
Keine großen Unterschiede, außer in der Zeit Gleichverteilung der Knoten → Θ(|V|)
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Planare Graphen
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3D Graphen
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Literatur
Thomas M. J. Fruchtermann and Edward M. Reingold„Graph Drawing by Force-directed Placement“Software-Practice and Experience, Vol. 21 (1 1), 1129-1164 (November 1991)
www.wikipedia.de– Hookes Gesetz
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Danke fürs Zuhören