literature review of force-based graphing algorithms

8/9/2019 Literature Review of Force-based Graphing Algorithms

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Outline

Goal of Force Graphing Algorithms

Review of Hookes Law

Three Main Algorithms

Spring Model

Simulated Annealing

The Sugiyama Algorithm

8 Variations

Review


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Goals

Aesthetics

Minimize edge crossings

Verities evenly distributed

Minimize drawing area

Maximize symmetries

Maximize angular resolution


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Goals


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3 Algorithms.


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Sugiyama Algorithm

Sugiyama, Tagawa, and Toda (1981)

The graph is assumed to be in layers.

Edges can only connect nodes in differentlayers.


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Sugiyama Algorithm

The algorithm is in four stages:

Stage 1: Refit the graph into layers (if not done soalready)

Stage 2: Reshape the vertices at each level toreduce edge crossings

Stage 3: Reshape the vertices at each level toreduce edge lengths

Stage 4: Display the graph


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Sugiyama Algorithm

If an edge stretches across multiple layers,placeholder nodes are added to in betweenlayers.

Barycenter for a node: the minimum distancefrom a level to every other node.

Computing the barycenter of each nodeautomatically reduces edge crossings.

Nodes are moved based on their barycenters.

Nodes with smaller barycenters are given ahigher priority.


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Spring Model

Peter Eades, 1984

Vertices are replaced with rings.

Edges are replaced with springs.

Each spring applies a force on the rings,moving the rings slightly.

After several iterations of slight movements, the

springs reach an equilibrium point


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Spring Model

Vertices that are connected by an edge attract.

C1 * log(D * C2)

D: length of the spring

C1, C2: Constants

Vertices that are not connected repel.

C3 / sqrt(D)

C3: Constant

All forced are multiplied by C4 and repeated100 times.


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Simulated Annealing

Davidson and Harel (1989)

Each node uses an cost function, which consistof:

Potential Energy for each node: aij= C1 / (d

ij)2

Borderline Energy C2 ( 1/r2 + 1/l2 + 1/t2 1/b2)

Energy of each edge: C3 * (dk)2

Distance from node to an edge: C4 / (gkl)2

d_ij: distance between two nodes. C1, C2, C3, C4 are constants. r, l, t, b: distance from right, left, top, bottom g_kl: distance from node k to edge l


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Simulated Annealing

In addition to the cost function, there is apenalty for each edge crossings: C5 (aconstant)

The cost function is multiplied by a temperature High temperature = greater movement

Low temperature = lesser movement


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Simulated Annealing

Cooling Function. Over time, the temperature ofthe area "cools".

Initial temperature is arbitrarily high # in degrees

Trials per degree: 30 * # of nodes

Cooling function = Tp+1

= C6 * Tp

Termination condition: little or no movement for

three iterations. Algorithm runs in O(V2 * E)


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8 Variations.


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Kamada and Kawai (1989)

Variation of Eades spring model approach.

Each edge has a starting length of lij

Each edge desires to have a length equal to L L: Usually a function of the size of the

graphing area.


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Kamada and Kawai (1989)

dij: the shortest path of edge lengths between

nodes i and j.

kij

: strength of the spring equal to K / dij

K: Constant


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Fruchterman & Reingold (1990)

Includes ideas of Eades and Kamada using aSpring Force graph

Includes ideas of Davidson and Harel using

Simulated Annealing Cost function is based on Hooke's Law

Optimal distance between vertices: C *

sqrt(area / number of verticies) Proposes an idea for a grid-variant partitioning

scheme.


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Tunkelang (1994)

Step 1: Compute a minimal height spanningtree of the graph.

The graph nodes are placed so that the root

node is in the center and subsequent nodessurround it in rings.

Edges strive for a target edge length that is aparameter set by the user.

Larger length: better graph.

Smaller length: faster graph.


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Tunkelang (1994)

Attraction forces are edges; repulsion forcesare other nodes.

Attraction force: wa

* d2

Repulsion force: wr/ d2

d: distance between two nodes

wa: weight given to attractive force.

wr: weight given to repulsive force.


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Tunkelang (1994)

Target edge length is suggested to be:

(wr/w

a)1/4

Using the uniform grid approach to countingedge crossings, nodes are pushed until aminimal arrangement is reached.


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Frick, Ludwig, Mehldau (1995)

Builds on the work of Davidson and Harel(1989).

Instead of having a global temperature for

simulated annealing, each node has a localtemperature.

Each node starts with the same temperature.

After each node moves, the temperaturedecreases based on the movement. If the nodeappears to be oscillating, the temperaturedecreases quickly. If not, the temperature

decreases slowly.


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Gansner and North (1998)

Purpose of this algorithm is to improve theresults of the Kamada and Kawai algorithm.

Phase 1: Perform the original Kamada and

Kawai algorithm. Phase 2: Magnify the placement of all vertices,

giving them sufficient space.

For each vertex, compute a Voronoi diagram. Move the vertex to the nearest center. Vertices

gradually move away from each other in anorderly fashion.


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Bertault (1999)

Initial placement of nodes is made by thegraph's owner.

Edge Crossings should be preserved, not

minimized. Cost functions is split between attractive forces

(edges), repulsive forces (nodes), and repulsingforces (nodes versus edges).


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Bertault (1999)

For each node, 8 "nodearcs" are computed.These are arcs equal to1/8th of a circle of

equal proportions. Foreach arch, a calculationis made regarding themaximum movement

that can be madewithout changing nodecrossings.


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Walshaw (2000)

Implements Frucherman and Reingold's grid-variant algorithm.

Cost function between a vertex and another

vertex (global cost): -Cwk2

/x C: Constant

k: natural length of the spring

w: Number of vertices x: current length of the spring


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Walshaw (2000)

Vertices in the same grid region are consideredlocal and use a different cost function: ((x - k) /d) - global cost

d: distance between the two vertices. k (natural edge length) is set to the average edge

length

The cooling function at each iteration is 0.95

times the previous temperature.


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Walshaw (2000)

If x (current length of spring) is sufficientlylarge, the authors say that the spring isstretched so far that it has no impact. This isused as excuse to speed up the algorithm by asmall factor.


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Gajer, Goodrich, and Kobourov(2004)

Differs from most algorithms in that they use analgorithm for initial placement.

Similar to Tunkelang in that distinction is made

between local and global points. Local is determined by a limited breadth-first

search algorithm.

Cost function is based on the Kamada andKawai cost function (1989)


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Review

Eades (1984)Davidson &Harel (1989)

Sugiyama, Tagawa,& Toda (1981)

Kamada &Kawai (1989)

Fruchterman &Reingold (1990)

Tunkelang (1994)Frick, Ludwig, &

Mehldau (1995)

Gansner &

North (1998)

Bertault (1999)Walshaw (2000)Gajer, Goodrich, &Kobourov (2004)


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References

F. Bertault. A force-directed algorithm that preserves edge crossing properties. InGraph Drawing, pages 351358. Springer, 1999.

R. Davidson and D. Harel. Drawing graphs nicely using simulated annealing. ACMTransactions on Graphics (TOG), 15(4):301331, 1996.

Eades. A heuristic for graph drawing. Congressus numerantium, 42(149160):194

202, 1984. A. Frick, A. Ludwig, and H. Mehldau. A fast adaptive layout algorithm for undirected

graphs. Graph Drawing (Proc. GD 039; 94), volume 894 of Lecture Notes inComputer Science, 1995.

T.M.J. Fruchterman and E.M. Reingold. Graph drawing by force-directed placement.Software- Practice and Experience, 21(11):11291164, 1991.

P. Gajer, M. Goodrich, and S. Kobourov. A multi-dimensional approach to force-directed layouts of large graphs. In Graph Drawing, pages 211221. Springer.


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References

E. Gansner and S. North. Improved force-directed layouts. In Graph Drawing, pages364373. Springer, 1998.

T. Kamada and S. Kawai. An algorithm for drawing general undirected graphs.Information processing letters, 31(1):715, 1989.

K. Sugiyama and K. Misue. Graph drawing by the magnetic spring model. Journal of

Visual Languages and Computing, 6(3):217231, 1995. D. Tunkelang. A practical approach to drawing undirected graphs. Carnegie Mellon

University, 1994.

C. Walshaw. A multilevel algorithm for force-directed graph drawing. In GraphDrawing, pages 3155. Springer, 2000.


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Additional References

F. Brandenburg, M. Himsolt, and C. Rohrer. An experimental comparison of force-directed andrandomized graph drawing algorithms. In Graph Drawing, pages 7687. Springer.

G. Di Battista, P. Eades, R. Tamassia, and I.G. Tollis. Algorithms for drawing graphs: anannotated bibliography. Computational Geometry-Theory and Application, 4(5):235282, 1994.

G. Di Battista, A. Garg, G. Liotta, R. Tamassia, E. Tassinari, and F. Vargiu. An experimentalcomparison of four graph drawing algorithms* 1. Computational Geometry, 7(5-6):303325,

1997.

S. Sim. Automatic graph drawing algorithms. Manuscript, available att, 1996.

literature review of force-based graphing algorithms

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