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Faster Force-Directed Graph Drawing with the Well-Separated Pair - - PowerPoint PPT Presentation

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Faster Force-Directed Graph Drawing with the Well-Separated Pair Decomposition Fabian Lipp Alexander Wolff Johannes Zink Julius-Maximilians-Universitt Wrzburg September 24, Graph Drawing 2015 Introduction Classical force-directed

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Faster Force-Directed Graph Drawing with the Well-Separated Pair Decomposition

Fabian Lipp Alexander Wolff Johannes Zink

Julius-Maximilians-Universität Würzburg

September 24, Graph Drawing 2015

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Introduction

Classical force-directed algorithm [Eades, 1984, Fruchterman & Reingold, 1991] Attracting forces: O(m) time with m = #edges Repulsive forces: Θ(n2) time with n = #vertices

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 1 / 11

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Introduction

Classical force-directed algorithm [Eades, 1984, Fruchterman & Reingold, 1991] Attracting forces: O(m) time with m = #edges Repulsive forces: Θ(n2) time with n = #vertices Many speed-up techniques for force-directed algorithms known: Hierarchical O(n log n) force-calculation [Barnes & Hut, 1986] Potential-field-based multilevel algorithm [Hachul & Jünger, 2005] Evaluation of multilevel layout methods [Bartel et al., 2011]

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 1 / 11

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Introduction

Classical force-directed algorithm [Eades, 1984, Fruchterman & Reingold, 1991] Attracting forces: O(m) time with m = #edges Repulsive forces: Θ(n2) time with n = #vertices Many speed-up techniques for force-directed algorithms known: Hierarchical O(n log n) force-calculation [Barnes & Hut, 1986] Potential-field-based multilevel algorithm [Hachul & Jünger, 2005] Evaluation of multilevel layout methods [Bartel et al., 2011] Our approach Using well-separated pair decomposition to speed up repulsive force calculation to O(n log n) time. Can be combined with other speed-up techniques.

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 1 / 11

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Well-Separated Pairs

A B Definition Let s > 0. A and B in R2 are s-well-separated iff there are two balls CA and CB with A ⊆ CA and B ⊆ CB, radius r := r(CA) = r(CB), and distance ≥ s · r. [Callahan & Kosaraju, 1995]

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 2 / 11

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Well-Separated Pairs

CA CB A B Definition Let s > 0. A and B in R2 are s-well-separated iff there are two balls CA and CB with A ⊆ CA and B ⊆ CB, radius r := r(CA) = r(CB), and distance ≥ s · r. [Callahan & Kosaraju, 1995]

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 2 / 11

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Well-Separated Pairs

CA CB r r A B Definition Let s > 0. A and B in R2 are s-well-separated iff there are two balls CA and CB with A ⊆ CA and B ⊆ CB, radius r := r(CA) = r(CB), and distance ≥ s · r. [Callahan & Kosaraju, 1995]

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 2 / 11

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Well-Separated Pairs

CA CB r r ≥ s · r A B Definition Let s > 0. A and B in R2 are s-well-separated iff there are two balls CA and CB with A ⊆ CA and B ⊆ CB, radius r := r(CA) = r(CB), and distance ≥ s · r. [Callahan & Kosaraju, 1995]

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 2 / 11

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SLIDE 9

Well-Separated Pairs

CA CB r r ≥ s · r A B Definition Let s > 0. A and B in R2 are s-well-separated iff there are two balls CA and CB with A ⊆ CA and B ⊆ CB, radius r := r(CA) = r(CB), and distance ≥ s · r. [Callahan & Kosaraju, 1995]

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 2 / 11

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Well-Separated Pair Decomposition (WSPD)

Definition Let S be a finite set in R2. A WSPD for S with respect to s > 0 is a sequence {A1, B1}, {A2, B2}, . . . , {Ak, Bk}

  • f s-well-separated pairs Ai, Bi such

that Ai, Bi ⊆ S, and ∀a = b ∈ S there is exactly one i with a ∈ Ai and b ∈ Bi or vice versa.

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 3 / 11

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Well-Separated Pair Decomposition (WSPD)

Definition Let S be a finite set in R2. A WSPD for S with respect to s > 0 is a sequence {A1, B1}, {A2, B2}, . . . , {Ak, Bk}

  • f s-well-separated pairs Ai, Bi such

that Ai, Bi ⊆ S, and ∀a = b ∈ S there is exactly one i with a ∈ Ai and b ∈ Bi or vice versa.

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 3 / 11

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Well-Separated Pair Decomposition (WSPD)

Definition Let S be a finite set in R2. A WSPD for S with respect to s > 0 is a sequence {A1, B1}, {A2, B2}, . . . , {Ak, Bk}

  • f s-well-separated pairs Ai, Bi such

that Ai, Bi ⊆ S, and ∀a = b ∈ S there is exactly one i with a ∈ Ai and b ∈ Bi or vice versa.

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 3 / 11

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SLIDE 13

Well-Separated Pair Decomposition (WSPD)

Definition Let S be a finite set in R2. A WSPD for S with respect to s > 0 is a sequence {A1, B1}, {A2, B2}, . . . , {Ak, Bk}

  • f s-well-separated pairs Ai, Bi such

that Ai, Bi ⊆ S, and ∀a = b ∈ S there is exactly one i with a ∈ Ai and b ∈ Bi or vice versa.

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 3 / 11

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Well-Separated Pair Decomposition (WSPD)

Definition Let S be a finite set in R2. A WSPD for S with respect to s > 0 is a sequence {A1, B1}, {A2, B2}, . . . , {Ak, Bk}

  • f s-well-separated pairs Ai, Bi such

that Ai, Bi ⊆ S, and ∀a = b ∈ S there is exactly one i with a ∈ Ai and b ∈ Bi or vice versa.

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 3 / 11

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Well-Separated Pair Decomposition (WSPD)

Definition Let S be a finite set in R2. A WSPD for S with respect to s > 0 is a sequence {A1, B1}, {A2, B2}, . . . , {Ak, Bk}

  • f s-well-separated pairs Ai, Bi such

that Ai, Bi ⊆ S, and ∀a = b ∈ S there is exactly one i with a ∈ Ai and b ∈ Bi or vice versa.

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 3 / 11

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SLIDE 16

Well-Separated Pair Decomposition (WSPD)

Definition Let S be a finite set in R2. A WSPD for S with respect to s > 0 is a sequence {A1, B1}, {A2, B2}, . . . , {Ak, Bk}

  • f s-well-separated pairs Ai, Bi such

that Ai, Bi ⊆ S, and ∀a = b ∈ S there is exactly one i with a ∈ Ai and b ∈ Bi or vice versa.

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 3 / 11

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Computation of WSPD

Theorem Let S be a set of n points in R2. For any s > 0, a WSPD consisting of O(n) pairs can be computed in O(n log n) time. [Callahan & Kosaraju, 1995]

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 4 / 11

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Repulsive Force Calculation

|B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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Repulsive Force Calculation

|B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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Repulsive Force Calculation

|B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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Repulsive Force Calculation

|B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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Repulsive Force Calculation

|B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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SLIDE 23

Repulsive Force Calculation

|B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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SLIDE 24

Repulsive Force Calculation

|B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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SLIDE 25

Repulsive Force Calculation

|B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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Repulsive Force Calculation

|B| · Frep |A| · Frep Compute forces for Θ(n2) pairs of vertices

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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Repulsive Force Calculation

|B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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SLIDE 28

Repulsive Force Calculation

|B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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Repulsive Force Calculation

|B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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Repulsive Force Calculation

|B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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Repulsive Force Calculation

A B |B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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SLIDE 32

Repulsive Force Calculation

A B |B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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SLIDE 33

Repulsive Force Calculation

|B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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SLIDE 34

Repulsive Force Calculation

|B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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SLIDE 35

Repulsive Force Calculation

|B| · Frep |A| · Frep

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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Repulsive Force Calculation

|B| · Frep |A| · Frep Compute forces for k ∈ O(n) well-separated pairs

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 11

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Using the Split Tree

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 6 / 11

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Using the Split Tree

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 6 / 11

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Using the Split Tree

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 6 / 11

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Using the Split Tree

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 6 / 11

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Using the Split Tree

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 6 / 11

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Using the Split Tree

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 6 / 11

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Running time

Compute the WSPD using a split tree O(n log n) Compute forces for every pair O(n) Propagate forces in the split tree O(n) Total O(n log n)

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 7 / 11

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Experiments

Implementation in Java based on FRLayout in Jung (Java Universal Network/Graph Framework) Comparison to similar speed-up techniques: Quadtree based [Barnes & Hut, 1986] Grid variant [Fruchterman & Reingold, 1991]

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 8 / 11

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Experiments: Running time

10 20 30 40 50 60 70 80 90 100 0.01 0.02 0.03 Number of vertices Running time in seconds FRLayout (JUNG) FR+Quad, Θ = 1.0 FR+Grid FR+WSPD, s = 0.1

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 9 / 11

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Experiments: Quality

better 50 100 150 200 250 300 0.3 0.35 Number of edge crossings Standard deviation of edge length FRLayout (JUNG) FR+Quad, Θ = 1.0 FR+Grid FR+WSPD, s = 0.1

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 10 / 11

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Experiments: Quality

better 50 100 150 200 250 300 0.3 0.35 Number of edge crossings Standard deviation of edge length FRLayout (JUNG) FR+Quad, Θ = 1.0 FR+Grid FR+WSPD, s = 0.1

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 10 / 11

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Experiments: Quality

better 50 100 150 200 250 300 0.3 0.35 Number of edge crossings Standard deviation of edge length FRLayout (JUNG) FR+Quad, Θ = 1.0 FR+Grid FR+WSPD, s = 0.1

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 10 / 11

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Experiments: Quality

better 50 100 150 200 250 300 0.3 0.35 Number of edge crossings Standard deviation of edge length FRLayout (JUNG) FR+Quad, Θ = 1.0 FR+Grid FR+WSPD, s = 0.1

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 10 / 11

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Experiments: Quality

better 50 100 150 200 250 300 0.3 0.35 Number of edge crossings Standard deviation of edge length FRLayout (JUNG) FR+Quad, Θ = 1.0 FR+Grid FR+WSPD, s = 0.1

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 10 / 11

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Summary

Asymptotic running time of O(n log n) for each iteration Notably faster than unmodified algorithm Similar to unmodfied algorithm in terms of quality Better quality than Grid variant of Fruchterman & Reingold

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 11 / 11

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Summary

Asymptotic running time of O(n log n) for each iteration Notably faster than unmodified algorithm Similar to unmodfied algorithm in terms of quality Better quality than Grid variant of Fruchterman & Reingold Demonstration during the poster session this afternoon

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 11 / 11

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Summary

Asymptotic running time of O(n log n) for each iteration Notably faster than unmodified algorithm Similar to unmodfied algorithm in terms of quality Better quality than Grid variant of Fruchterman & Reingold Demonstration during the poster session this afternoon Thank you!

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 11 / 11

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APPENDIX

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 1 / 7

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Example drawings

FRLayout (JUNG) FR+WSPD, s = 0.1

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 2 / 7

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Example drawings

FRLayout (JUNG) FR+WSPD, s = 0.1

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 3 / 7

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SLIDE 57

Experiments: Running time on large graphs

20,000 40,000 60,000 80,000 100,000 50 100 150 Number of vertices Running time in seconds FRLayout (JUNG) FR+Quad, Θ = 1.0 FR+Grid FR+WSPD, s = 0.1

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 4 / 7

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Experiments: Running times compared to OGDF

10 20 30 40 50 60 70 80 90 100 0.01 0.02 0.03 Number of vertices Running time in seconds FRLayout (JUNG) FRExact (OGDF) FR+Grid FM3 (OGDF) FR+Quad, Θ = 1.0 FM3 single (OGDF) FR+WSPD, s = 0.1 GEM (OGDF)

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 5 / 7

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Experiments: Running times compared to OGDF

20,000 40,000 60,000 80,000 100,000 50 100 150 Number of vertices Running time in seconds FRLayout (JUNG) FRExact (OGDF) FR+Grid FM3 (OGDF) FR+Quad, Θ = 1.0 FM3 single (OGDF) FR+WSPD, s = 0.1 GEM (OGDF)

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 6 / 7

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Experiments: Quality compared to OGDF

50 100 150 200 250 300 0.3 0.35 Number of edge crossings Standard deviation of edge length FR+WSPD, s = 0.1 FM3 (OGDF) GEM (OGDF) FRExact (OGDF) FM3 single (OGDF)

Fabian Lipp, Alexander Wolff, Johannes Zink Force-Directed Graph Drawing with WSPD 7 / 7