# Generating and Drawing Area-Proportional Euler and Venn Diagrams

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```					         Generating and Drawing
Area-Proportional Euler and Venn Diagrams

Stirling Chow

Department of Computer Science
University of Victoria

Ph.D. Examination
May 10, 2007
Chapter Overview
Area-Proportionality                         Topology
(ω-EDGP)           Chapters 1/2          (EDGP)
Background

Chapters 3                             Chapters 4/5
Preliminary Cases                     Graph-Theoretic Model
(n ≤ 3)
Chapters 6
Chapters 9                         Diagram Expressiveness
Diagrams with Fullset
(any n)
Chapters 7
Computational Complexity

Chapter 8
Composite Diagrams

Chapters 10
Conclusion
Chapters 1/2: Introduction

(B)irds            (C)arnivores
(A)nimals

152      13        8        64         11

752

(B)irds            (C)arnivores
(A)nimals

152       13      8    64       11

752
Chapters 1/2: Basic Deﬁnitions

c1
{1}

{1,2}                c3
c2                                           Euler† vs. Euler-like‡ Diagrams
{3}
Connected vs. Disconnected
{2}
Concurrent vs. Non-concurrent
Non-pairwise vs. Pairwise
c1                                         Non-simple vs. Simple
c2                  c3     (Simple ≡ Non-concurrent & Pairwise)
{2}                     {3}   † RWD: U. Brighton and U. Kent, UK
‡ AV Library: INA and INRIA, France
{1,2}
{1}
Chapters 1/2: Problem Deﬁnitions

Set System and Items
S = {∅, {1}, {1, 2}, {2}, {3}}
X = {1, 2, 3}

c1
{1}               Euler Diagram Generation Problem (EDGP)
INPUT:       A set system S.
{1,2}         c3
OUTPUT:      An Euler diagram with
c2
{3}                      regions S.
{2}
Area-Proportional EDGP (ω-EDGP)
INPUT:       A set system S and
function ω : S\{∅} → R+ .
OUTPUT:      An ω-proportional
Euler diagram with regions S.
Chapter Overview
Area-Proportionality                         Topology
(ω-EDGP)           Chapters 1/2          (EDGP)
Background

Chapters 3                             Chapters 4/5
Preliminary Cases                     Graph-Theoretic Model
(n ≤ 3)
Chapters 6
Chapters 9                         Diagram Expressiveness
Diagrams with Fullset
(any n)
Chapters 7
Computational Complexity

Chapter 8
Composite Diagrams

Chapters 10
Conclusion
Chapter 3: ω-proportional n-Euler Diagrams, n ≤ 3

low blood
pressure
nausea
Two circles with bisection†

0.45   0.10   0.20

† Solution to ω-EDGP.
Chapter 3: ω-proportional n-Euler Diagrams, n ≤ 3

c1                      Two circles with bisection†
Three circles with hillclimbing
(w/ Peter Rodgers)
d13
d12
† Solution to ω-EDGP.
d23
c3

c2
Chapter 3: ω-proportional n-Euler Diagrams, n ≤ 3

Two circles with bisection†
c1   {1}                          Three circles with hillclimbing
c3
(w/ Peter Rodgers)
{1,3}      {3}
Three orthogonal rectangles/6-gons‡
c2                    {1,2,3} {2,3}         † Solution to ω-EDGP.
{1,2}                            ‡ Solution to ω-EDGP with fullset.
{2}
Chapter 3: Simple 3-Venn Diagrams with Convex Curves

Theorem
There exists a weight function ω for which there is no
ω-proportional simple 3-Venn diagram whose curves are convex.
c1

c2

c3
Chapter Overview
Area-Proportionality                         Topology
(ω-EDGP)           Chapters 1/2          (EDGP)
Background

Chapters 3                             Chapters 4/5
Preliminary Cases                     Graph-Theoretic Model
(n ≤ 3)
Chapters 6
Chapters 9                         Diagram Expressiveness
Diagrams with Fullset
(any n)
Chapters 7
Computational Complexity

Chapter 8
Composite Diagrams

Chapters 10
Conclusion
Chapters 4/5: Euler Duals

c1 , c2

{1,2}
c3
{1,2,3}

{3,4} {4}

{3}
c4
{3,5}

{5}
c5
Chapters 4/5: Euler Duals

{1,2}

{1,2,3}         {3,4}    {4}

∅
{3}
{3,5}

{5}
Chapters 4/5: Euler Duals

{1,2}

{1,2,3}         {3,4}    {4}

∅
{3}
{3,5}

{5}
Chapters 4/5: Euler Duals

{1,2}

{1,2,3}         {3,4}    {4}

∅
{3}
{3,5}

{5}
Chapters 4/5: Euler Duals

{1,2}

{1,2,3}         {3,4}    {4}

∅
{3}
{3,5}

{5}
Chapters 4/5: Connectivity Graphs

Connectivity Graph
Let S be a set system on the items X . A graph G is said to be a
connectivity graph for S if and only if the following properties hold
for G :
1   G is a connected planar graph on S,
2   G is loop-free, and
3   for each item x ∈ X ,
1   the subgraph Gx induced by the vertices containing x is
connected, and
2   the subgraph Gx induced by the vertices not containing x is
connected.

An Euler dual is a connectivity graph for S.
Chapters 4/5: Existence Conditions

Theorem
Let S be a set system. There exists an Euler diagram representing
S if and only if there is a connectivity graph for S that has no
parallel edges.

Corollary
The EDDP is decidable and there is an exponential time algorithm
for constructing connectivity graphs.

Note
The proofs of the existence conditions are constructive.
Chapters 4/5: Existence Conditions

Euler Diagram Generation Problem (EDGP)
INPUT:     A set system S.
OUTPUT:    A simple connectivity graph for S.

Euler Diagram Decision Problem (EDDP)
INPUT:     A set system S.
OUTPUT:    “Yes” if there is a simple connectivity graph for S;
otherwise, “No”.
Chapter Overview
Area-Proportionality                         Topology
(ω-EDGP)           Chapters 1/2          (EDGP)
Background

Chapters 3                             Chapters 4/5
Preliminary Cases                     Graph-Theoretic Model
(n ≤ 3)
Chapters 6
Chapters 9                         Diagram Expressiveness
Diagrams with Fullset
(any n)
Chapters 7
Computational Complexity

Chapter 8
Composite Diagrams

Chapters 10
Conclusion
Chapter 6: Expressiveness of Diagram Types
Set Systems

Euler−like Diagrams
all Jordan curves
C

C

Connected
Euler Diagrams
B         Non−concurrent     B
Simple
B                                B

A                  all curves
non−concurrent
diagram
connected
all intersections
pairwise
Chapter 6: Expressiveness of Diagram Types

Set Systems

Euler−like,              Euler−like,
all intersections             diagram
pairwise                connected

Simple

all Jordan curves

all curves
non−concurrent
Chapter 6: Expressiveness of Diagram Types

{1,2,6,7,8}

{1}                 {1,2}               {1}      {1,3}       {2,3}           {2}
{5,8,10}    {1,2,3,4,5}        {3,6,9}

{1}

{1,2,3}             Set Systems
{1,4,7,9,10}
Euler−like,                 Euler−like,
all intersections                diagram
pairwise                   connected

{1,2}
{1,2,3}
{1,2,3,4}
{1}
{3,4}                                                       Simple
all Jordan curves
{1,2}           {1,3}
all curves
non−concurrent

{2}           {2,3}           {3}

{1}                                                               {1}
{1,2}

{1,2}
{2,3}
{1,2,3}

{2,3}
{1,2,3}
{1,4}                  {1,5}

{4}             {4,5}                 {5}
Chapter Overview
Area-Proportionality                         Topology
(ω-EDGP)           Chapters 1/2          (EDGP)
Background

Chapters 3                             Chapters 4/5
Preliminary Cases                     Graph-Theoretic Model
(n ≤ 3)
Chapters 6
Chapters 9                         Diagram Expressiveness
Diagrams with Fullset
(any n)
Chapters 7
Computational Complexity

Chapter 8
Composite Diagrams

Chapters 10
Conclusion
Chapter 7: Computational Complexity of Existence

Euler Diagram Decision Problem (EDDP)
INPUT:     A set system S.
OUTPUT:    “Yes” if there is a simple connectivity graph for S;
otherwise, “No”.

Theorem
The EDDP is NP-complete.
Chapter 7: Previous Hypergraph Planarity Result

Vertex-Based Hypergraph Planarity Problem (HPLANAR)
INPUT:     A hypergraph H = (V , E ).
OUTPUT:    “Yes” if there is a planar graph G on V
for which each hyperedge e ∈ E (where e ⊆ V ),
induces a connected subgraph of G ;
otherwise, “No”.

Theorem
The HPLANAR is NP-complete (Johnson and Pollak ’87).
Chapter 7: Hypergraphs to Euler-like Diagrams

Partial Connectivity Graph
Let S be a set system on the items X . A graph G is said to be a
partial connectivity graph for S if and only if the following
properties hold for G :
1   G is a connected planar graph on S,
2   G is loop-free, and
3   for each item x ∈ X ,
1   the subgraph Gx induced by the vertices containing x is
connected, and
2   the subgraph Gx induced by the vertices not containing x is
connected.
Chapter 7: Previous Euler-like Diagram Result

Euler-like Diagram Decision Problem (weak-EDDP)
INPUT:      A set system S.
OUTPUT:     “Yes” if there is a partial connectivity graph for S;
otherwise, “No”.

Theorem
The weak-EDDP is NP-complete (Verroust and Viaud ’04).
Chapter 7: Our Result

Theorem
The EDDP is NP-complete.

NPC Reduction
Extension of Johnson and Pollak’s gadget transformation from
Hamilton Path on cubic 3-connected planar graphs
Structured in terms of set systems rather than hypergraphs
Additional structure to ensure Gx connectivity

Basic Idea
Construct set system S based on cubic 3-connected planar graph
G so that any connectivity graph for S contains a subgraph
homeomorphic to G .
Chapter Overview
Area-Proportionality                         Topology
(ω-EDGP)           Chapters 1/2          (EDGP)
Background

Chapters 3                             Chapters 4/5
Preliminary Cases                     Graph-Theoretic Model
(n ≤ 3)
Chapters 6
Chapters 9                         Diagram Expressiveness
Diagrams with Fullset
(any n)
Chapters 7
Computational Complexity

Chapter 8
Composite Diagrams

Chapters 10
Conclusion
Chapter 8: Prime Factorization of Diagrams

∅

c8
c1
c4
c1                                    *
c2
c3
c8
c9   c10
c6   c5                                               c5
c4        c9   c10     c2
c12

c11
c7
c1 2
c11        c6        c7
c3
Chapter 8: Prime Factorization Algorithm

Factorization Problem
INPUT:        A set system S on the items X .
OUTPUT:       A decomposition of S into factorization tree.

Theorem
There is an O(|X |2 |S| + |X |3 ) algorithm for solving the
Factorization Problem.
Chapter Overview
Area-Proportionality                         Topology
(ω-EDGP)           Chapters 1/2          (EDGP)
Background

Chapters 3                             Chapters 4/5
Preliminary Cases                     Graph-Theoretic Model
(n ≤ 3)
Chapters 6
Chapters 9                         Diagram Expressiveness
Diagrams with Fullset
(any n)
Chapters 7
Computational Complexity

Chapter 8
Composite Diagrams

Chapters 10
Conclusion
Chapter 9: Redrawing Monotone Euler Diagrams

Theorem
Any monotone n-Euler diagram can be redrawn area-proportionally
in O(n|V |) time.

c1
c4

c2                    c3
Chapter Overview
Area-Proportionality                         Topology
(ω-EDGP)           Chapters 1/2          (EDGP)
Background

Chapters 3                             Chapters 4/5
Preliminary Cases                     Graph-Theoretic Model
(n ≤ 3)
Chapters 6
Chapters 9                         Diagram Expressiveness
Diagrams with Fullset
(any n)
Chapters 7
Computational Complexity

Chapter 8
Composite Diagrams

Chapters 10
Conclusion
Chapter 10: Conclusion

Main Contributions
First systematic approach to area-proportional drawing;
unique convex curve non-existence proof
Concise Euler diagram existence conditions
First mathematically-based analysis of expressiveness;
understanding functional and aesthetic properties
First NPC result related to Euler diagrams (not Euler-like
diagrams); motivating composite Euler diagram heuristic
All drawing algorithms implemented and in use by third parties

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