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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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venn diagrams, euler diagram, graph drawing, constraint diagrams, spider diagrams, the user, frank ruskey, three circles, data sets, venn diagram, the zone, error function, leonhard euler, visual languages, l. euler

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posted: | 4/13/2010 |

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