Team 1034 Page 2 of 21 Contents 1 Introduction 1.1 Current Models 1.2 Developing Our Approach 2 Notation and definitions 3 Theoretical evaluation of our model 4 Method Description 4.1 Voronoi Diagrams 4.1.1 Useful Features of Voronoi diagrams 4.2 Voronoiesque Diagrams 7889 4.3 Determining Generator Points Using Population Density Distributions 4.4 Procedure for Creating Regions using Voronoi and Voronoiesque Diagrams. 10 5 Redistricting in New York State opulation Density Map 5.2 Limitations of the Image-Based Density Map 11 5.3 Selecting Generator Points 5.4 Applying Voronoi Diagrams to NY 14 5.5 Applying Voronoiesque Diagrams to NY 14 5.6 Precisely Defining Boundary line 6 Analysis 6.1 New York State result 6.2 General results 7 Improving the Method 7.1 Boundary Refinement 72G Obstacles 8 Bulletin to the voters of the state of New york 9 Conclusion List of figures 1 Illustration of Voronoi diagram generated with Euclidean metric. Note the ompactness and simplicity of the regions. 2 Illustration of the process of growing a Voronoiesque diagram with respect to a population density. Only three three generator points are used. Figures from left to right iterate with timeTeam 1034 Page 2 of 21 Contents 1 Introduction 4 1.1 Current Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Developing Our Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Notation and Definitions 5 3 Theoretical Evaluation of our Model 6 4 Method Description 7 4.1 Voronoi Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.1.1 Useful Features of Voronoi Diagrams . . . . . . . . . . . . . . . . . . 8 4.2 Voronoiesque Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.3 Determining Generator Points Using Population Density Distributions . . . 9 4.4 Procedure for Creating Regions using Voronoi and Voronoiesque Diagrams . 10 5 Redistricting in New York State 10 5.1 Population Density Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2 Limitations of the Image-Based Density Map . . . . . . . . . . . . . . . . . 11 5.3 Selecting Generator Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.4 Applying Voronoi Diagrams to NY . . . . . . . . . . . . . . . . . . . . . . . 14 5.5 Applying Voronoiesque Diagrams to NY . . . . . . . . . . . . . . . . . . . . 14 5.6 Precisely Defining Boundary Lines . . . . . . . . . . . . . . . . . . . . . . . 17 6 Analysis 17 6.1 New York State Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.2 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7 Improving the Method 18 7.1 Boundary Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7.2 Geographic Obstacles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 8 Bulletin to the Voters of the State of New York 19 9 Conclusion 20 List of Figures 1 Illustration of Voronoi diagram generated with Euclidean metric. Note the compactness and simplicity of the regions. . . . . . . . . . . . . . . . . . . . 7 2 Illustration of the process of ‘growing’ a Voronoiesque diagram with respect to a population density. Only three three generator points are used. Figures from left to right iterate with time. . . . . . . . . . . . . . . . . . . . . . . . 9