Team 1034 Page 9 of 21 Figure 2: Illustration of the process ofgrowing a Voronoiesque diagram with respect to a population density. Only three three generator points are used. Figures from left to right terate with time for all vi, vi representing different regions. The manner in which the v (t)'s are grown radially from one iteration to the next is determined by the metric used What's useful about Voronoiesque d agran is that their growth can De c ontrolled by requiring that the area under the function f for each region is the same at every iteration In our model, we take f to be the population distribution of the state. Thus the abov equation is a statement of population equality. Also, when f is constant, the regions grow at a constant rate, so the resulting diagram is voronoi. The final consideration for using Voronoiesque diagrams is determining the locat generator points 4.3 Determining Generator Points Using Population Density Dis- For now, we have defined how to generate regions given a set of generator points. Here we consider how to define the generator points in order to create Voronoi and Vornoiesque diagrams. In the case of Voronoi diagrams, this is our only degree of freedom since gen- erator points generate unique Voronoi regions. We found no well defined algorithm to do this, but instead came up with a procedure that functions decently Our first approach is to place generator points at the m largest set of peaks that are well distributed throughout the state, (where m is the required number of districts in that state). By choosing generator points in this way, we keep larger cities within the boundaries we will generate with Voronoi or Vornoiesque diagrams and we make sure the generator points are well dispersed throughout the state. One problem that arises is when cities are so large that in order for districts to hold the same amount of people, a city must be divided into districts. A perfect example is New York City, which contains enough people to hold 13 districts. Taking large cities into account takes extra consideration. Our second approach is to choose the largest peaks in the population distribution and assign each peak with a weight. The weight for each generator point is the number of districts the population surrounding that peak needs to be divided into. We call this weight the degeneracy of the generator point. We begin assigning generator points to the highest populated cities with their corresponding degeneracies until the sum of all the generator points and their respective degeneracies is equal to m. In other words, untilTeam 1034 Page 9 of 21 Figure 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. for all Vi, Vj representing different regions. The manner in which the V(t) i ’s are grown radially from one iteration to the next is determined by the metric used. What’s useful about Voronoiesque diagrams is that their growth can be controlled by requiring that the area under the function f for each region is the same at every iteration. In our model, we take f to be the population distribution of the state. Thus the above equation is a statement of population equality. Also, when f is constant, the regions grow at a constant rate, so the resulting diagram is Voronoi. The final consideration for using Voronoiesque diagrams is determining the location for generator points. 4.3 Determining Generator Points Using Population Density Dis￾tributions For now, we have defined how to generate regions given a set of generator points. Here we consider how to define the generator points in order to create Voronoi and Vornoiesque diagrams. In the case of Voronoi diagrams, this is our only degree of freedom since gen￾erator points generate unique Voronoi regions. We found no well defined algorithm to do this, but instead came up with a procedure that functions decently. Our first approach is to place generator points at the m largest set of peaks that are well distributed throughout the state, (where m is the required number of districts in that state). By choosing generator points in this way, we keep larger cities within the boundaries we will generate with Voronoi or Vornoiesque diagrams and we make sure the generator points are well dispersed throughout the state. One problem that arises is when cities are so large that in order for districts to hold the same amount of people, a city must be divided into districts. A perfect example is New York City, which contains enough people to hold 13 districts. Taking large cities into account takes extra consideration. Our second approach is to choose the largest peaks in the population distribution and assign each peak with a weight. The weight for each generator point is the number of districts the population surrounding that peak needs to be divided into. We call this weight the degeneracy of the generator point. We begin assigning generator points to the highest populated cities with their corresponding degeneracies until the sum of all the generator points and their respective degeneracies is equal to m. In other words, until: