LAUSR

Significance Geographic maps are a popular means to visualize spatial statistics. Conventionally, each map region is displayed with an area proportional to the actual land area. But equal-area maps can grossly misrepresent demographic data: Densely populated cities should be given more prominence than large, but sparsely populated territories. Cartograms solve this problem by rescaling map regions in proportion to, for example, population or gross domestic products. Until now, it has generally been cumbersome or slow to calculate map projections for contiguous cartograms. Here we describe and benchmark a fast flow-based algorithm that computes cartograms in a matter of seconds, yet maintains the strengths of previous methods—a development which may lead to a more widespread adoption of cartograms. Cartograms are maps that rescale geographic regions (e.g., countries, districts) such that their areas are proportional to quantitative demographic data (e.g., population size, gross domestic product). Unlike conventional bar or pie charts, cartograms can represent correctly which regions share common borders, resulting in insightful visualizations that can be the basis for further spatial statistical analysis. Computer programs can assist data scientists in preparing cartograms, but developing an algorithm that can quickly transform every coordinate on the map (including points that are not exactly on a border) while generating recognizable images has remained a challenge. Methods that translate the cartographic deformations into physics-inspired equations of motion have become popular, but solving these equations with sufficient accuracy can still take several minutes on current hardware. Here we introduce a flow-based algorithm whose equations of motion are numerically easier to solve compared with previous methods. The equations allow straightforward parallelization so that the calculation takes only a few seconds even for complex and detailed input. Despite the speedup, the proposed algorithm still keeps the advantages of previous techniques: With comparable quantitative measures of shape distortion, it accurately scales all areas, correctly fits the regions together, and generates a map projection for every point. We demonstrate the use of our algorithm with applications to the 2016 US election results, the gross domestic products of Indian states and Chinese provinces, and the spatial distribution of deaths in the London borough of Kensington and Chelsea between 2011 and 2014.