Consistent thinning of large geographical data for map visualization

Large-scale map visualization systems play an increasingly important role in presenting geographic datasets to end-users. Since these datasets can be extremely large, a map rendering system often needs to select a small fraction of the data to visualize them in a limited space. This article addresses the fundamental challenge of thinning: determining appropriate samples of data to be shown on specific geographical regions and zoom levels. Other than the sheer scale of the data, the thinning problem is challenging because of a number of other reasons: (1) data can consist of complex geographical shapes, (2) rendering of data needs to satisfy certain constraints, such as data being preserved across zoom levels and adjacent regions, and (3) after satisfying the constraints, an optimal solution needs to be chosen based on objectives such as maximality, fairness, and importance of data. This article formally defines and presents a complete solution to the thinning problem. First, we express the problem as an integer programming formulation that efficiently solves thinning for desired objectives. Second, we present more efficient solutions for maximality, based on DFS traversal of a spatial tree. Third, we consider the common special case of point datasets, and present an even more efficient randomized algorithm. Fourth, we show that contiguous regions are tractable for a general version of maximality for which arbitrary regions are intractable. Fifth, we examine the structure of our integer programming formulation and show that for point datasets, our program is integral. Finally, we have implemented all techniques from this article in Google Maps [Google 2005] visualizations of fusion tables [Gonzalez et al. 2010], and we describe a set of experiments that demonstrate the trade-offs among the algorithms.