Algorithmic aspects of proportional symbol maps

Proportional symbol maps visualize numerical data associated with point locations by placing a scaled symbol—typically opaque disks or squares—at the corresponding point on a map. Overlapping symbols need to be drawn in such a way that the user can still judge their relative sizes accurately. We identify two types of suitable drawings: physically realizable drawings and stacking drawings. For these we study the following two problems: Max-Min—maximize the minimum visible boundary length of each symbol—and Max-Total—maximize the total visible boundary length over all symbols. We show that both problems are NP-hard for physically realizable drawings. Max-Min can be solved in O(n2 log n) time for stacking drawings, which can be improved to O(n log n) or O(n log2 n) time when the input has certain properties. We also experimented with four methods to compute stacking drawings: our solution to the Max-Min problem performs best on the data sets considered.