A Optimality of Clustering Properties of Space Filling Curves

Space filling curves have been used in the design of data structures for multidimensional data since many decades. A fundamental quality metric of a space filling curve is its “clustering number” with respect to a class of queries, which is the average number of contiguous segments on the space filling curve that a query region can be partitioned into. We present a characterization of the clustering number of a general class of space filling curves, as well as the first non-trivial lower bounds on the clustering number for any space filling curve. Our results answer questions that have been open for more than 15 years.