Expected-case Complexity of Approximate Nearest Neighbor Searching

Most research in algorithms for geometric query problems has focused on their worst-case performance. However, when information on the query distribution is available, the alternative paradigm of designing and analyzing algorithms from the perspective of expected-case performance appears more attractive. We study the approximate nearest neighbor problem from this perspective. As a first step in this direction, we assume that the query points are sampled uniformly from a hypercube that encloses all the data points; however, we make no assumption on the distribution of the data points. We show that with a simple partition tree, called the sliding-midpoint tree, it is possible to achieve linear space and logarithmic query time in the expected case; in contrast, the data structures known to achieve linear space and logarithmic query time in the worst case are complex, and algorithms on them run more slowly in practice. Moreover, we prove that the sliding-midpoint tree achieves optimal expected query time in a certain class of algorithms