Search costs in quadtrees and singularity perturbation asymptotics

Quadtrees constitute a classical data structure for storing and accessing collections of points in multidimensional space. It is proved that, in any dimension, the cost of a random search in a randomly grown quadtree has logarithmic mean and variance and is asymptotically ditributed as a normal variable. The limit distribution property extends to quadtrees of all dimensions a result only known so far to hold for binary search trees. The analysis is based on a technique of singularity perturbation that appears to be of some generality. For quadtrees, this technique is applied to linear differential equations satisfied by interventing bivariate generating functions This work was partly supported by the ESPRIT Basic Research Action No. 7141 (ALCOM II).