Analyzing Metric Space Indexes: What For?

It has been a long way since the beginnings of metric space searching, where people coming from algorithmics tried to apply their background to this new paradigm, obtaining variable, but especially difficult to explain, success or lack of it. Since then, some has been learned about the specifics of the problem, in particular regarding key aspects such as the intrinsic dimensionality, that were not well understood in the beginning. The inclusion of those aspects in the picture has led to the most important developments in the area. Similarly, researchers have tried to apply asymptotic analysis concepts to understand and predict the performance of the data structures. Again, it was soon clear that this was insufficient, and that the characteristics of the metric space itself could not be neglected. Although some progress has been made on understanding concepts such as the curse of dimensionality, modern researchers seem to have given up in using asymptotic analysis. They rely on experiments, or at best in detailed cost models that are good predictors but do not explain why the data structures perform in the way they do. In this paper I will argue that this is a big loss. Even if the predictive capability of asymptotic analysis is poor, it constitutes a great tool to understand the algorithmic concepts behind the different data structures, and gives powerful hints in the design of new ones. I will exemplify my view by recollecting what is known on asymptotic analysis of metric indexes, and will add some new results.