Graphs for Metric Space Searching

[Who doesn’t understand a glance, won’t understand a long explanation either.] – Arab proverb The problem of Similarity Searching consists in finding the elements from a set which are similar to a given query under some criterion. If the similarity is expressed by means of a metric, the problem is called Metric Space Searching. In this thesis we present new methodologies to solve this problem using graphs G(V,E) to represent the metric database. In G, the set V corresponds to the objects from the metric space and E to a small subset of edges from V × V , whose weights are computed according to the metric of the space under consideration. In particular, we study k-nearest neighbor graphs (knngs). The knng is a weighted graph connecting each element from V —or equivalently, each object from the metric space— to its k nearest neighbors. We develop algorithms both to construct knngs in general metric spaces, and to use them for proximity searching. These results allow us to use a graph to index a metric space, requiring a moderate amount of memory, with better search performance than that of classical pivot-based algorithms. Finally, we show that the graph-based approach offers a great potential for improvements, ranging from fully dynamic graph-based indices to optimizations tuned for metric space searching. The high amount of computational resources required to build and traverse these graphs also motivated us to research on fundamental algorithmic problems, such as incremental sorting and priority queues. We propose new algorithms for these problems which, in practice, improve upon the state-of-the-art solutions in many cases, and are useful in many other algorithmic scenarios. As a matter of fact, they yield one of the fastest Minimum Spanning Tree (MST) construction algorithms for random graphs. These basic algorithms not only open new research lines, for instance, MST construction algorithms for arbitrary graphs; but also they turn out to be appealing to be applied directly in production environments.