On the complexity of reverse similarity search

Two decision problems are presented that arise from reversing the operation of a distance-based indexing tree. Whereas similarity search finds points in the tree given a query point, reverse similarity search begins with a set of constraints like those defining a leaf and generates a point meeting the constraints. These problems derive from robust hashing, a technique used in similarity search and security applications. The problems are analysed for spaces of strings and vectors with a variety of metrics: strings with Hamming distance; the usual (Levenshtein) edit distance; an edit distance we introduce called Superghost distance; arbitrary weighted tree metrics; and real vectors with Minkowski LP metrics (of which the Euclidean distance is a special case). They are found to inhabit different complexity classes depending on the metric. In particular, the reverse similarity search problem derived from a VP- or GH-tree is NP-complete for any LP metric except that it is in P for a GH-tree with the Euclidean metric.