Efficient Searching for Multi—dimensional Data Made Simple

Abstract

We introduce an innovative decomposition technique which reduces a multi—dimensional searching problem to a sequence of one—dimensional problems, each one easily manageable in optimal time×space complexity using traditional searching strategies. The reduction has no additional storage requirement and the time complexity to reconstruct the result of the original multi—dimensional query is linear in the dimension.

More precisely, we showhowto preprocess a set of S ⊑ IN^d of multi—dimensional objects into a data structure requiring O(m log n) space, where m = {S} and n is the maximum number of different values for each coordinate. The obtained data structure is implicit, i.e. does not use pointers, and is able to answer the exact match query in 7(d - 1) steps. Additionally, the model of computation required for querying the data structure is very simple; the only arithmetic operation needed is the addition and no shift operation is used.

The technique introduced, overcoming the multi—dimensional bottleneck, can be also applied to non traditional models of computation as external memory, distributed, and hierarchical environments. Additionally, we will show how the proposed technique permits the effective realizability of the well known perfect hashing techniques on real data.

The algorithms for building the data structure are easy to implement and run in polynomial time.