On Farey table and its compression for space optimization with guaranteed error bounds

Abstract

Farey sequences, introduced by such renowned mathematicians as John Farey, Charles Haros, and Augustin-L. Cauchy over 200 years ago, are quite well- known by today in theory of fractions, but its computational perspectives are pos- sibly not yet explored up to its merit. In this paper, we present some novel theoret- ical results and e cient algorithms for representation of a Farey sequence through a Farey table. The ranks of the fractions in a Farey sequence are stored in the Farey table to provide an e cient solution to the rank problem, thereby aiding in and speeding up any application frequently requiring fraction ranks for computational speed-up. As the size of the Farey sequence grows quadratically with its order, the Farey table becomes inadvertently large, which calls for its (lossy) compression up to a permissible error. We have, therefore, proposed two compression schemes to obtain a compressed Farey table (CFT). The necessary analysis has been done in detail to derive the error bound in a CFT. As the nal step towards space opti- mization, we have also shown how a CFT can be stored in a 1-dimensional array. Experimental results have been furnished to demonstrate the characteristics and e ciency of a Farey table and its compressed form.