Tradeoff Between Label Space and Auxiliary Space for Representation of Equivalence Classes

Abstract

Given a set of $n$ elements that are partitioned into equivalence classes, we study the problem of assigning unique labels to these elements in order to support the query that asks whether the elements corresponding to two given labels belong to the same equivalence class. This problem has been studied by Katz et al [11], Alstrup et al  [1], and Lewenstein et al [12]. Lewenstein et al [12] showed that if the labels were to be assigned from the set $\{1,...,n\}$, a data structure of size $\Theta (\sqrt{n})$ bits is necessary and sufficient to represent the equivalence classes. They also showed that with no auxiliary data structure, a label space of size $\sum _{i=1}^{n}\lfloor \frac{n}{i}\rfloor$ is necessary and sufficient. Our main result is that if we allow a label space of size $cn$ for any constant $c > 1$, a data structure of size $\Theta (\log n)$ bits is necessary and sufficient. We also show that the equivalence query in such a data structure can be answered in $\Theta (1)$ time. We believe that our work can trigger further work on tradeoffs between label space and auxiliary data structure space for other labeling problems.

This work was sponsored by the NSERC of Canada and the Canada Research Chairs Program.