Compact Data Structures for Dedekind Groups and Finite Rings

Abstract

A group with n elements can be stored using $\mathcal {O}(n^2)$ space via its Cayley table which can answer a group multiplication query in $\mathcal {O}(1)$ time. Information theoretically it needs $\varOmega (n\log n)$ bits or $\varOmega (n)$ words in word-RAM model just to store a group (Farzan and Munro, ISSAC 2006).

For functions $s,t:\mathbb {N}\longrightarrow \mathbb {R}_{\ge 0}$, we say that a data structure is an $(\mathcal {O}(s),\mathcal {O}(t))$-data structure if it uses $\mathcal {O}(s)$ space and answers a query in $\mathcal {O}(t)$ time. Except for cyclic groups it was not known if we can design $(\mathcal {O}(n),\mathcal {O}(1))$-data structure for interesting classes of groups.

In this paper, we show that there exist $(\mathcal {O}(n),\mathcal {O}(1))$-data structures for several classes of groups and for any ring and thus achieve information theoretic lower bound asymptotically. More precisely, we show that there exist $(\mathcal {O}(n),\mathcal {O}(1))$-data structures for the following algebraic structures with n elements:

• Dedekind groups: This class contains abelian groups, Hamiltonian groups.

• Groups whose indecomposable factors admit $(\mathcal {O}(n),\mathcal {O}(1))$-data structures.

• Groups whose indecomposable factors are strongly indecomposable.

• Groups defined as a semidirect product of groups that admit $(\mathcal {O}(n),\mathcal {O}(1))$-data structures.

• Finite rings.