Succinct indexable dictionaries with applications to encoding k-ary trees and multisets

We consider the <i>indexable dictionary</i> problem which consists in storing a set <i>S</i> ⊆ {0,…, <i>m</i> - 1} for some integer <i>m,</i> while supporting the operations of <i>rank</i>(<i>x</i>), which returns the number of elements in <i>S</i> that are less than <i>x</i> if <i>x</i> ε <i>S,</i> and -1 otherwise; and <i>select</i>(<i>i</i>) which returns the <i>i</i>-th smallest element in <i>S.</i>We give a structure that supports both operations in <i>O</i>(1) time on the RAM model and requires <i>B</i>(<i>n,m</i>) + <i>o</i>(<i>n</i>) + <i>O</i>(lg lg <i>m</i>) bits to store a set of size <i>n,</i> where <i>B</i>(<i>n,m</i>) = ⌈lg (<inf><i>n</i></inf>^<i>m</i>)⌉ is the minimum number of bits required to store any <i>n</i>-element subset from a universe of size <i>m.</i> Previous dictionaries taking this space only supported (yes/no) membership queries in <i>O</i>(1) time. In the cell probe model we can remove the <i>O</i>(lg lg <i>m</i>) additive term in the space bound, answering a question raised by Fich and Miltersen, and Pagh.We also present two applications of our dictionary structure:• An information-theoretically optimal representation for <i>k-ary cardinal trees</i> (aka <i>k</i>-ary tries). Our structure uses <i>C</i>(<i>n,k</i>) + <i>o</i>(<i>n</i> + lg <i>k</i>) bits to store a <i>k</i>-ary tree with <i>n</i> nodes and can support parent, <i>i</i>-th child, child labeled <i>i,</i> and the degree of a node in constant time, where <i>C</i>(<i>n,k</i>) is the minimum number of bits to store any <i>n</i>-node <i>k</i>-ary tree. Previous space efficient representations for cardinal <i>k</i>-ary trees required <i>C</i>(<i>n,k</i>) + Ω(<i>n</i>) bits.• An optimal representation for multisets where (appropriate generalisations of) the <i>select</i> and <i>rank</i> operations can be supported in <i>O</i>(1) time. Our structure uses <i>B</i>(<i>n, m + n</i>) + <i>o</i>(<i>n</i>) + <i>O</i>(lg lg <i>m</i>) bits to represent a multiset of size <i>n</i> from an <i>m</i> element set; the first term is the minimum number of bits required to represent such a multiset.