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

• Authors: Rajeev Raman;Venkatesh Raman;Srinivasa Rao Satti
• Venue: ACM Transactions on Algorithms (TALG)
• Year: 2007

We consider the indexable dictionary problem, which consists of storing a set S ⊆ {0,…,m − 1} for some integer m while supporting the operations of rank(x), which returns the number of elements in S that are less than x if x ∈ S, and −1 otherwise; and select(i), which returns the ith smallest element in S. We give a data structure that supports both operations in O(1) time on the RAM model and requires B(n, m) + o(n) + O(lg lg m) bits to store a set of size n, where B(n, m) = ⌊lg (m/n)⌋ is the minimum number of bits required to store any n-element subset from a universe of size m. Previous dictionaries taking this space only supported (yes/no) membership queries in O(1) time. In the cell probe model we can remove the O(lg lg m) additive term in the space bound, answering a question raised by Fich and Miltersen [1995] and Pagh [2001]. We present extensions and applications of our indexable dictionary data structure, including: —an information-theoretically optimal representation of a k-ary cardinal tree that supports standard operations in constant time; —a representation of a multiset of size n from {0,…,m − 1} in B(n, m + n) + o(n) bits that supports (appropriate generalizations of) rank and select operations in constant time; and + O(lg lg m) —a representation of a sequence of n nonnegative integers summing up to m in B(n, m + n) + o(n) bits that supports prefix sum queries in constant time.