Cell-Probe Lower Bounds for Succinct Partial Sums

The partial sums problem in succinct data structures asks to preprocess an array A[1 . . n] of bits into a data structure using as close to n bits as possible, and answer queries of the form Rank(k) = k i=1 A[i]. The problem has been intensely studied, and features as a subroutine in a number of succinct data structures. We show that, if we answer Rank(k) queries by probing t cells of w bits, then the space of the data structure must be at least n + n/wO(t) bits. This redundancy/probe trade-off is essentially optimal: Patrascu [FOCS’08] showed how to achieve n+n (w/t)Ω(t) bits. We also extend our lower bound to the closely related Select queries, and to the case of sparse arrays.

• Type: Journal
• Year: 2010
• Where: SODA
• Authors: Mihai Patrascu, Emanuele Viola
• Keywords: Algorithms, Data Structure, Partial Sums Problem, SODA 2010, Succinct Data