The cell probe complexity of succinct data structures

We consider time-space tradeoffs for static data structure problems in the cell probe model with word size 1 (the bit probe model). In this model, the goal is to represent nn-bit data with s=n+rs=n+r bits such that queries (of a certain type) about the data can be answered by reading at most tt bits of the representation. Ideally, we would like to keep both ss and tt small, but there are tradeoffs between the values of ss and tt that limit the possibilities of keeping both parameters small.In this paper, we consider the case of succinct   representations, where s=n+rs=n+r for some redundancy  r≪nr≪n. For a Boolean version of the problem of polynomial evaluation with preprocessing of coefficients, we show a lower bound on the redundancy–query time tradeoff of the form (r+1)t≥Ω(n/logn).(r+1)t≥Ω(n/logn). In particular, for very small redundancies rr, we get an almost optimal lower bound stating that the query algorithm has to inspect almost the entire data structure (up to a logarithmic factor). We show similar lower bounds for problems satisfying a certain combinatorial properties of a coding theoretic flavor, and obtain (r+1)t≥Ω(n)(r+1)t≥Ω(n) for certain problems. Previously, no ω(m)ω(m) lower bounds were known on tt in the general model for explicit Boolean problems, even for very small redundancies.By restricting our attention to systematic or index   structures ϕϕ satisfying ϕ(x)=x⋅ϕ∗(x)ϕ(x)=x⋅ϕ∗(x) for some map ϕ∗ϕ∗ (where ⋅⋅ denotes concatenation), we show similar lower bounds on the redundancy–query time tradeoff for the natural data structuring problems of Prefix Sum and Substring Search.