TR24-054 Authors: Karthik Gajulapalli, Alexander Golovnev, Samuel King

Publication: 20th March 2024 12:19

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We study the problem of function inversion with preprocessing where, given a function $f : [N] \to [N]$ and a point $y$ in its image, the goal is to find an $x$ such that $f(x) = y$ using at most $T$ oracle queries to $f$ and $S$ bits of preprocessed advice that depend on $f$.

The seminal work of Corrigan-Gibbs and Kogan [TCC 2019] initiated a line of research that shows many exciting connections between the non-adaptive setting of this problem and other areas of theoretical computer science. Specifically, they introduced a very weak class of algorithms (strongly non-adaptive) where the points queried by the oracle depend only on the inversion point $y$, and are independent of the answers to the previous queries and the $S$ bits of advice. They showed that proving even mild lower bounds on strongly non-adaptive algorithms for function inversion would imply a breakthrough result in circuit complexity.

We prove that every strongly non-adaptive algorithm for function inversion (and even for its special case of permutation inversion) must have $ST = \Omega(N \log (N) \log (T))$. This gives the first improvement to the long-standing lower bound of $ST = \Omega(N \log N)$ due to Yao [STOC 90]. As a corollary, we conclude the first separation between strongly non-adaptive and adaptive algorithms for permutation inversion, where the adaptive algorithm by Hellman [TOIT 80] achieves the trade-off $ST = O( N \log N)$.

Additionally, we show equivalence between lower bounds for strongly non-adaptive data structures and the one-way communication complexity of certain partial functions. As an example, we recover our lower bound on function inversion in the communication complexity framework.