Sampling Permutations with Cell Probes is Hard

TL;DR

This paper proves that generating permutations with limited cell probes is computationally hard, establishing tight lower bounds on the number of probes needed, especially for nonadaptive algorithms, impacting permutation data structures.

Contribution

It establishes tight lower bounds on the probe complexity for sampling permutations, confirming Viola's conjecture up to a constant in the exponent and improving bounds for nonadaptive probes.

Findings

Lower bound: $d igg( ext{probes}igg) ext{ is at least } ( ext{log } n)^{ ext{constant}}$

Nonadaptive probes require polynomially many probes, i.e., $d igg( ext{probes}igg) ext{ is at least } n^{ ext{constant}}$

Implications for lower bounds in succinct permutation data structures

Abstract

Suppose we are given an infinite sequence of input cells, each initialized with a uniform random symbol from $[n]$. How hard is it to output a sequence in $[n]^n$ that is close to a uniform random permutation? Viola (SICOMP 2020) conjectured that if each output cell is computed by making $d$ probes to input cells, then $d\geq\omega(1)$. Our main result shows that, in fact, $d\geq (\log n)^{\Omega(1)}$, which is tight up to the constant in the exponent. Our techniques also show that if the probes are nonadaptive, then $d\geq n^{\Omega(1)}$, which is an exponential improvement over the previous nonadaptive lower bound due to Yu and Zhan (ITCS 2024). Our results also imply lower bounds against succinct data structures for storing permutations.