Randomized vs. Deterministic Separation in Time-Space Tradeoffs of Multi-Output Functions

TL;DR

This paper demonstrates the first polynomial separation between randomized and deterministic time-space tradeoffs for multi-output functions, revealing fundamental differences in computational complexity under different models.

Contribution

It introduces a total function exhibiting a polynomial gap in time-space tradeoffs between randomized and deterministic algorithms, with a novel proof method distinct from prior probabilistic approaches.

Findings

Randomized algorithms achieve efficient computation with low space.

Deterministic algorithms require significantly more time and space.

The lower bound implies limitations on derandomization of space-efficient algorithms.

Abstract

We prove the first polynomial separation between randomized and deterministic time-space tradeoffs of multi-output functions. In particular, we present a total function that on the input of $n$ elements in $[n]$, outputs $O(n)$ elements, such that: (1) There exists a randomized oblivious algorithm with space $O(\log n)$, time $O(n\log n)$ and one-way access to randomness, that computes the function with probability $1-O(1/n)$; (2) Any deterministic oblivious branching program with space $S$ and time $T$ that computes the function must satisfy $T^2S\geq\Omega(n^{2.5}/\log n)$. This implies that logspace randomized algorithms for multi-output functions cannot be black-box derandomized without an $\widetilde{\Omega}(n^{1/4})$ overhead in time. Since previously all the polynomial time-space tradeoffs of multi-output functions are proved via the Borodin-Cook method, which is a probabilistic method that inherently gives the same lower bound for randomized and deterministic branching programs, our lower bound proof is intrinsically different from previous works. We also examine other natural candidates for proving such separations, and show that any polynomial separation for these problems would resolve the long-standing open problem of proving $n^{1+\Omega(1)}$ time lower bound for decision problems with $\mathrm{polylog}(n)$ space.