XOR-Functions and Beyond in the Random Query Model
Sumegha Garg, Janani Sundaresan, Santhoshini Velusamy
We study time--space tradeoffs for computing Boolean functions $f:\{0,1\}^n \to \{0,1\}$ in the random query model. On input $x$, a branching program receives independent samples $(i,x(i))$, where $i$ is uniform in $[n]$. We want to evaluate $f(x)$ while minimizing the width $2^S$ and the length $T$ of the branching program. There is a simple upper bound of $T = \tilde{O}(n)$, $S = O(n)$: coupon-collector shows that $\tilde{O}(n)$ samples suffice, and they can be stored in $O(n)$ space. Raz and Zhan (ITCS 2020) introduced this model and studied the zero-error setting. In the bounded-error setting, Dinur (SODA 2024) proved $TS = \tildeΩ(n \cdot inf[f])$, where $inf[f]$ is the total influence of $f$. For many functions with low influence such as Equality and Tribes, this lower bound can leave a substantial gap from the best known upper bounds. Our main result is a stronger lower bound of $T(S+\log T) = Ω(n \cdot sens[g])$ for any XOR-function $g(x,y)= f(x \oplus y)$, where $sens[g]$ is the sensitivity of $g$. Through simple corollaries, we get tight nearly-quadratic time--space lower bounds for Tribes and Equality, despite their low influence. Our proof introduces a new notion of information complexity tailored to the random query model along with a corresponding direct-sum argument which may be of independent interest. Dinur also proved a tight bound of $TS = Ω(n^2)$ for Majority, whose influence is only $Θ(\sqrt{n})$, and asked whether an alternative characterization of Boolean functions could give a unified proof of tight results for a broader class of functions, including Majority. We answer this open question by proving $TS = \tildeΩ(n \cdot dgn[f])$, where $dgn[f]$ is the degeneracy of the sensitivity graph of $f$ (for Majority, this is $Θ(n)$). We also prove that this bound is tight for all symmetric functions up to logarithmic factors.
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