Tight Space Bounds for Pivot-Based Distance Oracles in Sparse Graphs

Abstract

An approximate distance oracle (ADO) is a compact data structure that answers distance queries within a multiplicative \emph{stretch} $k$. The classical oracle of Thorup and Zwick~[STOC'01, JACM'05] has stretch $k=2\ell-1$ and uses $O(\ell n^{1+1/\ell})$ space, for every integer $\ell\ge1$, and this trade-off is optimal for dense graphs under the Erdős girth conjecture. However, for sparse graphs ($m=O(n)$), their lower bound yields nothing beyond $Ω(m)$ space. A pivot-based $1$-hop ADO gives every vertex $\tilde{O}(1)$ pivots and returns as an estimate the length of a walk of the form $u\to p_u \to p_v \to v$, where $p_u$ is a pivot of $u$, $p_v$ is a pivot of $v$, and $p_u\to p_v$ is a stored distance. We observe that all the state-of-the-art ADOs with $\tilde{O}(1)$ query time are in fact pivot-based $1$-hop ADOs. Our main result is that, under the Erdős girth conjecture for graphs with constant degree, there are sparse graphs on which every \emph{pivot-based} $1$-hop ADO with stretch $k$ requires $\widetildeΩ(m^{1+2/(k+1)})$ space, for every \emph{real} $k\ge1$. This matches the best known upper bounds with $\tilde{O}(1)$ query time, up to polylogarithmic factors, at every stretch where they are known. We also extend our lower bound to pivot-based ADOs with any constant number $β$ of hops, showing that they require $\widetildeΩ(m^{1+2/(β(k+1))})$ space. In addition, we obtain a tight lower bound for two-hop hopsets. Under the Erdős girth conjecture, for every integer $\ell\ge3$, $(2\ell-3,2)$-hopsets on sparse graphs require size $Ω(m^{1+1/\ell})$, which matches a construction based on the oracles of [Kadria and Roditty, ISAAC'25]. Using the recent explicit high-girth graphs of Double Agent~[doubleAI'26], we also obtain unconditional versions of our ADO lower bounds, with the exponent above $1$ multiplied by $4/5$.