Path-Reporting Distance Oracles with Logarithmic Stretch and Size O(n loglog n)

TL;DR

This work advances path-reporting distance oracles by intertwining novel sparse distance preservers, small-support hopsets, and interactive structures to approach non-path-reporting lower bounds while preserving path outputs. It introduces two main PRDO constructions with near-linear base size and linear-in-k stretch, achieving $S(n,k)=O\left(\left\lceil\frac{k\log\log n}{\log n}\right\rceil\,n^{1+\frac{1}{k}}\right)$ and $f(k)=O(k)$, $q(k)=O\left(\log\left\lceil\frac{k\log\log n}{\log n}\right\rceil\right)$, plus regimes with $S(n,k)=O(n^{1+\frac{1}{k}})$ and comparable query times. By constructing $(1+\varepsilon)$-preservers with near-linear size and integrating them with hopsets of small support, the authors obtain interactive preservers, emulators, and spanners that compose into PRDOs with attractive size-stretch-time trade-offs, including ultra-sparse variants. The results push PRDOs closer to the bounds known for non-path-reporting distance oracles, while providing practical path outputs and modular construction that can adapt to different regimes of $k$, $n$, and error tolerance. The work also furnishes a framework for ultra-sparse and linear-size spanners in the interactive setting, with broad implications for distance queries in large graphs.

Abstract

Given an $n$-vertex undirected graph $G=(V,E,w)$, and a parameter $k\geq1$, a path-reporting distance oracle (or PRDO) is a data structure of size $S(n,k)$, that given a query $(u,v)\in V^2$, returns an $f(k)$-approximate shortest $u-v$ path $P$ in $G$ within time $q(k)+O(|P|)$. Here $S(n,k)$, $f(k)$ and $q(k)$ are arbitrary functions. A landmark PRDO due to Thorup and Zwick, with an improvement of Wulff-Nilsen, has $S(n,k)=O(k\cdot n^{1+\frac{1}{k}})$, $f(k)=2k-1$ and $q(k)=O(\log k)$. The size of this oracle is $Ω(n\log n)$ for all $k$. Elkin and Pettie and Neiman and Shabat devised much sparser PRDOs, but their stretch was polynomially larger than the optimal $2k-1$. On the other hand, for non-path-reporting distance oracles, Chechik devised a result with $S(n,k)=O(n^{1+\frac{1}{k}})$, $f(k)=2k-1$ and $q(k)=O(1)$. In this paper we make a dramatic progress in bridging the gap between path-reporting and non-path-reporting distance oracles. We devise a PRDO with size $S(n,k)=O(\lceil\frac{k\log\log n}{\log n}\rceil\cdot n^{1+\frac{1}{k}})$, stretch $f(k)=O(k)$ and query time $q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)$. We can also have size $O(n^{1+\frac{1}{k}})$, stretch $O(k\cdot\lceil\frac{k\log\log n}{\log n}\rceil)$ and query time $q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)$. Our results on PRDOs are based on novel constructions of approximate distance preservers, that we devise in this paper. Specifically, we show that for any $ε>0$, any $k=1,2,...$, and any graph $G$ and a collection $\mathcal{P}$ of $p$ vertex pairs, there exists a $(1+ε)$-approximate preserver with $O(γ(ε,k)\cdot p+n\log k+n^{1+\frac{1}{k}})$ edges, where $γ(ε,k)=(\frac{\log k}ε)^{O(\log k)}$. These new preservers are significantly sparser than the previous state-of-the-art approximate preservers due to Kogan and Parter.

Figures (4)

• Figure 1: An illustration for the way that the oracle $D$ acts on a query $(u,v)$. First, it invokes the partial TZ oracle $D_0$ on the query $(u,v)$. In case that $D_0$ did not output a $u-v$ path, it outputs two vertices $u',v'\in S$ and two paths - $P_{u,u'}$ from $u$ to $u'$, and $P_{v,v'}$ from $v$ to $v'$. Each of these paths (depicted by black lines) has weight at most $h\cdot d_G(u,v)$, and thus, by the triangle inequality, $d_G(u',v')\leq(2h+1)d_G(u,v)$. Then, the oracle $D$ invokes the interactive emulator $D_E$ on the query $(u',v')$ to produce a $u'-v'$ path $P_{u',v'}$ (depicted by blue lines), that is not necessarily contained in the graph $G$, but consists only of vertices of $S$, and has weight at most $\alpha_Ek_1\cdot(2h+1)d_G(u,v)$. Finally, on each edge $(x,y)$ of $P_{u',v'}$, the oracle $D$ invokes the interactive distance preserver $D_P$, to obtain an $x-y$ path in $G$ (depicted by orange lines), with weight at most $\alpha_Pd_G(x,y)$. Concatenating all these paths, for every edge $(x,y)$ in $P_{u',v'}$, results in a path in $G$, with weight at most $\alpha_P\cdot\alpha_Ek_1\cdot(2h+1)d_G(u,v)$. Together with $P_{u,u'}$ and $P_{v,v'}$, we obtain a $u-v$ path in $G$, with weight at most $\left(2h+\alpha_P\cdot\alpha_Ek_1\cdot(2h+1)\right)d_G(u,v)$.
• Figure 2: If the dashed line between $p_i(a),p_i(b')$ represents a hop-edge in $H$, then $[x,y]$ is labeled as Good. Otherwise, it is labeled as Bad.
• Figure 3: A branching event. When we assume that $d(u,v)\leq d(y,z)$, it follows that the weight of the sub-paths from each of $y,z,u,v$ to $x$ is at most $d(y,z)\leq\frac{1}{2}d(y,p_{i+1}(y))$.
• Figure 4: The returned path. The dashed lines are the paths $P_i$. Notice that they do not always pass through the roots of their respective clusters $C_i$.

Theorems & Definitions (66)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1: EP15
• Definition 4: Interactive Spanner
• Definition 5: Interactive Emulator
• Definition 6: Interactive Distance Preserver
• Claim 1
• Lemma 1
• proof