Approximate Distance Sensitivity Oracles in Subquadratic Space

TL;DR

This work advances fault-tolerant distance sensitivity oracles by achieving subquadratic-space, near-optimal stretch results for constant $f$ on unweighted graphs. It combines short-path handling via Thorup–Zwick-style DOs with long-path solutions built from fault-tolerant trees, granularity, and novel expath representations to yield a $(3+\varepsilon)$-approximate $f$-DSO and sublinear-query-time variants. A key contribution is the introduction of FT-trees with granularity and a hierarchical, pyramid-like replacement-path framework that maintains subquadratic space while delivering strong approximation guarantees. Additionally, the paper refines the preprocessing of existing $f$-DSOs, notably Chechik–Cohen–Fiat–Kaplan, reducing their preprocessing time to $\tilde{O}(mn^{2+o(1)}/\varepsilon^{f})$, thereby broadening practical applicability. The results collectively push the boundary between space, query time, and stretch in fault-tolerant distance querying for large graphs.

Abstract

An $f$-edge fault-tolerant distance sensitive oracle ($f$-DSO) with stretch $σ\ge 1$ is a data structure that preprocesses a given undirected, unweighted graph $G$ with $n$ vertices and $m$ edges, and a positive integer $f$. When queried with a pair of vertices $s, t$ and a set $F$ of at most $f$ edges, it returns a $σ$-approximation of the $s$-$t$-distance in $G-F$. We study $f$-DSOs that take subquadratic space. Thorup and Zwick [JACM 2005] showed that this is only possible for $σ\ge 3$. We present, for any constant $f \ge 1$ and $α\in (0, \frac{1}{2})$, and any $\varepsilon > 0$, a randomized $f$-DSO with stretch $ 3 + \varepsilon$ that w.h.p. takes $\widetilde{O}(n^{2-\fracα{f+1}}) \cdot O(\log n/\varepsilon)^{f+2}$ space and has an $O(n^α/\varepsilon^2)$ query time. The time to build the oracle is $\widetilde{O}(mn^{2-\fracα{f+1}}) \cdot O(\log n/\varepsilon)^{f+1}$. We also give an improved construction for graphs with diameter at most $D$. For any positive integer $k$, we devise an $f$-DSO with stretch $2k-1$ that w.h.p. takes $O(D^{f+o(1)} n^{1+1/k})$ space and has $\widetilde{O}(D^{o(1)})$ query time, with a preprocessing time of $O(D^{f+o(1)} mn^{1/k})$. Chechik, Cohen, Fiat, and Kaplan [SODA 2017] devised an $f$-DSO with stretch $1{+}\varepsilon$ and preprocessing time $O(n^{5+o(1)}/\varepsilon^f)$, albeit with a super-quadratic space requirement. We show how to reduce their preprocessing time to $O(mn^{2+o(1)}/\varepsilon^f)$.

Figures (4)

• Figure 4: A visualization of the trapezoid $\mathop{\mathrm{tr}}\nolimits^{\varepsilon/9}_{G-F}(P)$ in \ref{['lem:not_too_far_off']} for the case $u=x$. The vertices $u,v$ are endpoints of failing edges in $F$ or the query vertices $s$ or $t$, they are not part of $\mathop{\mathrm{tr}}\nolimits^{\varepsilon/9}_{G-F}(P)$. Vertex $y$ lies on the path $P$ and vertex $z$ is in $V(F)$. The replacement path from $y$ to $z$ has length at most $\frac{\varepsilon}{9} \space d_{G-F}(u,y)$. The smaller trapezoid around $P[u..y] \circ P(y,z,F)$ (red dashed line) does not contain any vertex from $V(F)$.
• Figure 5: The red, blue and yellow vertices are some of the netpoints contained in $p_\text{left}$ for the path $(u,b)$. The set $p_\text{right}$ is created in the same way but with reversed distance (hinted below in grey), i.e., from $b$ to $u$.
• Figure 7: The exploded graph for computing shortest $\ell$-decomposable paths.
• Figure 8: Example construction of graph $G_i$ (right) from $G$ (left) for the algorithm to compute shortest $i$-partial $\ell$-expaths (\ref{['lem:l-expath']}). Each layer $V_{i,j}$ is connected to its neighboring layers in the same way as the first two layers. If the red path is the shortest path from $s_0$ to $a_\ell$, then the entry nodes for $a_\ell$ are $w_{a_\ell} = b_0$ and $x_{a_\ell} = a_\ell$.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• corollary 1.3
• Theorem 1.4
• Lemma 4.1: Inheritance property
• Lemma 4.1
• Lemma 4.2
• Lemma 5.1
• definition 5.2: $\frac{\varepsilon}{9}$-trapezoid
• Lemma 5.3: Lemma 2.6 in ChCoFiKa17