Distributed Distance Sensitivity Oracles

TL;DR

This work introduces the first nontrivial Distance Sensitivity Oracle (DSO) constructions for directed graphs in the CONGEST model, along with unconditional lower bounds and near-optimal results for related problems. It develops two main DSO approaches: (i) fast-query DSOs with $O(k+D)$ query rounds after $\tilde{O}(n^{3/2})$ preprocessing and $\tilde{O}(n)$ space per node, and (ii) fast-preprocessing DSOs achieving $\tilde{O}(n)$ preprocessing rounds and $\tilde{O}(k\sqrt{n}+D)$ query rounds, both with $\tilde{O}(n^2)$ total oracle size. A key distributed primitive is the Excluded Shortest Paths computation, used to build replacement-path-aware structures, and the results extend to seb-DSO with tight or near-tight bounds linked to $k$-SSP. The paper also gives near-optimal upper and lower bounds for All-Pairs Second Simple Shortest Paths (2-APSiSP), with a $\tilde{O}(n)$-round algorithm and $\tilde{\Omega}(n)$-round lower bound in undirected unweighted graphs, even when APSP distances are known. Collectively, these results advance fault-tolerant routing in distributed networks by enabling efficient reconfiguration under edge failures and by clarifying fundamental limits in CONGEST.

Abstract

We present results for the distance sensitivity oracle (DSO) problem, where one needs to preprocess a given directed weighted graph $G=(V,E)$ in order to answer queries about the shortest path distance in $G$ from vertex $s$ to vertex $t$ avoiding edge $e$, for any $s,t \in V, e \in E$. DSO enables optimal re-routing under a link failure, and can serve as a key component for fault tolerance in a distributed setting. However, no non-trivial results for DSO are known in the distributed CONGEST model. We present DSO algorithms with different tradeoffs between preprocessing and query cost: one that optimizes query response rounds, and another that prioritizes preprocessing rounds. We complement these algorithms with unconditional CONGEST lower bounds for DSO. Our DSO lower bounds build on a lower bound we present for the $k$-source shortest paths problem ($k$-SSP), which may be of independent interest. Additionally, we present almost-optimal upper and lower bounds for the related all pairs second simple shortest path (2-APSiSP) problem.

Figures (6)

• Figure 1: Example graph $G$ with out-shortest path tree $T_x$ for source vertex $x$. Blue path $P=\langle a,b,c \rangle$ denotes an example excluded path. $T_x(P)$ denotes the subtree rooted at key vertex $b$ of $P$ (second vertex on $P$). Solid edges denote tree edges in $T_x$, dashed $v$-$z$ edge denotes non-tree edge in $G$.
• Figure 2: Example graph $G$ where path $P=\langle a,b,c \rangle$ is removed from $T_x$. Green edges denote the shortest $x$ to $z$ path through $v$ when $P$ is excluded. Dotted red edge denotes an added $x$-$z$ edge, assigned the same weight as the green path from $x$ to $z$ in $G$.
• Figure 3: Lower bound construction for directed unweighted $k$-SSP
• Figure 4: Lower bound construction for undirected weighted $k$-SSP. Thick edges (between tree leaves and path vertices) all have weight $5nW$, and dashed edges all have weight $W$ ($W = O(n)$ is a weight parameter).
• Figure 5: Lower bound for answering $k$ DSO queries in directed unweighted graphs

Theorems & Definitions (46)

• Definition 1: Shortest path trees and independent paths
• Definition 2: Excluded Shortest Paths Problem
• Definition 3: Distance Sensitivity Oracles Problem (DSO)
• Definition 4: Single-Edge Batched DSO (seb-DSO)
• Definition 5: General Batched DSO
• Definition 6: $k$-Source Shortest Paths Problem ($k$-SSP)
• Definition 7: All Pairs Second Simple Shortest Path Problem (2-APSiSP)
• Theorem 1
• Theorem 2
• Theorem 3