[Paper Review] Preserving Distances in Very Faulty Graphs

Greg Bodwin, Fabrizio Grandoni+2 more|arXiv (Cornell University)|Mar 30, 2017

Complexity and Algorithms in Graphs36 references4 citations

TL;DR

This paper presents the first truly sub-quadratic size single-pair distance preservers for unweighted graphs under any fixed number of edge faults, achieving $O(n^{2-g(f)})$ size for $f \geq 1$ with a positive decreasing function $g(f)$. It establishes tight bounds showing that subquadratic size is only possible with additive error $\Omega(\varepsilon f)$, and provides matching upper and lower bounds for weighted graphs, proving $\Theta(n^2)$ size for $f \geq 2$ and $\Theta(n)$ for $f=1$ in undirected graphs.

ABSTRACT

Preservers and additive spanners are sparse (hence cheap to store) subgraphs that preserve the distances between given pairs of nodes exactly or with some small additive error, respectively. Since real-world networks are prone to failures, it makes sense to study fault-tolerant versions of the above structures. This turns out to be a surprisingly difficult task. For every small but arbitrary set of edge or vertex failures, the preservers and spanners need to contain {\em replacement paths} around the faulted set. In this paper we make substantial progress on fault tolerant preservers and additive spanners: (1) We present the first truly sub-quadratic size single-pair preservers in unweighted (possibly directed) graphs for \emph{any} fixed number $f$ of faults. Our result indeed generalizes to the single-source case, and can be used to build new fault-tolerant additive spanners (for all pairs). (2) The size of the above single-pair preservers is $O(n^{2-g(f)})$ for some positive function $g$, and grows to $O(n^2)$ for increasing $f$. We show that this is necessary even in undirected unweighted graphs, and even if you allow for a small additive error: If you aim at size $O(n^{2-ε})$ for $ε>0$, then the additive error has to be $Ω(\eps f)$. This surprisingly matches known upper bounds in the literature. (3) For weighted graphs, we provide matching upper and lower bounds for the single pair case. Namely, the size of the preserver is $Θ(n^2)$ for $f\geq 2$ in both directed and undirected graphs, while for $f=1$ the size is $Θ(n)$ in undirected graphs. For directed graphs, we have a superlinear upper bound and a matching lower bound. Most of our lower bounds extend to the distance oracle setting, where rather than a subgraph we ask for any compact data structure.

Motivation & Objective

• To resolve the long-open question of the worst-case size of a single-pair distance preserver under $f \geq 3$ edge faults.
• To close the gap in understanding fault-tolerant spanners and preservers, especially for $f > 2$, where no nontrivial bounds existed prior.
• To establish tight lower and upper bounds for both unweighted and weighted graphs, including the first nontrivial sub-quadratic constructions for $f \geq 3$.
• To show that subquadratic preservers are only possible with a nontrivial additive error, and that this error is tightly bounded by $\Omega(\varepsilon f)$ for $O(n^{2-\varepsilon})$ size.

Proposed method

• Construct a reduction from $f$-fault-tolerant single-pair preservers to $f$-fault-tolerant $n$-pair preservers in a modified graph, leveraging replacement paths around faulty edges.
• Use a modular construction where a path $Y$ connects source $s$ to target $t$, and intermediate nodes link to a base graph $K$ that encodes the $n$-pair distance preserver problem.
• Prove lower bounds by showing that any $1$-fault-tolerant $s$-$t$ preserver must preserve all shortest paths in $K$, thus requiring $\Omega(\texttt{DP}(n))$ edges.
• Establish that for $f \geq 2$, the size of any $f$-FT $s$-$t$ preserver in undirected unweighted graphs must be $\Omega(n^{2-\varepsilon})$ for any $\varepsilon > 0$, unless the additive error is $\Omega(f)$.
• Derive upper bounds via a new construction that computes $f$-FT preservers in $O(fmn)$ time, generalizing to single-source and all-pairs settings.
• Use the fact that replacement paths for $f$ faults require preserving paths that detour through $f$-fault-avoiding structures, and bound the number of such paths via combinatorial arguments.

Experimental results

Research questions

• RQ1 What is the worst-case size of a single-pair distance preserver in unweighted graphs under $f \geq 3$ edge faults?
• RQ2 Can sub-quadratic size $f$-fault-tolerant preservers exist for $f \geq 3$, and if so, what is the trade-off with additive error?
• RQ3 What are the tight bounds for $f$-fault-tolerant distance preservers in weighted graphs, particularly for $f=1$ and $f \geq 2$?
• RQ4 Is there a super-linear lower bound for $f$-FT additive spanners in unweighted graphs when $f \geq 2$, even with small additive error?
• RQ5 Can the $f$-FT single-source preserver size be bounded below by $\Omega(n^{2-1/(f+1)})$ for $f \geq 3$, as conjectured?

Key findings

• For unweighted graphs, the paper constructs $f$-fault-tolerant single-pair preservers of size $O(n^{2-g(f)})$ for any fixed $f \geq 1$, with $g(f)$ a positive decreasing function.
• In unweighted graphs, any $f$-FT $s$-$t$ preserver of size $O(n^{2-\varepsilon})$ must have additive error $\Omega(\varepsilon f)$, and this bound is tight.
• For weighted graphs, the size of a $1$-FT $s$-$t$ preserver is $\Theta(n)$ in undirected graphs and $\Theta(\texttt{DP}(n))$ in directed graphs, where $\texttt{DP}(n)$ denotes the sparsity of $n$-pair preservers.
• For $f \geq 2$, the size of a $f$-FT $s$-$t$ preserver in weighted graphs is $\Theta(n^2)$ even in undirected graphs, showing a sharp threshold at $f=1$.
• The paper proves that $\Omega(n^{2-\varepsilon})$ size preservers require $\Omega(\varepsilon f)$ additive error, matching known upper bounds in the literature.
• Lower bounds extend to distance oracles, showing that compact data structures for $f$-FT distances must also have $\Omega(n^{2-\varepsilon})$ size if error is $o(f)$.