Simple Dynamic Spanners with Near-optimal Recourse against an Adaptive Adversary

TL;DR

This paper presents new dynamic algorithms for maintaining graph spanners against adaptive adversaries, achieving near-optimal size, stretch, and recourse, thus closing significant gaps in the field.

Contribution

It introduces algorithms that match the best known size-stretch trade-offs with near-optimal recourse against adaptive adversaries, a major advancement over previous work.

Findings

Achieves $(2k-1)$-spanners with $O(n^{1+1/k} ext{log} n)$ size and $O( ext{log} n)$ recourse.

Provides a 3-spanner with $ ilde O(n^{1.5})$ size, polylogarithmic recourse, and $ ilde O( ext{sqrt}(n))$ worst-case update time.

Closes the gap between oblivious and adaptive adversary algorithms in dynamic graph spanners.

Abstract

Designing dynamic algorithms against an adaptive adversary whose performance match the ones assuming an oblivious adversary is a major research program in the field of dynamic graph algorithms. One of the prominent examples whose oblivious-vs-adaptive gap remains maximally large is the \emph{fully dynamic spanner} problem; there exist algorithms assuming an oblivious adversary with near-optimal size-stretch trade-off using only $\operatorname{polylog}(n)$ update time [Baswana, Khurana, and Sarkar TALG'12; Forster and Goranci STOC'19; Bernstein, Forster, and Henzinger SODA'20], while against an adaptive adversary, even when we allow infinite time and only count recourse (i.e. the number of edge changes per update in the maintained spanner), all previous algorithms with stretch at most $\log^{5}(n)$ require at least $\Omega(n)$ amortized recourse [Ausiello, Franciosa, and Italiano ESA'05]. In this paper, we completely close this gap with respect to recourse by showing algorithms against an adaptive adversary with near-optimal size-stretch trade-off and recourse. More precisely, for any $k\ge1$, our algorithm maintains a $(2k-1)$-spanner of size $O(n^{1+1/k}\log n)$ with $O(\log n)$ amortized recourse, which is optimal in all parameters up to a $O(\log n)$ factor. As a step toward algorithms with small update time (not just recourse), we show another algorithm that maintains a $3$-spanner of size $\tilde O(n^{1.5})$ with $\operatorname{polylog}(n)$ amortized recourse \emph{and} simultaneously $\tilde O(\sqrt{n})$ worst-case update time.