arXiv subjects

Arnold Filtser

Publications and source records attributed to Arnold Filtser.

At least 19 recordsLinked to original sources

arXiv2026

Fully Dynamic All-Pairs Min Cuts on Planar Graphs via Planar Pattern Emulators

We study the fully-dynamic all-pairs minimum cut problem on planar graphs. Given an $n$-vertex undirected planar graph $G=(V,E)$ undergoing edge insertions and deletions, the goal is to efficiently process these updates while supporting $s$-$t$ minimum cut queries for any pair of vertices $s$ and $t$. We give a $(1+\varepsilon)$-approximate dynamic algorithm that supports edge updates and minimum cut queries in $n^{o(1)}$ time, for any $1/\text{poly}(\log n) < \varepsilon < 1$. For general graphs, previous results either are limited to partially dynamic settings or achieve only logarithmic-factor approximations. The main conceptual and technical contribution is to show that a suitable adaptation of the planar emulator of Chang, Krauthgamer, and Tan~[STOC'22] satisfies the stronger pattern emulator introduced by Chen and Tan~[ICALP'25]. This leads almost immediately to a nearly linear-time construction of pattern emulators, which, when combined with the recent fully dynamic $(1+\varepsilon)$-approximate all-pairs shortest paths algorithm of Filtser, Goranci, Patel, and Probst Gutenberg~[FOCS'24], yields our result.

cs.DS↗

arXiv2026

How to Walk a Dog in Parallel: On Parallel Computation of the Discrete Frechet Distance

The Fréchet distance (also known as the dog-leash distance) is a well-studied and prevalent distance measure for polygonal curves. In this paper we present parallel algorithms for computing the discrete \frechet\ distance between two curves, in both the PRAM and MPC models. For two polygonal curves of length $n$, we show that the distance between them can be computed in the PRAM model in $O(\log n/ σ)$ rounds and $\tilde{O}(n^{2+σ})$ total work for any $0<σ<1$. In the sublinear MPC model with $n^ρ$ space per machine, it can be computed in $O(\log n / ρ^2)$ rounds and $\tilde{O}(n^{2+ρ})$ total space. For $c$-packed curves, we match our bounds to the sequential algorithms, showing that a $(1+\eps)$-approximation of the distance can be computed in the PRAM model in $O(\log{n}+1/σ^3)$ rounds and $\tilde{O}(n^{1+σ})$ total work, and in the sublinear MPC model in $O(1 /ρ^3)$ rounds and $\tilde{O}(n^{1+ρ})$ total space, for any constant $\eps>0$. As part of our algorithms for $c$-packed curves, we also present parallel algorithms for computing curve simplifications, which may be of independent interest.

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arXiv2026

Geometric Hashing in MPC with Applications to Clustering and Stochastic Embeddings

In this paper we study consistent hashing (a.k.a. sparse partitions) for doubling metrics in the Massively Parallel Computation (MPC) model. A $(Γ,Λ)$-hash with diameter bound $Δ$ partitions a metric space into clusters of diameter at most $Δ$ such that every ball of radius $Δ/Γ$ intersects at most $Λ$ clusters in expectation. Consistent hashing has emerged as a central tool in geometric MPC algorithms. However, so far it has been solely used in Euclidean spaces, where it can be pre-computed. On the other hand, in cases where the metric space is received as input and unknown in advance, no previous parallel construction was known. We present the first fully scalable MPC algorithm over machines with local space $n^σ$, that given an $n$-point metric space with doubling dimension $d\leq O(σ\log n)$, samples a randomized $(Γ,2^{O(d/Γ)})$-hash in $O(d/σ^3)$ rounds. There is a dichotomy: if the doubling dimension is $d=Ω(σ\cdot\log n)$, a polynomial number of rounds is required to compute such a hash. We conclude by showing that this MPC primitive yields $O(d/σ^3)$-round fully scalable MPC algorithms for a broad class of fundamental geometric problems in doubling metrics, including stochastic tree embeddings with $O(\log n)$ expected distortion, constant factor approximation for facility location, bicriteria $(O(ε^{-2}),1+ε)$-approximation for $k$-median and $k$-means.

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arXiv2026

Highway Dimension: a Metric View

Realistic metric spaces (such as road/transportation networks) tend to be much more algorithmically tractable than general metrics. In an attempt to formalize this intuition, Abraham et al. (SODA 2010, JACM 2016) introduced the notion of highway dimension. A weighted graph $G$ has highway dimension $h$ if for every ball $B$ of radius $\approx4r$, there is a hitting set of size $h$ hitting all the shortest paths of length $>r$ in $B$. Unfortunately, this definition fails to incorporate some very natural metric spaces such as the grid graph, and the Euclidean plane. We relax the definition of highway dimension by demanding to hit only approximate shortest paths. In addition to generalizing the original definition, this new definition also incorporates all doubling spaces (in particular the grid graph and the Euclidean plane). We then construct a PTAS for TSP under this new definition (improving a QPTAS w.r.t. the original more restrictive definition of Feldmann et al. (SICOMP 2018)). Finally, we develop a basic metric toolkit for spaces with small highway dimension by constructing padded decompositions, sparse covers/partitions, and tree covers. An abundance of applications follow.

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arXiv2026

DAG Covers for Structured Graphs: The Steiner Point Effect

Given a weighted digraph $G$, a $(t,g,μ)$-DAG cover is a collection of $g$ dominating DAGs $D_1,\dots,D_g$ such that all distances are approximately preserved: for every pair $(u,v)$ of vertices, $\min_id_{D_i}(u,v)\le t\cdot d_{G}(u,v)$, and the total number of non-$G$ edges is bounded by $|(\cup_i D_i)\setminus G|\le μ$. Assadi, Hoppenworth, and Wein [STOC 25] and Filtser [SODA 26] studied DAG covers for general digraphs. This paper initiates the study of \emph{Steiner} DAG cover, where the DAGs are allowed to contain Steiner points. We obtain Steiner DAG covers on the important classes of planar digraphs and low-treewidth digraphs. Specifically, we show that any digraph with treewidth tw admits a $(1,2,\tilde{O}(n\cdot tw))$-Steiner DAG cover. For planar digraphs we provide a $(1+\varepsilon,2,\tilde{O}_\varepsilon(n))$-Steiner DAG cover. We also demonstrate a stark difference between Steiner and non-Steiner DAG covers. As a lower bound, we show that any non-Steiner DAG cover for graphs with treewidth $1$ with stretch $t<2$ and sub-quadratic number of extra edges requires $Ω(\log n)$ DAGs.

cs.DS↗

arXiv2026

Fatness and Flatness

Fat minors are the metric analog of graph minors that are tailored to the analysis of metric (edge-weighted) graphs and, more generally, metric spaces having a suitable notion of shortest paths. Despite a large interest in this notion, not much is known about the structure of metric graphs excluding a fixed fat minor. We prove that if a metric graph $G$ excludes a fixed graph $H$ as a $δ$-fat minor, for some $δ>0$, then $G$ enjoys the metric analog of flatness (aka uniform quasi-wideness) - a structural property from the field of Sparsity. In essence, our flatness result says that for any $α\geq β$ large enough compared to $δ$, in every large enough set $A$ in $G$ one can find a sizable subset $B$ that becomes $α$-scattered after removing a bounded number of balls of radius $β$. We call this property drill-flatness. Notably, the proof only relies on excluding shallow fat minors: every branch set has radius at most $2α$. As a corollary, we prove that metric graphs that exclude a fixed $δ$-fat minor have bounded $\varepsilon$-scatter dimension if we consider only $\varepsilon$-scatters at distances large enough compared to $δ$. By combining this with the results of Abbasi et al. [FOCS 2023], we infer that the $k$-Center problem on instances excluding $H$ as a $δ$-fat minor admits an approximation algorithm that finds a solution of cost at most $(1+\varepsilon)\cdot\mathsf{OPT}+{\cal O}(δ/\varepsilon^2)$ in time ${\cal O}_{H,\varepsilon}(n^{{\cal O}(1)})$. This is one of the first algorithmic results for general fat-minor-free metrics. We also study drill-flatness in hereditary classes of (unweighted) graphs, where we obtain a characterization equating drill-flatness with excluding shallow induced minors. This is an induced analog of the equivalence between flatness and nowhere denseness - one of central results of Sparsity.

math.CO↗

arXiv2026

Bi-Lipschitz extensions and outlier embeddings into trees

We develop low distortion embeddings with outliers from arbitrary metrics into hierarchically separated trees (HSTs). In particular, we develop an efficient algorithm that for any $ε>0$, given an input metric $(X,d)$, and a probabilistic embedding of all but $k$ points from $X$ into HSTs with distortion $c$, samples from a probabilistic embedding of all but $O(\frac{k}ε\log k)$ points into HSTs that achieves distortion at most $(32+ε)c$. Our results are based on two key technical components. First, we extend an algorithm of Munagala et al. [2023] for minimizing the distortion of embeddings without outliers into HSTs to the setting with outliers. We combine this with new results on bi-Lipschitz extensions into trees and $\ell_1$ space. In particular, we show that any probabilistic embedding into HSTs can be extended to $k$ additional points with only a factor $O(\log k)$ of additional distortion. This bi-Lipschitz extension result utilizes a new probabilistic partitioning scheme that we call onion partitioning.

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arXiv2026

Hop-Constrained Metric Embeddings and their Applications

In network design problems, such as compact routing, the goal is to route packets between nodes using the (approximated) shortest paths. A desirable property of these routes is a small number of hops, which makes them more reliable, and reduces the transmission costs. Following the overwhelming success of stochastic tree embeddings for algorithmic design, Haeupler, Hershkowitz, and Zuzic (STOC'21) studied hop-constrained Ramsey-type metric embeddings into trees. Specifically, embedding $f:G(V,E)\rightarrow T$ has Ramsey hop-distortion $(t,M,β,h)$ (here $t,β,h\ge1$ and $M\subseteq V$) if $\forall u,v\in M$, $d_G^{(β\cdot h)}(u,v)\le d_T(u,v)\le t\cdot d_G^{(h)}(u,v)$. $t$ is called the distortion, $β$ is called the hop-stretch, and $d_G^{(h)}(u,v)$ denotes the minimum weight of a $u-v$ path with at most $h$ hops. Haeupler {\em et al.} constructed embedding where $M$ contains $1-ε$ fraction of the vertices and $β=t=O(\frac{\log^2 n}ε)$. They used their embedding to obtain multiple bicriteria approximation algorithms for hop-constrained network design problems. In this paper, we first improve the Ramsey-type embedding to obtain parameters $t=β=\frac{\tilde{O}(\log n)}ε$, and generalize it to arbitrary distortion parameter $t$ (in the cost of reducing the size of $M$). This embedding immediately implies polynomial improvements for all the approximation algorithms from Haeupler {\em et al.}. Further, we construct hop-constrained clan embeddings (where each vertex has multiple copies), and use them to construct bicriteria approximation algorithms for the group Steiner tree problem, matching the state of the art of the non constrained version. Finally, we use our embedding results to construct hop constrained distance oracles, distance labeling, and most prominently, the first hop constrained compact routing scheme with provable guarantees.

cs.DS↗

arXiv2026

Directed Low Diameter Decomposition for Structured Digraphs

Low diameter decompositions, or LDDs for short, are a fundamental primitive in the design of efficient graph algorithms. Roughly speaking, an LDD is a distribution over partitions of the vertices into bounded-diameter clusters such that nearby vertices are likely to be clustered together. Recently, there has been growing interest in lifting the notion of LDDs into \emph{directed graphs}. In particular, there are two natural directed analogues. The first is a directed LDD, where after removing a random subset of edges, every strongly connected component has a small diameter. The second is a quasipartition, which imposes the stronger requirement that whenever one vertex can still reach another after the edge removal, the two vertices must be close in the original directed metric. Every quasipartition yields an LDD, but the converse does not necessarily hold. In this work, we initiate the systematic study of LDDs in structured directed graphs. As our first main result, we show that any directed graph with pathwidth $\mathsf{pw}$ admits an $(O(\mathsf{pw}), Δ)$-LDD. This improves upon the previous best-known $(2^{O(\mathsf{pw}^2)}, Δ)$-LDD construction, which was implicitly derived from the quasipartition result of Salmasi, Sidiropoulos, and Sridhar [SODA'19]. As our second result, we show that the integrality gap of the Directed Non-Bipartite Sparsest-Cut LP relaxation on an $n$-vertex graph with treewidth $\mathsf{tw}$ is $O(\mathsf{tw} \log n)$. This improves upon the $O(\mathsf{tw}\log^2 n)$ bound of Mémoli, Sidiropoulos, and Sridhar [ICALP'16, Algorithmica'18]. We obtain this result through the refined analysis of the quasipartition construction of Mémoli et al. for bounded treewidth graphs.

cs.DS↗

arXiv2026

Near-Resolution of the Tradeoff Conjecture in Distributed Proof Labeling Schemes

In the $t$-Proof Labeling Scheme model ($t$-PLS model), our goal is to certify that a network of nodes satisfies a given property $P$. A prover assigns a label to each node, and each node decides to accept or reject based on its labeled $t$-hop neighborhood. If $P$ holds, there exists a labeling that makes all nodes accept. If $P$ does not hold, in all labelings at least one node rejects. The cost of a scheme is its maximum label size. The Tradeoff Conjecture [Feuilloley, Fraigniaud, Hirvonen, Paz, and Perry, DISC 18, Dist. Comput.~21] hypothesizes that the existence of a $1$-PLS for a property $P$ with cost $p$ implies the existence of a $t$-PLS for $P$ with cost $O(\lceil p/t \rceil)$. The conjecture was initially shown to hold for specific graph classes, such as trees, cycles, and grids. Later, a weaker $\widetilde{O}(\lceil Δp/\sqrt{t} \rceil)$ cost was shown for fixed minor-free graphs, where $Δ$ is the maximum degree. In this work we resolve the Tradeoff Conjecture, up to a single logarithmic factor. In general graphs, we show that the existence of a $1$-PLS with cost $p$ implies the existence of an $O(t\log{n})$-PLS with cost $O(\lceil p/t \rceil)$ for the same property. For fixed minor-free graphs (which include e.g. planar graphs), we show that the existence of a $1$-PLS with cost $p$ implies the existence of a $t$-PLS with cost $O(\lceil p/t \rceil+\log{n})$ for the same property. We also refute a previously suggested stronger variant of the Tradeoff Conjecture, and show that having very large $t$-hop neighborhoods is an insufficient condition for obtaining a tradeoff better than $O(\lceil p/t \rceil)$.

cs.DS↗

arXiv2026

Dynamic Light Spanners in Doubling Metrics

A $t$-spanner of a point set $X$ in a metric space $(\mathcal{X}, δ)$ is a graph $G$ with vertex set $P$ such that, for any pair of points $u,v \in X$, the distance between $u$ and $v$ in $G$ is at most $t$ times $δ(u,v)$. We study the problem of maintaining a spanner for a dynamic point set $X$ -- that is, when $X$ undergoes a sequence of insertions and deletions -- in a metric space of constant doubling dimension. For any constant $\varepsilon>0$, we maintain a $(1+\varepsilon)$-spanner of $P$ whose total weight remains within a constant factor of the weight of the minimum spanning tree of $X$. Each update (insertion or deletion) can be performed in $\operatorname{poly}(\log Φ)$ time, where $Φ$ denotes the aspect ratio of $X$. Prior to our work, no efficient dynamic algorithm for maintaining a light-weight spanner was known even for point sets in low-dimensional Euclidean space.

cs.CG↗

arXiv2025

The Squishy Grid Problem

In this paper we consider the problem of approximating Euclidean distances by the infinite integer grid graph. Although the topology of the graph is fixed, we have control over the edge-weight assignment $w:E\to \mathbb{R}_{\ge 0}$, and hope to have grid distances be asymptotically isometric to Euclidean distances, that is, for all grid points $u,v$, $\mathrm{dist}_w(u,v) = (1\pm o(1))\|u-v\|_2$. We give three methods for solving this problem, each attractive in its own way. * Our first construction is based on an embedding of the recursive, non-periodic pinwheel tiling of Radin and Conway into the integer grid. Distances in the pinwheel graph are asymptotically isometric to Euclidean distances, but no explicit bound on the rate of convergence was known. We prove that the multiplicative distortion of the pinwheel graph is $(1+1/Θ(\log^ξ\log D))$, where $D$ is the Euclidean distance and $ξ=Θ(1)$. The pinwheel tiling approach is conceptually simple, but can be improved quantitatively. * Our second construction is based on a hierarchical arrangement of "highways." It is simple, achieving stretch $(1 + 1/Θ(D^{1/9}))$, which converges doubly exponentially faster than the pinwheel tiling approach. * The first two methods are deterministic. An even simpler approach is to sample the edge weights independently from a common distribution $\mathscr{D}$. Whether there exists a distribution $\mathscr{D}^*$ that makes grid distances Euclidean, asymptotically and in expectation, is major open problem in the theory of first passage percolation. Previous experiments show that when $\mathscr{D}$ is a Fisher distribution, grid distances are within 1\% of Euclidean. We demonstrate experimentally that this level of accuracy can be achieved by a simple 2-point distribution that assigns weights 0.41 or 4.75 with probability 44\% and 56\%, respectively.

cs.CG↗

arXiv2025

Optimal Padded Decomposition For Bounded Treewidth Graphs

A $(β,δ,Δ)$-padded decomposition of an edge-weighted graph $G = (V,E,w)$ is a stochastic decomposition into clusters of diameter at most $Δ$ such that for every vertex $v\in V$, the probability that $\rm{ball}_G(v,γΔ)$ is entirely contained in the cluster containing $v$ is at least $e^{-βγ}$ for every $γ\in [0,δ]$. Padded decompositions have been studied for decades and have found numerous applications, including metric embedding, multicommodity flow-cut gap, multicut, and zero extension problems, to name a few. In these applications, parameter $β$, called the padding parameter, is the most important parameter since it decides either the distortion or the approximation ratios. For general graphs with $n$ vertices, $β= Θ(\log n)$. Klein, Plotkin, and Rao showed that $K_r$-minor-free graphs have padding parameter $β= O(r^3)$, which is a significant improvement over general graphs when $r$ is a constant. A long-standing conjecture is to construct a padded decomposition for $K_r$-minor-free graphs with padding parameter $β= O(\log r)$. Despite decades of research, the best-known result is $β= O(r)$, even for graphs with treewidth at most $r$. In this work, we make significant progress toward the aforementioned conjecture by showing that graphs with treewidth $\rm{tw}$ admit a padded decomposition with padding parameter $O(\log \rm{tw})$, which is tight. As corollaries, we obtain an exponential improvement in dependency on treewidth in a host of algorithmic applications: $O(\sqrt{ \log n \cdot \log(\rm{tw})})$ flow-cut gap, max flow-min multicut ratio of $O(\log(\rm{tw}))$, an $O(\log(\rm{tw}))$ approximation for the 0-extension problem, an $\ell^{O(\log n)}_\infty$ embedding with distortion $O(\log \rm{tw})$, and an $O(\log \rm{tw})$ bound for integrality gap for the uniform sparsest cut.

cs.DS↗

arXiv2025

Stochastic Embedding of Digraphs into DAGs

Given a weighted digraph $G=(V,E,w)$, a stochastic embedding into DAGs is a distribution $\mathcal{D}$ over pairs of DAGs $(D_1,D_2)$ such that for every $u,v$: (1) the reachability is preserved: $u\rightsquigarrow_G v$ (i.e., $v$ is reachable from $u$ in $G$) implies that $u\rightsquigarrow_{D_1} v$ or $u\rightsquigarrow_{D_2} v$ (but not both), and (2) distances are dominated: $d_G(u,v)\le\min\{d_{D_1}(u,v),d_{D_2}(u,v)\}$. The stochastic embedding $\mathcal{D}$ has expected distortion $t$ if for every $u,v\in V$, \[ \mathbb{E}_{(D_{1},D_{2})\sim\mathcal{D}}\left[d_{D_{1}}(u,v)\cdot\boldsymbol{1}[u\rightsquigarrow_{D_{1}}v]+d_{D_{2}}(u,v)\cdot\boldsymbol{1}[u\rightsquigarrow_{D_{2}}v]\right]\le t\cdot d_{G}(u,v)~. \] Finally, the sparsity of $\mathcal{D}$ is the maximum number of edges in any of the DAGs in its support. Given an $n$ vertex digraph with $m$ edges, we construct a stochastic embedding into DAGs with expected distortion $\tilde{O}(\log n)$ and $\tilde{O}(m)$ sparsity, improving a previous result by Assadi, Hoppenworth, and Wein [STOC 25], which achieved expected distortion $\tilde{O}(\log^3 n)$. Further, we can sample DAGs from this distribution in $\tilde{O}(m)$ time.

cs.DS↗

arXiv2025

FPT approximations for Capacitated Sum of Radii and Diameters

The Capacitated Sum of Radii problem involves partitioning a set of points $P$, where each point $p\in P$ has capacity $U_p$, into $k$ clusters that minimize the sum of cluster radii, such that the number of points in the cluster centered at point $p$ is at most $U_p$. We begin by showing that the problem is APX-hard, and that under gap-ETH there is no parameterized approximation scheme (FPT-AS). We then construct a $\approx5.83$-approximation algorithm in FPT time (improving a previous $\approx7.61$ approximation in FPT time). Our results also hold when the objective is a general monotone symmetric norm of radii. We also improve the approximation factors for the uniform capacity case, and for the closely related problem of Capacitated Sum of Diameters.

cs.DS↗

arXiv2025

How to Protect Yourself from Threatening Skeletons: Optimal Padded Decompositions for Minor-Free Graphs

Roughly, a metric space has padding parameter $β$ if for every $Δ>0$, there is a stochastic decomposition of the metric points into clusters of diameter at most $Δ$ such that every ball of radius $γΔ$ is contained in a single cluster with probability at least $e^{-γβ}$. The padding parameter is an important characteristic of a metric space with vast algorithmic implications. In this paper we prove that the shortest path metric of every $K_r$-minor-free graph has padding parameter $O(\log r)$, which is also tight. This resolves a long standing open question, and exponentially improves the previous bound. En route to our main result, we construct sparse covers for $K_r$-minor-free graphs with improved parameters, and we prove a general reduction from sparse covers to padded decompositions.

cs.DS↗

arXiv2025

Faster Approximation Algorithms for k-Center via Data Reduction

We study efficient algorithms for the Euclidean $k$-Center problem, focusing on the regime of large $k$. We take the approach of data reduction by considering $α$-coreset, which is a small subset $S$ of the dataset $P$ such that any $β$-approximation on $S$ is an $(α+ β)$-approximation on $P$. We give efficient algorithms to construct coresets whose size is $k \cdot o(n)$, which immediately speeds up existing approximation algorithms. Notably, we obtain a near-linear time $O(1)$-approximation when $k = n^c$ for any $0 < c < 1$. We validate the performance of our coresets on real-world datasets with large $k$, and we observe that the coreset speeds up the well-known Gonzalez algorithm by up to $4$ times, while still achieving similar clustering cost. Technically, one of our coreset results is based on a new efficient construction of consistent hashing with competitive parameters. This general tool may be of independent interest for algorithm design in high dimensional Euclidean spaces.

cs.DS↗

arXiv2024

Online Duet between Metric Embeddings and Minimum-Weight Perfect Matchings

Low-distortional metric embeddings are a crucial component in the modern algorithmic toolkit. In an online metric embedding, points arrive sequentially and the goal is to embed them into a simple space irrevocably, while minimizing the distortion. Our first result is a deterministic online embedding of a general metric into Euclidean space with distortion $O(\log n)\cdot\min\{\sqrt{\logΦ},\sqrt{n}\}$ (or, $O(d)\cdot\min\{\sqrt{\logΦ},\sqrt{n}\}$ if the metric has doubling dimension $d$), solving a conjecture by Newman and Rabinovich (2020), and quadratically improving the dependence on the aspect ratio $Φ$ from Indyk et al.\ (2010). Our second result is a stochastic embedding of a metric space into trees with expected distortion $O(d\cdot \logΦ)$, generalizing previous results (Indyk et al.\ (2010), Bartal et al.\ (2020)). Next, we study the \emph{online minimum-weight perfect matching} problem, where a sequence of $2n$ metric points arrive in pairs, and one has to maintain a perfect matching at all times. We allow recourse (as otherwise the order of arrival determines the matching). The goal is to return a perfect matching that approximates the \emph{minimum-weight} perfect matching at all times, while minimizing the recourse. Our third result is a randomized algorithm with competitive ratio $O(d\cdot \log Φ)$ and recourse $O(\log Φ)$ against an oblivious adversary, this result is obtained via our new stochastic online embedding. Our fourth result is a deterministic algorithm against an adaptive adversary, using $O(\log^2 n)$ recourse, that maintains a matching of weight at most $O(\log n)$ times the weight of the MST, i.e., a matching of lightness $O(\log n)$. We complement our upper bounds with a strategy for an oblivious adversary that, with recourse $r$, establishes a lower bound of $Ω(\frac{\log n}{r \log r})$ for both competitive ratio and lightness.

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