arXiv subjects

Shay Solomon

Publications and source records attributed to Shay Solomon.

At least 55 records · Page 3Linked to original sources

arXiv2021

Local Algorithms for Bounded Degree Sparsifiers in Sparse Graphs

In graph sparsification, the goal has almost always been of {global} nature: compress a graph into a smaller subgraph ({sparsifier}) that maintains certain features of the original graph. Algorithms can then run on the sparsifier, which in many cases leads to improvements in the overall runtime and memory. This paper studies sparsifiers that have bounded (maximum) degree, and are thus {locally} sparse, aiming to improve local measures of runtime and memory. To improve those local measures, it is important to be able to compute such sparsifiers {locally}. We initiate the study of local algorithms for bounded degree sparsifiers in unweighted sparse graphs, focusing on the problems of vertex cover, matching, and independent set. Let $ε> 0$ be a slack parameter and $α\ge 1$ be a density parameter. We devise local algorithms for computing: (1) A $(1+ε)$-vertex cover sparsifier of degree $O(α/ ε)$, for any graph of {arboricity} $α$. (2) A $(1+ε)$-maximum matching sparsifier and also a $(1+ε)$-maximal matching sparsifier of degree $O(α/ ε)$, for any graph of arboricity $α$. (3) A $(1+ε)$-independent set sparsifier of degree $O(α^2 / ε)$, for any graph of average degree $α$. Our algorithms require only a single communication round in the standard message passing models of distributed computing, and moreover, they can be simulated locally in a trivial way. As an immediate application we can extend results from distributed computing and local computation algorithms that apply to graphs of degree bounded by $d$ to graphs of arboricity $O(d / ε)$ or average degree $O(d^2 / ε)$, at the expense of increasing the approximation guarantee by a factor of $(1+ε)$. In particular, we can extend the plethora of recent local computation algorithms [...]

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arXiv2021

Truly Optimal Euclidean Spanners

Euclidean spanners are important geometric structures, having found numerous applications over the years. Cornerstone results in this area from the late 80s and early 90s state that for any $d$-dimensional $n$-point Euclidean space, there exists a $(1+ε)$-spanner with $nO(ε^{-d+1})$ edges and lightness $O(ε^{-2d})$. Surprisingly, the fundamental question of whether or not these dependencies on $ε$ and $d$ for small $d$ can be improved has remained elusive, even for $d = 2$. This question naturally arises in any application of Euclidean spanners where precision is a necessity. The state-of-the-art bounds $nO(ε^{-d+1})$ and $O(ε^{-2d})$ on the size and lightness of spanners are realized by the {\em greedy} spanner. In 2016, Filtser and Solomon proved that, in low dimensional spaces, the greedy spanner is near-optimal. The question of whether the greedy spanner is truly optimal remained open to date. The contribution of this paper is two-fold. We resolve these longstanding questions by nailing down the exact dependencies on $ε$ and $d$ and showing that the greedy spanner is truly optimal. Specifically, for any $d= O(1), ε= Ω({n}^{-\frac{1}{d-1}})$: - We show that any $(1+ε)$-spanner must have $n Ω(ε^{-d+1})$ edges, implying that the greedy (and other) spanners achieve the optimal size. - We show that any $(1+ε)$-spanner must have lightness $Ω(ε^{-d})$, and then improve the upper bound on the lightness of the greedy spanner from $O(ε^{-2d})$ to $O(ε^{-d})$. We then complement our negative result for the size of spanners with a rather counterintuitive positive result: Steiner points lead to a quadratic improvement in the size of spanners! Our bound for the size of Steiner spanners is tight as well (up to lower-order terms).

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arXiv2020

Light Euclidean Spanners with Steiner Points

The FOCS'19 paper of Le and Solomon, culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy $(1+ε)$-spanner in $\mathbb{R}^d$ is $\tilde{O}(ε^{-d})$ for any $d = O(1)$ and any $ε= Ω(n^{-\frac{1}{d-1}})$ (where $\tilde{O}$ hides polylogarithmic factors of $\frac{1}ε$), and also shows the existence of point sets in $\mathbb{R}^d$ for which any $(1+ε)$-spanner must have lightness $Ω(ε^{-d})$. Given this tight bound on the lightness, a natural arising question is whether a better lightness bound can be achieved using Steiner points. Our first result is a construction of Steiner spanners in $\mathbb{R}^2$ with lightness $O(ε^{-1} \log Δ)$, where $Δ$ is the spread of the point set. In the regime of $Δ\ll 2^{1/ε}$, this provides an improvement over the lightness bound of Le and Solomon [FOCS 2019]; this regime of parameters is of practical interest, as point sets arising in real-life applications (e.g., for various random distributions) have polynomially bounded spread, while in spanner applications $ε$ often controls the precision, and it sometimes needs to be much smaller than $O(1/\log n)$. Moreover, for spread polynomially bounded in $1/ε$, this upper bound provides a quadratic improvement over the non-Steiner bound of Le and Solomon [FOCS 2019], We then demonstrate that such a light spanner can be constructed in $O_ε(n)$ time for polynomially bounded spread, where $O_ε$ hides a factor of $\mathrm{poly}(\frac{1}ε)$. Finally, we extend the construction to higher dimensions, proving a lightness upper bound of $\tilde{O}(ε^{-(d+1)/2} + ε^{-2}\log Δ)$ for any $3\leq d = O(1)$ and any $ε= Ω(n^{-\frac{1}{d-1}})$.

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arXiv2020

Improved Dynamic Graph Coloring

This paper studies the fundamental problem of graph coloring in fully dynamic graphs. Since the problem of computing an optimal coloring, or even approximating it to within $n^{1-ε}$ for any $ε> 0$, is NP-hard in static graphs, there is no hope to achieve any meaningful computational results for general graphs in the dynamic setting. It is therefore only natural to consider the combinatorial aspects of dynamic coloring, or alternatively, study restricted families of graphs. Towards understanding the combinatorial aspects of this problem, one may assume a black-box access to a static algorithm for $C$-coloring any subgraph of the dynamic graph, and investigate the trade-off between the number of colors and the number of recolorings per update step. In WADS'17, Barba et al. devised two complementary algorithms: For any $β> 0$ the first (respectively, second) maintains an $O(C βn^{1/β})$ (resp., $O(C β)$)-coloring while recoloring $O(β)$ (resp., $O(βn^{1/β})$) vertices per update. Our contribution is two-fold: - We devise a new algorithm for general graphs that improves significantly upon the first trade-off in a wide range of parameters: For any $β> 0$, we get a $\tilde{O}(\frac{C}β\log^2 n)$-coloring with $O(β)$ recolorings per update, where the $\tilde{O}$ notation supresses polyloglog$(n)$ factors. In particular, for $β=O(1)$ we get constant recolorings with polylog$(n)$ colors; this is an exponential improvement over the previous bound. - For uniformly sparse graphs, we use low out-degree orientations to strengthen the above result by bounding the update time of the algorithm rather than the number of recolorings. Then, we further improve this result by introducing a new data structure that refines bounded out-degree edge orientations and is of independent interest.

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arXiv2020

When Algorithms for Maximal Independent Set and Maximal Matching Run in Sublinear-Time

Maximal independent set (MIS), maximal matching (MM), and $(Δ+1)$-coloring in graphs of maximum degree $Δ$ are among the most prominent algorithmic graph theory problems. They are all solvable by a simple linear-time greedy algorithm and up until very recently this constituted the state-of-the-art. In SODA 2019, Assadi, Chen, and Khanna gave a randomized algorithm for $(Δ+1)$-coloring that runs in $\widetilde{O}(n\sqrt{n})$ time, which even for moderately dense graphs is sublinear in the input size. The work of Assadi et al. however contained a spoiler for MIS and MM: neither problems provably admits a sublinear-time algorithm in general graphs. In this work, we dig deeper into the possibility of achieving sublinear-time algorithms for MIS and MM. The neighborhood independence number of a graph $G$, denoted by $β(G)$, is the size of the largest independent set in the neighborhood of any vertex. We identify $β(G)$ as the ``right'' parameter to measure the runtime of MIS and MM algorithms: Although graphs of bounded neighborhood independence may be very dense (clique is one example), we prove that carefully chosen variants of greedy algorithms for MIS and MM run in $O(nβ(G))$ and $O(n\log{n}\cdotβ(G))$ time respectively on any $n$-vertex graph $G$. We complement this positive result by observing that a simple extension of the lower bound of Assadi et.al. implies that $Ω(nβ(G))$ time is also necessary for any algorithm to either problem for all values of $β(G)$ from $1$ to $Θ(n)$. We note that our algorithm for MIS is deterministic while for MM we use randomization which we prove is unavoidable: any deterministic algorithm for MM requires $Ω(n^2)$ time even for $β(G) = 2$.

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arXiv2020

A Generalized Matching Reconfiguration Problem

The goal in {\em reconfiguration problems} is to compute a {\em gradual transformation} between two feasible solutions of a problem such that all intermediate solutions are also feasible. In the {\em Matching Reconfiguration Problem} (MRP), proposed in a pioneering work by Ito et al.\ from 2008, we are given a graph $G$ and two matchings $M$ and $M'$, and we are asked whether there is a sequence of matchings in $G$ starting with $M$ and ending at $M'$, each resulting from the previous one by either adding or deleting a single edge in $G$, without ever going through a matching of size $< \min\{|M|,|M'|\}-1$. Ito et al.\ gave a polynomial time algorithm for the problem. In this paper we introduce a natural generalization of the MRP that depends on an integer parameter $Δ\ge 1$: here we are allowed to make $Δ$ changes to the current solution rather than 1 at each step of the {transformation procedure}. There is always a valid sequence of matchings transforming $M$ to $M'$ if $Δ$ is sufficiently large, and naturally we would like to minimize $Δ$. We first devise an optimal transformation procedure for unweighted matching with $Δ= 3$, and then extend it to weighted matchings to achieve asymptotically optimal guarantees. The running time of these procedures is linear. We further demonstrate the applicability of this generalized problem to dynamic graph matchings. In this area, the number of changes to the maintained matching per update step (the \emph{recourse bound}) is an important quality measure. Nevertheless, the \emph{worst-case} recourse bounds of almost all known dynamic matching algorithms are prohibitively large, much larger than the corresponding update times. We fill in this gap via a surprisingly simple black-box reduction: Any dynamic algorithm for maintaining [...]

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arXiv2020

The Greedy Spanner is Existentially Optimal

The greedy spanner is arguably the simplest and most well-studied spanner construction. Experimental results demonstrate that it is at least as good as any other spanner construction, in terms of both the size and weight parameters. However, a rigorous proof for this statement has remained elusive. In this work we fill in the theoretical gap via a surprisingly simple observation: The greedy spanner is \emph{existentially optimal} (or existentially near-optimal) for several important graph families, in terms of both the size and weight. Roughly speaking, the greedy spanner is said to be existentially optimal (or near-optimal) for a graph family $\mathcal G$ if the worst performance of the greedy spanner over all graphs in $\mathcal G$ is just as good (or nearly as good) as the worst performance of an optimal spanner over all graphs in $\mathcal G$. Focusing on the weight parameter, the state-of-the-art spanner constructions for both general graphs (due to Chechik and Wulff-Nilsen [SODA'16]) and doubling metrics (due to Gottlieb [FOCS'15]) are complex. Plugging our observation on these results, we conclude that the greedy spanner achieves near-optimal weight guarantees for both general graphs and doubling metrics, thus resolving two longstanding conjectures in the area. Further, we observe that approximate-greedy spanners are existentially near-optimal as well. Consequently, we provide an $O(n \log n)$-time construction of $(1+ε)$-spanners for doubling metrics with constant lightness and degree. Our construction improves Gottlieb's construction, whose runtime is $O(n \log^2 n)$ and whose number of edges and degree are unbounded, and remarkably, it matches the state-of-the-art Euclidean result (due to Gudmundsson et al.\ [SICOMP'02]) in all the involved parameters (up to dependencies on $ε$ and the dimension).

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arXiv2019

Fully Dynamic $(Δ+1)$-Coloring in Constant Update Time

The problem of (vertex) $(Δ+1)$-coloring a graph of maximum degree $Δ$ has been extremely well-studied over the years in various settings and models. Surprisingly, for the dynamic setting, almost nothing was known until recently. In SODA'18, Bhattacharya, Chakrabarty, Henzinger and Nanongkai devised a randomized data structure for maintaining a $(Δ+1)$-coloring with $O(\log Δ)$ expected amortized update time. In this paper, we present a $(Δ+1)$-coloring data structure that achieves a constant amortized update time and show that this time bound holds not only in expectation but also with high probability.

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arXiv2018

Fully Dynamic MIS in Uniformly Sparse Graphs

We consider the problem of maintaining a maximal independent set (MIS) in a dynamic graph subject to edge insertions and deletions. Recently, Assadi, Onak, Schieber and Solomon (STOC 2018) showed that an MIS can be maintained in sublinear (in the dynamically changing number of edges) amortized update time. In this paper we significantly improve the update time for uniformly sparse graphs. Specifically, for graphs with arboricity $α$, the amortized update time of our algorithm is $O(α^2 \cdot \log^2 n)$, where $n$ is the number of vertices. For low arboricity graphs, which include, for example, minor-free graphs as well as some classes of `real world' graphs, our update time is polylogarithmic. Our update time improves the result of Assadi et al. for all graphs with arboricity bounded by $m^{3/8 - ε}$, for any constant $ε> 0$. This covers much of the range of possible values for arboricity, as the arboricity of a general graph cannot exceed $m^{1/2}$.

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arXiv2018

Fully Dynamic Maximal Independent Set with Sublinear in n Update Time

The first fully dynamic algorithm for maintaining a maximal independent set (MIS) with update time that is sublinear in the number of edges was presented recently by the authors of this paper [Assadi et.al. STOC'18]. The algorithm is deterministic and its update time is $O(m^{3/4})$, where $m$ is the (dynamically changing) number of edges. Subsequently, Gupta and Khan and independently Du and Zhang [arXiv, April 2018] presented deterministic algorithms for dynamic MIS with update times of $O(m^{2/3})$ and $O(m^{2/3} \sqrt{\log m})$, respectively. Du and Zhang also gave a randomized algorithm with update time $\widetilde{O}(\sqrt{m})$. Moreover, they provided some partial (conditional) hardness results hinting that update time of $m^{1/2-ε}$, and in particular $n^{1-ε}$ for $n$-vertex dense graphs, is a natural barrier for this problem for any constant $ε>0$, for both deterministic and randomized algorithms that satisfy a certain natural property. In this paper, we break this natural barrier and present the first fully dynamic (randomized) algorithm for maintaining an MIS with update time that is always sublinear in the number of vertices, namely, an $\widetilde{O}(\sqrt{n})$ expected amortized update time algorithm. We also show that a simpler variant of our algorithm can already achieve an $\widetilde{O}(m^{1/3})$ expected amortized update time, which results in an improved performance over our $\widetilde{O}(\sqrt{n})$ update time algorithm for sufficiently sparse graphs, and breaks the $m^{1/2}$ barrier of Du and Zhang for all values of $m$.

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arXiv2018

Fully Dynamic Almost-Maximal Matching: Breaking the Polynomial Barrier for Worst-Case Time Bounds

Despite significant research efforts, the state-of-the-art algorithm for maintaining an approximate matching in fully dynamic graphs has a polynomial {worst-case} update time, even for very poor approximation guarantees. In a recent breakthrough, Bhattacharya, Henzinger and Nanongkai showed how to maintain a constant approximation to the minimum vertex cover, and thus also a constant-factor estimate of the maximum matching size, with polylogarithmic worst-case update time. Later (in SODA'17 Proc.) they improved the approximation factor all the way to $2+ε$. Nevertheless, the longstanding fundamental problem of {maintaining} an approximate matching with sub-polynomial worst-case time bounds remained open. We present a randomized algorithm for maintaining an {almost-maximal} matching in fully dynamic graphs with polylogarithmic worst-case update time. Such a matching provides $(2+ε)$-approximations for both the maximum matching and the minimum vertex cover, for any $ε> 0$. Our result was done independently of the $(2+ε)$-approximation result of Bhattacharya et al., so it provides the first $(2+ε)$-approximation for minimum vertex cover (together with Bhattacharya et al.'s result) and the first $(2+ε)$-approximation for maximum (integral) matching. The polylogarithmic worst-case update time of our algorithm holds deterministically, while the almost-maximality guarantee holds with high probability. This result not only settles the aforementioned problem on dynamic matchings, but also provides essentially the best possible approximation guarantee for dynamic vertex cover (assuming the unique games conjecture).

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arXiv2018

Fully Dynamic Maximal Independent Set with Sublinear Update Time

A maximal independent set (MIS) can be maintained in an evolving $m$-edge graph by simply recomputing it from scratch in $O(m)$ time after each update. But can it be maintained in time sublinear in $m$ in fully dynamic graphs? We answer this fundamental open question in the affirmative. We present a deterministic algorithm with amortized update time $O(\min\{Δ,m^{3/4}\})$, where $Δ$ is a fixed bound on the maximum degree in the graph and $m$ is the (dynamically changing) number of edges. We further present a distributed implementation of our algorithm with $O(\min\{Δ,m^{3/4}\})$ amortized message complexity, and $O(1)$ amortized round complexity and adjustment complexity (the number of vertices that change their output after each update). This strengthens a similar result by Censor-Hillel, Haramaty, and Karnin (PODC'16) that required an assumption of a non-adaptive oblivious adversary.

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arXiv2018

Representations of Sparse Distributed Networks: A Locality-Sensitive Approach

In 1999, Brodal and Fagerberg (BF) gave an algorithm for maintaining a low outdegree orientation of a dynamic uniformly sparse graph. Specifically, for a dynamic graph on $n$-vertices, with arboricity bounded by $α$ at all times, the BF algorithm supports edge updates in $O(\log n)$ amortized update time, while keeping the maximum outdegree in the graph bounded by $O(α)$. Such an orientation provides a basic data structure for uniformly sparse graphs, which found applications to a plethora of dynamic graph algorithms. A significant weakness of the BF algorithm is the possible \emph{temporary} blowup of the maximum outdegree, following edge insertions. Although BF eventually reduces all outdegrees to $O(α)$, local memory usage at the vertices, which is an important quality measure in distributed systems, cannot be bounded. We show how to modify the BF algorithm to guarantee that the outdegrees of all vertices are bounded by $O(α)$ at all times, without hurting any of its other properties, and present an efficient distributed implementation of the modified algorithm. This provides the \emph{first} representation of distributed networks in which the local memory usage at all vertices is bounded by the arboricity (which is essentially the average degree of the densest subgraph) rather than the maximum degree. For settings where there are no local memory constraints, we take the temporary outdegree blowup to the extreme and allow a permanent outdegree blowup. This allows us to address the second significant weakness of the BF algorithm -- its inherently \emph{global} nature: An insertion of an edge $(u,v)$ may trigger changes in the orientations of edges that are far away from $u$ and $v$. We suggest an alternative \emph{local} scheme, which does not guarantee any outdegree bound on the vertices, yet is just as efficient as the BF scheme for various applications.

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arXiv2018

Wireless Expanders

This paper introduces an extended notion of expansion suitable for radio networks. A graph $G=(V,E)$ is called an $(α_w, β_w)$-{wireless expander} if for every subset $S \subseteq V$ s.t. $|S|\leq α_w \cdot |V|$, there exists a subset $S'\subseteq S$ s.t. there are at least $β_w \cdot |S|$ vertices in $V\backslash S$ adjacent in $G$ to exactly one vertex in $S'$. The main question we ask is the following: to what extent are ordinary expanders also good {wireless} expanders? We answer this question in a nearly tight manner. On the positive side, we show that any $(α, β)$-expander with maximum degree $Δ$ and $β\geq 1/Δ$ is also a $(α_w, β_w)$ wireless expander for $β_w = Ω(β/ \log (2 \cdot \min\{Δ/ β, Δ\cdot β\}))$. Thus the wireless expansion is smaller than the ordinary expansion by at most a factor logarithmic in $\min\{Δ/ β, Δ\cdot β\}$, which depends on the graph \emph{average degree} rather than maximum degree; e.g., for low arboricity graphs, the wireless expansion matches the ordinary expansion up to a constant. We complement this positive result by presenting an explicit construction of a "bad" $(α, β)$-expander for which the wireless expansion is $β_w = O(β/ \log (2 \cdot \min\{Δ/ β, Δ\cdot β\})$. We also analyze the theoretical properties of wireless expanders and their connection to unique neighbor expanders, and demonstrate their applicability: Our results yield improved bounds for the {spokesmen election problem} that was introduced in the seminal paper of Chlamtac and Weinstein (1991) to devise efficient broadcasting for multihop radio networks. Our negative result yields a significantly simpler proof than that from the seminal paper of Kushilevitz and Mansour (1998) for a lower bound on the broadcast time in radio networks.

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arXiv2016

Fully Dynamic Maximal Matching in Constant Update Time

Baswana, Gupta and Sen [FOCS'11] showed that fully dynamic maximal matching can be maintained in general graphs with logarithmic amortized update time. More specifically, starting from an empty graph on $n$ fixed vertices, they devised a randomized algorithm for maintaining maximal matching over any sequence of $t$ edge insertions and deletions with a total runtime of $O(t \log n)$ in expectation and $O(t \log n + n \log^2 n)$ with high probability. Whether or not this runtime bound can be improved towards $O(t)$ has remained an important open problem. Despite significant research efforts, this question has resisted numerous attempts at resolution even for basic graph families such as forests. In this paper, we resolve the question in the affirmative, by presenting a randomized algorithm for maintaining maximal matching in general graphs with \emph{constant} amortized update time. The optimal runtime bound $O(t)$ of our algorithm holds both in expectation and with high probability. As an immediate corollary, we can maintain 2-approximate vertex cover with constant amortized update time. This result is essentially the best one can hope for (under the unique games conjecture) in the context of dynamic approximate vertex cover, culminating a long line of research. Our algorithm builds on Baswana et al.'s algorithm, but is inherently different and arguably simpler. As an implication of our simplified approach, the space usage of our algorithm is linear in the (dynamic) graph size, while the space usage of Baswana et al.'s algorithm is always at least $Ω(n \log n)$. Finally, we present applications to approximate weighted matchings and to distributed networks.

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arXiv2014

Light Spanners

A $t$-spanner of a weighted undirected graph $G=(V,E)$, is a subgraph $H$ such that $d_H(u,v)\le t\cdot d_G(u,v)$ for all $u,v\in V$. The sparseness of the spanner can be measured by its size (the number of edges) and weight (the sum of all edge weights), both being important measures of the spanner's quality -- in this work we focus on the latter. Specifically, it is shown that for any parameters $k\ge 1$ and $ε>0$, any weighted graph $G$ on $n$ vertices admits a $(2k-1)\cdot(1+ε)$-stretch spanner of weight at most $w(MST(G))\cdot O_ε(kn^{1/k}/\log k)$, where $w(MST(G))$ is the weight of a minimum spanning tree of $G$. Our result is obtained via a novel analysis of the classic greedy algorithm, and improves previous work by a factor of $O(\log k)$.

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arXiv2014

Light spanners for snowflake metrics

A classic result in the study of spanners is the existence of light low-stretch spanners for Euclidean spaces. These spanners ahve arbitrary low stretch, and weight only a constant factor greater than that of the minimum spanning tree of the points (with dependence on the stretch and Euclidean dimention). A central open problem in this field asks whether other spaces admit low weight spanners as well - for example metric space with low intrinsic dimension - yet only a handful of results of this type are known. In this paper, we consider snowflake metric spaces of low intrinsic dimension. The α-snowflake of a metric (X,δ) is the metric (X,$δ^α$) for 0<α<1. By utilizing an approach completely different than those used for Euclidean spaces, we demonstrate that snowflake metrics admit light spanners. Further, we show that the spanner is of diameter O($\log$n), a result not possible for Euclidean spaces. As an immediate corollary to our spanner, we obtain dramatic improvments in algorithms for the traveling salesman problem in this setting, achieving a polynomial-time approximation scheme with near-linear runtime. Along the way, we show that all ${\ell}_p$ spaces admit light spanners, a result of interest in its own right.

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arXiv2013

Orienting Fully Dynamic Graphs with Worst-Case Time Bounds

In edge orientations, the goal is usually to orient (direct) the edges of an undirected $n$-vertex graph $G$ such that all out-degrees are bounded. When the graph $G$ is fully dynamic, i.e., admits edge insertions and deletions, we wish to maintain such an orientation while keeping a tab on the update time. Low out-degree orientations turned out to be a surprisingly useful tool, with several algorithmic applications involving static or dynamic graphs. Brodal and Fagerberg (1999) initiated the study of the edge orientation problem in terms of the graph's arboricity, which is very natural in this context. They provided a solution with constant out-degree and \emph{amortized} logarithmic update time for all graphs with constant arboricity, which include all planar and excluded-minor graphs. However, it remained an open question (first proposed by Brodal and Fagerberg, later by others) to obtain similar bounds with worst-case update time. We resolve this 15 year old question in the affirmative, by providing a simple algorithm with worst-case bounds that nearly match the previous amortized bounds. Our algorithm is based on a new approach of a combinatorial invariant, and achieves a logarithmic out-degree with logarithmic worst-case update times. This result has applications in various dynamic graph problems such as maintaining a maximal matching, where we obtain $O(\log n)$ worst-case update time compared to the $O(\frac{\log n}{\log\log n})$ amortized update time of Neiman and Solomon (2013).

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