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Priority queues are one of the most fundamental and widely used data structures in computer science. Their primary objective is to efficiently support the insertion of new elements with assigned priorities and the extraction of the highest priority element. In this study, we investigate the design of priority queues within the learning-augmented framework, where algorithms use potentially inaccurate predictions to enhance their worst-case performance. We examine three prediction models spanning different use cases, and show how the predictions can be leveraged to enhance the performance of priority queue operations. Moreover, we demonstrate the optimality of our solution and discuss some possible applications., Comment: Accepted as a conference paper at NeurIPS 2024

Introduced by Papadimitriou and Yannakakis in 1989, layered graph traversal is a central problem in online algorithms and mobile computing that has been studied for several decades, and which now is essentially resolved in its original formulation. In this paper, we demonstrate that what appears to be an innocuous modification of the problem actually leads to a drastic (exponential) reduction of the competitive ratio. Specifically, we present an algorithm that is $O(\log^2 w)$-competitive for traversing unweighted layered graphs of width $w$. Our algorithm chooses the agent's position simply according to the probability distribution over the current layer that maximizes the sum of entropies of the induced distributions in the preceding layers.

FM1/NM/FM2 trilayers have garnered considerable attention because of their potential in spintronic applications. A thorough investigation of the spin transport properties of these trilayers is therefore important. Asymmetric trilayers, particularly those including Platinum (Pt) as a spacer are less explored. Pt mediates exchange coupling between the two FM layers and thus offers a unique platform to investigate the spin-transport properties under indirect exchange coupling conditions through the spin-pumping mechanism. Our analytical focus on the acoustic mode of the ferromagnetic resonance spectrum, facilitated by the distinct magnetizations of the Ni80Fe20 and Co layers, allows for the isolation of individual layer resonances. The derived spin-pumping induced damping of the Ni80Fe20 and Co layers reveals a direct dependence on the Pt spacer thickness. Furthermore, fitting of the weighted average of the damping parameters to the spin-pumping induced damping of acoustic mode reveals that the observed FMR spectra is indeed a result of the in-phase precession of the magnetizations in two FM layers. The extracted effective spin-mixing conductance varies with the FM/NM interface, specifically 1.72x10^19 m^(-2) at the Ni80Fe20/Pt and 4.07x10^19 m^(-2) at the Co/Pt interface, indicating a strong correlation with interfacial characteristics. Additionally, we deduce the spin diffusion length in Pt to be between 1.02 and 1.55 nm and calculate the interfacial spin transparency and spin current densities, highlighting significant disparities between the Ni80Fe20/Pt and Co/Pt interfaces. This detailed analysis enhances our understanding of spin transport in Ni80Fe20/Pt/Co trilayers. It offers insights important for advancing spintronic device design and lays the groundwork for future theoretical investigations of trilayer system., Comment: 16 pages, 11 figures

Uncontrolled hypertension is a global problem that needs to be addressed. Despite the many mHealth solutions in the market, the nonadherence relative to intended use jeopardizes treatment success. Although investigating user experience is one of the most important mechanisms for understanding mHealth discontinuance, surprisingly, the core determinants of overall user experience (i.e., positive and negative) about mHealth apps for hypertension are unknown. To address the mentioned gap in knowledge, this study adopts the computational grounded theory methodological framework and employs advanced deep learning algorithms to predict core quality criteria that affect overall user experience of hypertension apps published in the Apple App Store. This study contributes to theory and practice of designing evidence-based interventions for hypertension in the form of propositions and provide valuable managerial implications and recommendations for manufacturers., Comment: 17 pages, 6 Figures, 1 Table

We explore the fundamental problem of sorting through the lens of learning-augmented algorithms, where algorithms can leverage possibly erroneous predictions to improve their efficiency. We consider two different settings: In the first setting, each item is provided a prediction of its position in the sorted list. In the second setting, we assume there is a "quick-and-dirty" way of comparing items, in addition to slow-and-exact comparisons. For both settings, we design new and simple algorithms using only $O(\sum_i \log \eta_i)$ exact comparisons, where $\eta_i$ is a suitably defined prediction error for the $i$th element. In particular, as the quality of predictions deteriorates, the number of comparisons degrades smoothly from $O(n)$ to $O(n\log n)$. We prove that the comparison complexity is theoretically optimal with respect to the examined error measures. An experimental evaluation against existing adaptive and non-adaptive sorting algorithms demonstrates the potential of applying learning-augmented algorithms in sorting tasks.

Using analog beamforming in mmWave frequency bands we can focus the energy towards a receiver to achieve high throughput. However, this requires the network to quickly find the best downlink beam configuration in the face of non-IID data. We propose a personalized Federated Learning (FL) method to address this challenge, where we learn a mapping between uplink Sub-6GHz channel estimates and the best downlink beam in heterogeneous scenarios with non-IID characteristics. We also devise FedLion, a FL implementation of the Lion optimization algorithm. Our approach reduces the signaling overhead and provides superior performance, up to 33.6% higher accuracy than a single FL model and 6% higher than a local model., Comment: 10 pages, to be published in IEEE Future Networks World Forum 2023

A major technique in learning-augmented online algorithms is combining multiple algorithms or predictors. Since the performance of each predictor may vary over time, it is desirable to use not the single best predictor as a benchmark, but rather a dynamic combination which follows different predictors at different times. We design algorithms that combine predictions and are competitive against such dynamic combinations for a wide class of online problems, namely, metrical task systems. Against the best (in hindsight) unconstrained combination of $\ell$ predictors, we obtain a competitive ratio of $O(\ell^2)$, and show that this is best possible. However, for a benchmark with slightly constrained number of switches between different predictors, we can get a $(1+\epsilon)$-competitive algorithm. Moreover, our algorithms can be adapted to access predictors in a bandit-like fashion, querying only one predictor at a time. An unexpected implication of one of our lower bounds is a new structural insight about covering formulations for the $k$-server problem.

We prove a few new lower bounds on the randomized competitive ratio for the $k$-server problem and other related problems, resolving some long-standing conjectures. In particular, for metrical task systems (MTS) we asympotically settle the competitive ratio and obtain the first improvement to an existential lower bound since the introduction of the model 35 years ago (in 1987). More concretely, we show: 1. There exist $(k+1)$-point metric spaces in which the randomized competitive ratio for the $k$-server problem is $\Omega(\log^2 k)$. This refutes the folklore conjecture (which is known to hold in some families of metrics) that in all metric spaces with at least $k+1$ points, the competitive ratio is $\Theta(\log k)$. 2. Consequently, there exist $n$-point metric spaces in which the randomized competitive ratio for MTS is $\Omega(\log^2 n)$. This matches the upper bound that holds for all metrics. The previously best existential lower bound was $\Omega(\log n)$ (which was known to be tight for some families of metrics). 3. For all $k

Federated Learning (FL) is a promising framework for distributed learning when data is private and sensitive. However, the state-of-the-art solutions in this framework are not optimal when data is heterogeneous and non-Independent and Identically Distributed (non-IID). We propose a practical and robust approach to personalization in FL that adjusts to heterogeneous and non-IID data by balancing exploration and exploitation of several global models. To achieve our aim of personalization, we use a Mixture of Experts (MoE) that learns to group clients that are similar to each other, while using the global models more efficiently. We show that our approach achieves an accuracy up to 29.78 % and up to 4.38 % better compared to a local model in a pathological non-IID setting, even though we tune our approach in the IID setting., Comment: To appear in International Workshop on Trustworthy Federated Learning in Conjunction with IJCAI 2022 (FL-IJCAI'22). 8 pages

We study the block-aware caching problem, a generalization of classic caching in which fetching (or evicting) pages from the same block incurs the same cost as fetching (or evicting) just one page from the block. Given a cache of size $k$, and a sequence of requests from $n$ pages partitioned into given blocks of size $\beta\leq k$, the goal is to minimize the total cost of fetching to (or evicting from) cache. We show the following results: $\bullet$ For the eviction cost model, we show an $O(\log k)$-approximate offline algorithm, a $k$-competitive deterministic online algorithm, and an $O(\log^2 k)$-competitive randomized online algorithm. $\bullet$ For the fetching cost model, we show an integrality gap of $\Omega(\beta)$ for the natural LP relaxation of the problem, and an $\Omega(\beta + \log k)$ lower bound for randomized online algorithms. The strategy of ignoring the block-structure and running a classical paging algorithm trivially achieves an $O(\beta)$ approximation and an $O(\beta \log k)$ competitive ratio respectively for the offline and online-randomized setting. $\bullet$ For both fetching and eviction models, we show improved bounds for the $(h,k)$-bicriteria version of the problem. In particular, when $k=2h$, we match the performance of classical caching algorithms up to constant factors. Our results establish a separation between the tractability of the fetching and eviction cost models, which is interesting since fetching/evictions costs are the same up to an additive term for classic caching. Previous work only studied online deterministic algorithms for the fetching cost model when $k > h$. Our insight is to relax the block-aware caching problem to a submodular covering LP. The main technical challenge is to maintain a competitive fractional solution, and to round it with bounded loss, as the constraints of this LP are revealed online.

In a 1989 paper titled "shortest paths without a map", Papadimitriou and Yannakakis introduced an online model of searching in a weighted layered graph for a target node, while attempting to minimize the total length of the path traversed by the searcher. This problem, later called layered graph traversal, is parametrized by the maximum cardinality $k$ of a layer of the input graph. It is an online setting for dynamic programming, and it is known to be a rather general and fundamental model of online computing, which includes as special cases other acclaimed models. The deterministic competitive ratio for this problem was soon discovered to be exponential in $k$, and it is now nearly resolved: it lies between $\Omega(2^k)$ and $O(k2^k)$. Regarding the randomized competitive ratio, in 1993 Ramesh proved, surprisingly, that this ratio has to be at least $\Omega(k^2 / \log^{1+\epsilon} k)$ (for any constant $\epsilon > 0$). In the same paper, Ramesh also gave an $O(k^{13})$-competitive randomized online algorithm. Between 1993 and the results obtained in this paper, no progress has been reported on the randomized competitive ratio of layered graph traversal. In this work we show how to apply the mirror descent framework on a carefully selected evolving metric space, and obtain an $O(k^2)$-competitive randomized online algorithm. This matches asymptotically an improvement of the aforementioned lower bound (Bubeck, Coester, Rabani; STOC 2023), which we announced (among other results) after the initial publication of the results here., Comment: FOCS '22 and accepted at SICOMP

We introduce a natural online allocation problem that connects several of the most fundamental problems in online optimization. Let $M$ be an $n$-point metric space. Consider a resource that can be allocated in arbitrary fractions to the points of $M$. At each time $t$, a convex monotone cost function $c_t: [0,1]\to\mathbb{R}_+$ appears at some point $r_t\in M$. In response, an algorithm may change the allocation of the resource, paying movement cost as determined by the metric and service cost $c_t(x_{r_t})$, where $x_{r_t}$ is the fraction of the resource at $r_t$ at the end of time $t$. For example, when the cost functions are $c_t(x)=\alpha x$, this is equivalent to randomized MTS, and when the cost functions are $c_t(x)=\infty\cdot 1_{x<1/k}$, this is equivalent to fractional $k$-server. We give an $O(\log n)$-competitive algorithm for weighted star metrics. Due to the generality of allowed cost functions, classical multiplicative update algorithms do not work for the metric allocation problem. A key idea of our algorithm is to decouple the rate at which a variable is updated from its value, resulting in interesting new dynamics. This can be viewed as running mirror descent with a time-varying regularizer, and we use this perspective to further refine the guarantees of our algorithm. The standard analysis techniques run into multiple complications when the regularizer is time-varying, and we show how to overcome these issues by making various modifications to the default potential function. We also consider the problem when cost functions are allowed to be non-convex. In this case, we give tight bounds of $\Theta(n)$ on tree metrics, which imply deterministic and randomized competitive ratios of $O(n^2)$ and $O(n\log n)$ respectively on arbitrary metrics. Our algorithm is based on an $\ell_2^2$-regularizer.

We study the online problem of minimizing power consumption in systems with multiple power-saving states. During idle periods of unknown lengths, an algorithm has to choose between power-saving states of different energy consumption and wake-up costs. We develop a learning-augmented online algorithm that makes decisions based on (potentially inaccurate) predicted lengths of the idle periods. The algorithm's performance is near-optimal when predictions are accurate and degrades gracefully with increasing prediction error, with a worst-case guarantee almost identical to the optimal classical online algorithm for the problem. A key ingredient in our approach is a new algorithm for the online ski rental problem in the learning augmented setting with tight dependence on the prediction error. We support our theoretical findings with experiments.

The $k$-server conjecture, first posed by Manasse, McGeoch and Sleator in 1988, states that a $k$-competitive deterministic algorithm for the $k$-server problem exists. It is conjectured that the work function algorithm (WFA) achieves this guarantee, a multi-purpose algorithm with applications to various online problems. This has been shown for several special cases: $k=2$, $(k+1)$-point metrics, $(k+2)$-point metrics, the line metric, weighted star metrics, and $k=3$ in the Manhattan plane. The known proofs of these results are based on potential functions tied to each particular special case, thus requiring six different potential functions for the six cases. We present a single potential function proving $k$-competitiveness of WFA for all these cases. We also use this potential to show $k$-competitiveness of WFA on multiray spaces and for $k=3$ on trees. While the DoubleCoverage algorithm was known to be $k$-competitive for these latter cases, it has been open for WFA. Our potential captures a type of lazy adversary and thus shows that in all settled cases, the worst-case adversary is lazy. Chrobak and Larmore conjectured in 1992 that a potential capturing the lazy adversary would resolve the $k$-server conjecture. To our major surprise, this is not the case, as we show (using connections to the $k$-taxi problem) that our potential fails for three servers on the circle. Thus, our potential highlights laziness of the adversary as a fundamental property that is shared by all settled cases but violated in general. On the one hand, this weakens our confidence in the validity of the $k$-server conjecture. On the other hand, if the $k$-server conjecture holds, then we believe it can be proved by a variant of our potential.

We consider the online $k$-taxi problem, a generalization of the $k$-server problem, in which $k$ servers are located in a metric space. A sequence of requests is revealed one by one, where each request is a pair of two points, representing the start and destination of a travel request by a passenger. The goal is to serve all requests while minimizing the distance traveled without carrying a passenger. We show that the classic Double Coverage algorithm has competitive ratio $2^k-1$ on HSTs, matching a recent lower bound for deterministic algorithms. For bounded depth HSTs, the competitive ratio turns out to be much better and we obtain tight bounds. When the depth is $d\ll k$, these bounds are approximately $k^d/d!$. By standard embedding results, we obtain a randomized algorithm for arbitrary $n$-point metrics with (polynomial) competitive ratio $O(k^c\Delta^{1/c}\log_{\Delta} n)$, where $\Delta$ is the aspect ratio and $c\ge 1$ is an arbitrary positive integer constant. The only previous known bound was $O(2^k\log n)$. For general (weighted) tree metrics, we prove the competitive ratio of Double Coverage to be $\Theta(k^d)$ for any fixed depth $d$, but unlike on HSTs it is not bounded by $2^k-1$. We obtain our results by a dual fitting analysis where the dual solution is constructed step-by-step backwards in time. Unlike the forward-time approach typical of online primal-dual analyses, this allows us to combine information from the past and the future when assigning dual variables. We believe this method can be useful also for other problems. Using this technique, we also provide a dual fitting proof of the $k$-competitiveness of Double Coverage for the $k$-server problem on trees.

We consider a natural semi-online model for weighted paging, where at any time the algorithm is given predictions, possibly with errors, about the next arrival of each page. The model is inspired by Belady's classic optimal offline algorithm for unweighted paging, and extends the recently studied model for learning-augmented paging (Lykouris and Vassilvitskii, 2018) to the weighted setting. For the case of perfect predictions, we provide an $\ell$-competitive deterministic and an $O(\log \ell)$-competitive randomized algorithm, where $\ell$ is the number of distinct weight classes. Both these bounds are tight, and imply an $O(\log W)$- and $O(\log \log W)$-competitive ratio, respectively, when the page weights lie between $1$ and $W$. Previously, it was not known how to use these predictions in the weighted setting and only bounds of $k$ and $O(\log k)$ were known, where $k$ is the cache size. Our results also generalize to the interleaved paging setting and to the case of imperfect predictions, with the competitive ratios degrading smoothly from $O(\ell)$ and $O(\log \ell)$ to $O(k)$ and $O(\log k)$, respectively, as the prediction error increases. Our results are based on several insights on structural properties of Belady's algorithm and the sequence of page arrival predictions, and novel potential functions that incorporate these predictions. For the case of unweighted paging, the results imply a very simple potential function based proof of the optimality of Belady's algorithm, which may be of independent interest.

We consider a generalization of the fundamental online metrical service systems (MSS) problem where the feasible region can be transformed between requests. In this problem, which we call T-MSS, an algorithm maintains a point in a metric space and has to serve a sequence of requests. Each request is a map (transformation) $f_t\colon A_t\to B_t$ between subsets $A_t$ and $B_t$ of the metric space. To serve it, the algorithm has to go to a point $a_t\in A_t$, paying the distance from its previous position. Then, the transformation is applied, modifying the algorithm's state to $f_t(a_t)$. Such transformations can model, e.g., changes to the environment that are outside of an algorithm's control, and we therefore do not charge any additional cost to the algorithm when the transformation is applied. The transformations also allow to model requests occurring in the $k$-taxi problem. We show that for $\alpha$-Lipschitz transformations, the competitive ratio is $\Theta(\alpha)^{n-2}$ on $n$-point metrics. Here, the upper bound is achieved by a deterministic algorithm and the lower bound holds even for randomized algorithms. For the $k$-taxi problem, we prove a competitive ratio of $\tilde O((n\log k)^2)$. For chasing convex bodies, we show that even with contracting transformations no competitive algorithm exists. The problem T-MSS has a striking connection to the following deep mathematical question: Given a finite metric space $M$, what is the required cardinality of an extension $\hat M\supseteq M$ where each partial isometry on $M$ extends to an automorphism? We give partial answers for special cases.

Machine-learned predictors, although achieving very good results for inputs resembling training data, cannot possibly provide perfect predictions in all situations. Still, decision-making systems that are based on such predictors need not only to benefit from good predictions but also to achieve a decent performance when the predictions are inadequate. In this paper, we propose a prediction setup for arbitrary metrical task systems (MTS) (e.g., caching, k-server and convex body chasing) and online matching on the line. We utilize results from the theory of online algorithms to show how to make the setup robust. Specifically for caching, we present an algorithm whose performance, as a function of the prediction error, is exponentially better than what is achievable for general MTS. Finally, we present an empirical evaluation of our methods on real world datasets, which suggests practicality.

We study the resource augmented version of the $k$-server problem, also known as the $k$-server problem against weak adversaries or the $(h,k)$-server problem. In this setting, an online algorithm using $k$ servers is compared to an offline algorithm using $h$ servers, where $h\le k$. For uniform metrics, it has been known since the seminal work of Sleator and Tarjan (1985) that for any $\epsilon>0$, the competitive ratio drops to a constant if $k=(1+\epsilon) \cdot h$. This result was later generalized to weighted stars (Young 1994) and trees of bounded depth (Bansal et al. 2017). The main open problem for this setting is whether a similar phenomenon occurs on general metrics. We resolve this question negatively. With a simple recursive construction, we show that the competitive ratio is at least $\Omega(\log \log h)$, even as $k\to\infty$. Our lower bound holds for both deterministic and randomized algorithms. It also disproves the existence of a competitive algorithm for the infinite server problem on general metrics., Comment: To appear in STOC 2020

We show that on every $n$-point HST metric, there is a randomized online algorithm for metrical task systems (MTS) that is $1$-competitive for service costs and $O(\log n)$-competitive for movement costs. In general, these refined guarantees are optimal up to the implicit constant. While an $O(\log n)$-competitive algorithm for MTS on HST metrics was developed by Bubeck et al. (SODA 2019), that approach could only establish an $O((\log n)^2)$-competitive ratio when the service costs are required to be $O(1)$-competitive. Our algorithm can be viewed as an instantiation of online mirror descent with the regularizer derived from a multiscale conditional entropy. In fact, our algorithm satisfies a set of even more refined guarantees; we are able to exploit this property to combine it with known random embedding theorems and obtain, for any $n$-point metric space, a randomized algorithm that is $1$-competitive for service costs and $O((\log n)^2)$-competitive for movement costs., Comment: COLT 2019

We report on the determination of the anomalous spin Hall angle in the ferromagnetic metal alloy cobalt-iron (Co$_{25}$Fe$_{75}$, CoFe). This is accomplished by measuring the spin injection/detection efficiency in a multiterminal device with nanowires of platinum (Pt) and CoFe deposited onto the magnetic insulator yttrium iron garnet (Y$_3$Fe$_5$O$_{12}$, YIG). Applying a spin-resistor model to our multiterminal spin transport data, we determine the magnon conductivity in YIG, the spin conductance at the YIG/CoFe interface and finally the anomalous spin Hall angle of CoFe as a function of its spin diffusion length in a single device. Our experiments clearly reveal a negative anomalous spin Hall angle of the ferromagnetic metal CoFe, but a vanishing ordinary spin Hall angle. This is in contrast to the results reported for the ferromagnetic metals Co and permalloy., Comment: 4 pages, 4 figures

We study online competitive algorithms for the \emph{line chasing problem} in Euclidean spaces $\reals^d$, where the input consists of an initial point $P_0$ and a sequence of lines $X_1,X_2,...,X_m$, revealed one at a time. At each step $t$, when the line $X_t$ is revealed, the algorithm must determine a point $P_t\in X_t$. An online algorithm is called $c$-competitive if for any input sequence the path $P_0, P_1,...,P_m$ it computes has length at most $c$ times the optimum path. The line chasing problem is a variant of a more general convex body chasing problem, where the sets $X_t$ are arbitrary convex sets. To date, the best competitive ratio for the line chasing problem was $28.1$, even in the plane. We significantly improve this bound, by providing a~$3$-competitive algorithm for any dimension $d$. We also improve the lower bound on the competitive ratio, from $1.412$ to $1.5358$.

We consider the online $k$-taxi problem, a generalization of the $k$-server problem, in which $k$ taxis serve a sequence of requests in a metric space. A request consists of two points $s$ and $t$, representing a passenger that wants to be carried by a taxi from $s$ to $t$. The goal is to serve all requests while minimizing the total distance traveled by all taxis. The problem comes in two flavors, called the easy and the hard $k$-taxi problem: In the easy $k$-taxi problem, the cost is defined as the total distance traveled by the taxis; in the hard $k$-taxi problem, the cost is only the distance of empty runs. The hard $k$-taxi problem is substantially more difficult than the easy version with at least an exponential deterministic competitive ratio, $\Omega(2^k)$, admitting a reduction from the layered graph traversal problem. In contrast, the easy $k$-taxi problem has exactly the same competitive ratio as the $k$-server problem. We focus mainly on the hard version. For hierarchically separated trees (HSTs), we present a memoryless randomized algorithm with competitive ratio $2^k-1$ against adaptive online adversaries and provide two matching lower bounds: for arbitrary algorithms against adaptive adversaries and for memoryless algorithms against oblivious adversaries. Due to well-known HST embedding techniques, the algorithm implies a randomized $O(2^k\log n)$-competitive algorithm for arbitrary $n$-point metrics. This is the first competitive algorithm for the hard $k$-taxi problem for general finite metric spaces and general $k$. For the special case of $k=2$, we obtain a precise answer of $9$ for the competitive ratio in general metrics. With an algorithm based on growing, shrinking and shifting regions, we show that one can achieve a constant competitive ratio also for the hard $3$-taxi problem on the line (abstracting the scheduling of three elevators).

Context-free games on strings are two-player rewriting games based on a set of production rules and a regular target language. In each round, the first player selects a position of the current string; then the second player replaces the symbol at that position according to one of the production rules. The first player wins as soon as the current string belongs to the target language. In this paper the one-pass setting for context-free games is studied, where the knowledge of the first player is incomplete, she selects positions in a left-to-right fashion and only sees the current symbol and the symbols from previous rounds. The paper studies conditions under which dominant or undominated strategies for the first player exist and investigates the complexity of some related algorithmic problems.

Spontaneous symmetry-breaking quantum phase transitions play an essential role in condensed matter physics. The collective excitations in the broken-symmetry phase near the quantum critical point can be characterized by fluctuations of phase and amplitude of the order parameter. The phase oscillations correspond to the massless Nambu$-$Goldstone modes whereas the massive amplitude mode, analogous to the Higgs boson in particle physics, is prone to decay into a pair of low-energy Nambu$-$Goldstone modes in low dimensions. Especially, observation of a Higgs amplitude mode in two dimensions is an outstanding experimental challenge. Here, using the inelastic neutron scattering and applying the bond-operator theory, we directly and unambiguously identify the Higgs amplitude mode in a two-dimensional S=1/2 quantum antiferromagnet C$_9$H$_{18}$N$_2$CuBr$_4$ near a quantum critical point in two dimensions. Owing to an anisotropic energy gap, it kinematically prevents such decay and the Higgs amplitude mode acquires an infinite lifetime., Comment: 12 pages, 4 figures in the main text+3 figures in Supplementary Information

We study a variant of the $k$-server problem, the infinite server problem, in which infinitely many servers reside initially at a particular point of the metric space and serve a sequence of requests. In the framework of competitive analysis, we show a surprisingly tight connection between this problem and the $(h,k)$-server problem, in which an online algorithm with $k$ servers competes against an offline algorithm with $h$ servers. Specifically, we show that the infinite server problem has bounded competitive ratio if and only if the $(h,k)$-server problem has bounded competitive ratio for some $k=O(h)$. We give a lower bound of $3.146$ for the competitive ratio of the infinite server problem, which implies the same lower bound for the $(h,k)$-server problem even when $k/h \to \infty$ and holds also for the line metric; the previous known bounds were 2.4 for general metric spaces and 2 for the line. For weighted trees and layered graphs we obtain upper bounds, although they depend on the depth. Of particular interest is the infinite server problem on the line, which we show to be equivalent to the seemingly easier case in which all requests are in a fixed bounded interval away from the original position of the servers. This is a special case of a more general reduction from arbitrary metric spaces to bounded subspaces. Unfortunately, classical approaches (double coverage and generalizations, work function algorithm, balancing algorithms) fail even for this special case.