Worst-case Guarantees Research Articles

Pathological Cases for a Class of Reachability-Based Garbage Collectors

Although existing garbage collectors (GCs) perform extremely well on typical programs, there still exist pathological programs for which modern GCs significantly degrade performance. This observation begs the question: might there exist a ‘holy grail’ GC algorithm, as yet undiscovered, guaranteeing both constant-length pause times and that memory is collected promptly upon becoming unreachable? For decades, researchers have understood that such a GC is not always possible, i.e., some pathological behavior is unavoidable when the program can make heap cycles and operates near the memory limit, regardless of the GC algorithm used. However, this understanding has until now been only informal, lacking a rigorous formal proof. This paper complements that informal understanding with a rigorous proof, showing with mathematical certainty that every GC algorithm that can implement a realistic mutator-observer interface has some pathological program that forces it to either introduce a long GC pause into program execution or reject an allocation even though there is available space. Hence, language designers must either accept these pathological scenarios and design heuristic approaches that minimize their impact (e.g., generational collection), or restrict programs and environments to a strict subset of the behaviors allowed by our mutator-observer–style interface (e.g., by enforcing a type system that disallows cycles or overprovisioning memory). We do not expect this paper to have any effect on garbage collection practice. Instead, it provides the first mathematically rigorous answers to these interesting questions about the limits of garbage collection. We do so via rigorous reductions between GC and the dynamic graph connectivity problem in complexity theory, so future algorithms and lower bounds from either community transfer to the other via our reductions. We end by describing how to adapt techniques from the graph data structures community to build a garbage collector making worst-case guarantees that improve performance on our motivating, pathologically memory-constrained scenarios, but in practice find too much overhead to recommend for typical use.

On Hill’s Worst-Case Guarantee for Indivisible Bads

We evaluate the fairness of a rule allocating among agents with equal rights by computing their utility for a hypothetical worst-case share, which only depends on their own valuation and the number of agents. For indivisible goods, Budish proposed the maximin share : the least utility of a bundle in the best partition of the objects; unfortunately maximin share is not always satisfiable. Earlier, Hill proposed the worst maximin share over all utilities with the same largest possible single-object value. More conservative than maximin share, it is guaranteed to be satisfiable for any possible profile of utilities and its computation is elementary, whereas it is NP-hard to compute maximin share. We apply Hill’s approach to the allocation of indivisible bads (objects with disutilities) and compute in closed form the worst-case minimax share for a given value of the worst single bad. We show that the worst-case minimax share is close to the original minimax share and that its monotonic closure is the best guaranteed share for all allocation instances with a common upper bound on the value of the worst single bad.

On Scheduling Mechanisms Beyond the Worst Case

The problem of scheduling unrelated machines has been studied since the inception of algorithmic mechanism design (Nisan and Ronen, Algorithmic mechanism design(extended abstract). In: Proceedings of the Thirty First Annual ACM Symposium on Theory of Computing (STOC), pp. 129–140, 1999. It is a resource allocation problem that entails assigning m tasks to n machines for execution. Machines are regarded as strategic agents who may lie about their execution costs so as to minimize their time cost. To address the situation when monetary payment is not an option to compensate the machines’ costs, Koutsoupias (Theory Comput Syst 54:375–387, 2014) devised two truthful mechanisms, K and P respectively, that achieves an approximation ratio of n+12 and n, for social cost minimization. In addition, no truthful mechanism can achieve an approximation ratio better than n+12. Hence, mechanism K is optimal. While the approximation ratio provides a strong worst-case guarantee, it also limits us to a comprehensive understanding of mechanism performance on various inputs. This paper investigates these two scheduling mechanisms beyond the worst case. We first show that mechanism K achieves a smaller social cost than mechanism P on every input. That is, mechanism K is pointwise better than mechanism P. Next, for each task, when machines’ execution costs are independent and identically drawn from a task-specific distribution, we show that the average-case approximation ratio of mechanism K converges to a constant determined by the task-specific distribution. This bound is tight for mechanism K. For a better understanding of this distribution-dependent constant, on the one hand, we estimate its value by plugging in a few common distributions; on the other, we show that this converging bound improves a known bound (Zhang in Algorithmica 83(6):1638–1652, 2021)) which only captures the single-task setting. Last, we find that the average-case approximation ratio of mechanism P converges to the same constant.

A Neural Network Approach to Physical Information Embedding for Optimal Power Flow

With the increasing share of renewable energy in the power system, traditional power flow calculation methods are facing challenges of complexity and efficiency. To address these issues, this paper proposes a new framework for AC optimal power flow analysis based on a physics-informed convolutional neural network (PICNN) approach, which enables the neural network to learn solutions that follow physical laws by embedding the power flow equations and other physical constraints into the loss function of the network. Compared with the traditional power flow calculation method, the calculation speed of this method is improved by 10–30 times. Compared to traditional neural network models, the method provides higher accuracy, with an average increase in accuracy of up to 2.5–10 times. In addition, this paper introduces a methodology to extract worst-case guarantees for violations of the neural network’s predicted power generation constraints, determining the worst possible violation that could result from any neural network output across the entire input domain, and taking appropriate measures to reduce the violation. The method is experimentally shown to be highly accurate and reliable for the AC optimal power flow (AC-OPF) analysis problem, while reducing the dependence on a large amount of labelled data.

Automated Category Tree Construction: Hardness Bounds and Algorithms

Category trees, or taxonomies, are rooted trees where each node, called a category, corresponds to a set of related items. The construction of taxonomies has been studied in various domains, including e-commerce, document management, and question answering. Multiple algorithms for automating construction have been proposed, employing a variety of clustering approaches and crowdsourcing. However, no formal model to capture such categorization problems has been devised, and their complexity has not been studied. To address this, we propose in this work a combinatorial model that captures many practical settings and show that the aforementioned empirical approach has been warranted, as we prove strong inapproximability bounds for various problem variants and special cases when the goal is to produce a categorization of the maximum utility. In our model, the input is a set of n weighted item sets that the tree would ideally contain as categories. Each category, rather than perfectly match the corresponding input set, is allowed to exceed a given threshold for a given similarity function. The goal is to produce a tree that maximizes the total weight of the sets for which it contains a matching category. A key parameter is an upper bound on the number of categories an item may belong to, which produces the hardness of the problem, as initially each item may be contained in an arbitrary number of input sets. For this model, we prove inapproximability bounds, of order \(\tilde{\Theta }(\sqrt {n}) \) or \(\tilde{\Theta }(n) \) , for various problem variants and special cases, loosely justifying the aforementioned heuristic approach. Our work includes reductions based on parameterized randomized constructions that highlight how various problem parameters and properties of the input may affect the hardness. Moreover, for the special case where the category must be identical to the corresponding input set, we devise an algorithm whose approximation guarantee depends solely on a more granular parameter, allowing improved worst-case guarantees, as well as the application of practical exact solvers. We further provide efficient algorithms with much improved approximation guarantees for practical special cases where the cardinalities of the input sets or the number of input sets each items belongs to are not too large. Finally, we also generalize our results to DAG-based and non-hierarchical categorization.

An auction approach to aircraft bandwidth scheduling in non-terrestrial networks

An auction approach to aircraft bandwidth scheduling in non-terrestrial networks

Optimal inexactness schedules for tunable oracle-based methods

Several recent works address the impact of inexact oracles in the convergence analysis of modern first-order optimization techniques, e.g. Bregman Proximal Gradient and Prox-Linear methods as well as their accelerated variants, extending their field of applicability. In this paper, we consider situations where the oracle's inexactness can be chosen upon demand, more precision coming at a computational price counterpart. Our main motivations arise from oracles requiring the solving of auxiliary subproblems or the inexact computation of involved quantities, e.g. a mini-batch stochastic gradient as a full-gradient estimate. We propose optimal inexactness schedules according to presumed oracle cost models and patterns of worst-case guarantees, covering among others convergence results of the aforementioned methods under the presence of inexactness. Specifically, we detail how to choose the level of inexactness at each iteration to obtain the best trade-off between convergence and computational investments. Furthermore, we highlight the benefits one can expect by tuning those oracles' accuracy instead of keeping it constant throughout. Finally, we provide extensive numerical experiments that support the practical interest of our approach, both in offline and online settings, applied to the Fast Gradient algorithm.

Learning-Augmented Mechanism Design: Leveraging Predictions for Facility Location

In this work, we introduce an alternative model for the design and analysis of strategyproof mechanisms that is motivated by the recent surge of work in “learning-augmented algorithms.” Aiming to complement the traditional worst-case analysis approach in computer science, this line of work has focused on the design and analysis of algorithms that are enhanced with machine-learned predictions. The algorithms can use the predictions as a guide to inform their decisions, aiming to achieve much stronger performance guarantees when these predictions are accurate (consistency), while also maintaining near-optimal worst-case guarantees, even if these predictions are inaccurate (robustness). We initiate the design and analysis of strategyproof mechanisms that are augmented with predictions regarding the private information of the participating agents. To exhibit the important benefits of this approach, we revisit the canonical problem of facility location with strategic agents in the two-dimensional Euclidean space. We study both the egalitarian and utilitarian social cost functions, and we propose new strategyproof mechanisms that leverage predictions to guarantee an optimal trade-off between consistency and robustness. Furthermore, we also prove parameterized approximation results as a function of the prediction error, showing that our mechanisms perform well, even when the predictions are not fully accurate. Funding: The work of E. Balkanski was supported in part by the National Science Foundation [Grants CCF-2210501 and IIS-2147361]. The work of V. Gkatzelis and X. Tan was supported in part by the National Science Foundation [Grant CCF-2210502] and [CAREER Award CCF-2047907].

Bounded incentives in manipulating the probabilistic serial rule

Bounded incentives in manipulating the probabilistic serial rule

From computer systems to power systems: using stochastic network calculus for flexibility analysis in power systems

As power systems transition from controllable fossil fuel plants to variable renewable sources, managing power supply and demand fluctuations becomes increasingly important. Novel approaches are required to balance these fluctuations. The problem of determining the optimal deployment of flexibility options, considering factors such as timing and location, shares similarities with scheduling problems encountered in computer networks. In both cases, the objective is to coordinate various distributed units and manage the flow of either data or power. Among the methods for scheduling and resource allocation in computer networks, stochastic network calculus (SNC) is a promising approach that estimates worst-case guarantees for Quality of Service (QoS) indicators of computer networks, such as delay and backlog. Promising QoS indicators in the power system are given by the amount of stored energy, the serviced demand, and the demand elasticity. In this work, we investigate SNC for its capabilities and limitations to quantify flexibility service guarantees in power systems. We generate and aggregate stochastic envelopes for random processes, which was found useful for modeling flexibility in power systems at multiple time scales. In a case study on the reliability of a solar-powered car charging station, we obtain similar results as from a mixed-integer linear programming problem, which provides confidence that the chosen SNC approach is suitable for modeling power system flexibility.

Transportation Asset Acquisition under a Newsvendor Model with Cutting-Stock Restrictions: Approximation and Decomposition Algorithms

Logistics service providers use transportation assets to offer services to their customers. To cope with demand variability, they may acquire additional assets on a one-off (spot) basis. The planner’s problem is to determine the optimal level of assets acquired upfront, such that their cost is minimized, for a given planning horizon. Our formulation captures a nontrivial complication: Although ordering quantities are pertinent to asset acquisition, customer demand is in the form of service requests. Not only does each request have a stochastic duration, but also the total number of requests per customer is uncertain. We introduce a two-stage newsvendor model where demand for spot assets is derived through optimal cutting-stock patterns. Leveraging results from bin-packing, we propose polynomial algorithms that have worst-case guarantees for upper and lower bounds. Our method finds optimal solutions to instances intractable by commercial solvers. We investigate demand variability by means of a factorial experiment. We find that, whereas variability in the number of requests leads to higher costs, variability in each request’s duration can reduce costs. Finally, we demonstrate the modularity of our approach with two extensions: asset routing and outsourcing. Our results provide a practical approach to transportation asset acquisition and offer insights on the differing impact of demand uncertainty on the total acquisition cost. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.1201 .

The splay-list: a distribution-adaptive concurrent skip-list

The design and implementation of efficient concurrent data structures has seen significant attention. However, most of this work has focused on concurrent data structures providing good worst-case guarantees, although, in real workloads, objects are often accessed at different rates. Efficient distribution-adaptive data structures, such as splay-trees, are known in the sequential case; however, they often are hard to translate efficiently to the concurrent case. We investigate distribution-adaptive concurrent data structures, and propose a new design called the splay-list. At a high level, the splay-list is similar to a standard skip-list, with the key distinction that the height of each element adapts dynamically to its access rate: popular elements “move up,” whereas rarely-accessed elements decrease in height. We show that the splay-list provides order-optimal amortized complexity bounds for a subset of operations, while being amenable to efficient concurrent implementation. Experiments show that the splay-list can leverage distribution-adaptivity for performance, and can outperform the only previously-known distribution-adaptive concurrent design in certain workloads.

Sharp Performance Bounds for PASTA

LiDAR is a standard sensor choice for self-localization and SLAM on indoor autonomous robots. While there are many methods to estimate a robot’s location using LiDAR measurements, most rely on algorithms that solve a generic LiDAR scan matching problem. When safety is a concern, these algorithms must provide a bound on the localization error to enable safety enforcing controllers, such as those based on Control Barrier Functions. Unfortunately, most existing scan matching algorithms offer no formal guarantees and are tailored to structured, high-resolution 3D point clouds. In this paper, we present an improved theoretical analysis for a low-cost alternative to these methods named Pasta (Provably Accurate Simple Transformation Alignment), originally introduced in marchi2022lidar. We provide a formal worst-case guarantee on the localization error and show, experimentally, that it is tight. This characterization of the localization error simplifies the interface between high-dimensional perception data and safety-critical control.

Principled analyses and design of first-order methods with inexact proximal operators

Proximal operations are among the most common primitives appearing in both practical and theoretical (or high-level) optimization methods. This basic operation typically consists in solving an intermediary (hopefully simpler) optimization problem. In this work, we survey notions of inaccuracies that can be used when solving those intermediary optimization problems. Then, we show that worst-case guarantees for algorithms relying on such inexact proximal operations can be systematically obtained through a generic procedure based on semidefinite programming. This methodology is primarily based on the approach introduced by Drori and Teboulle (Math Program 145(1–2):451–482, 2014) and on convex interpolation results, and allows producing non-improvable worst-case analyses. In other words, for a given algorithm, the methodology generates both worst-case certificates (i.e., proofs) and problem instances on which they are achieved. Relying on this methodology, we study numerical worst-case performances of a few basic methods relying on inexact proximal operations including accelerated variants, and design a variant with optimized worst-case behavior. We further illustrate how to extend the approach to support strongly convex objectives by studying a simple relatively inexact proximal minimization method.

Privacy pitfalls of releasing in-vehicle network data

The ever-increasing volume of vehicular data has enabled different service providers to access and monetize in-vehicle network data of millions of drivers. However, such data often carry personal or even potentially sensitive information, and therefore service providers either need to ask for drivers' consent or anonymize such data in order to comply with data protection regulations. In this paper, we show that both fine-grained consent control as well as the adequate anonymization of in-network vehicular data are very challenging. First, by exploiting that in-vehicle sensor measurements are inherently interdependent, we are able to effectively i) re-identify a driver even from the raw, unprocessed CAN data with 97% accuracy, and ii) reconstruct the vehicle's complete location trajectory knowing only its speed and steering wheel position. Since such signal interdependencies are hard to identify even for data controllers, drivers' consent will arguably not be informed and hence may become invalid. Second, we show that the non-systematic application of different standard anonymization techniques (e.g., aggregation, suppression, signal distortion) often results in volatile, empirical privacy guarantees to the population as a whole but fails to provide a strong, worst-case privacy guarantee to every single individual. Therefore, we advocate the application of principled privacy models (such as Differential Privacy) to anonymize data with strong worst-case guarantee.

Data-Independent Structured Pruning of Neural Networks via Coresets.

Model compression is crucial for the deployment of neural networks on devices with limited computational and memory resources. Many different methods show comparable accuracy of the compressed model and similar compression rates. However, the majority of the compression methods are based on heuristics and offer no worst case guarantees on the tradeoff between the compression rate and the approximation error for an arbitrarily new sample. We propose the first efficient structured pruning algorithm with a provable tradeoff between its compression rate and the approximation error for any future test sample. Our method is based on the coreset framework, and it approximates the output of a layer of neurons/filters by a coreset of neurons/filters in the previous layer and discards the rest. We apply this framework in a layer-by-layer fashion from the bottom to the top. Unlike previous works, our coreset is data-independent, meaning that it provably guarantees the accuracy of the function for any input [Formula: see text], including an adversarial one.

Black-Box Acceleration of Monotone Convex Program Solvers

When and where was the study conducted: This work was done in 2018, 2019 and 2020 when Palma London was a PhD student at Caltech and Shai Vardi was a postdoc at Caltech. This work was also done in part while Palma London was visiting Purdue University, and while Reza Eghbali was a postdoctoral fellow the Simons Institute for the Theory of Computing. Adam Wierman is a professor at Caltech. Article Summary and Talking Points: Please describe the primary purpose/findings of your article in 3 sentences or less. This paper presents a framework for accelerating (speeding up) existing convex program solvers. Across engineering disciplines, a fundamental bottleneck is the availability of fast, efficient, accurate solvers. We present an acceleration method that speeds up linear programing solvers such as Gurobi and convex program solvers such as the Splitting Conic Solver by two orders of magnitude. Please include 3-5 short bullet points of “Need to Know” items regarding this research and your findings. - Optimizations problems arise in many engineering and science disciplines, and developing efficient optimization solvers is key to future innovation. - We speed up linear programing solver Gurobi by two orders of magnitude. - This work applies to optimization problems with monotone objective functions and packing constraints, which is a common problem formulation across many disciplines. Please identify 2 pull quotes from your article that best capture the novelty and impact of your research. “We propose a framework for accelerating exact and approximate convex programming solvers for packing linear programming problems and a family of convex programming problems with linear constraints. Analytically, we provide worst-case guarantees on the run time and the quality of the solution produced. Numerically, we demonstrate that our framework speeds up Gurobi and the Splitting Conic Solver by two orders of magnitude, while maintaining a near-optimal solution.” “Our focus in this paper is on a class of packing problems for which data is either very costly or hard to obtain. In these situations, the number of data points available is much smaller than the number of variables. In a machine-learning setting, this regime is increasingly prevalent because it is often advantageous to consider larger and larger feature spaces, while not necessarily obtaining proportionally more data.” Article Implications - Please describe in 5 sentences or less the innovative takeaway(s) of your research. This framework applies to optimization problems with monotone objective functions and packing constraints, which is a common problem formulation across many disciplines, including machine learning, inference, and resource allocation. Providing fast solvers for these problems is crucial. We exploit characteristics of the problem structure and leverage statistical properties of the problem constraints to allow us to speed up optimization solvers. We present worst-case guarantees on run-time, and empirically demonstrate speedups of two orders of magnitude. - Please describe in 5 sentences or less why your findings would be of interest to the general public. Many problems in engineering, science, math, and machine learning involve solving an optimization problem. Fast, efficient optimization solvers are key to future innovation in science and engineering. This work presents a tool to accelerate existing convex solvers, and thus can also be applied to future solvers. As the size of datasets grow it is even more crucial to have fast solvers. - Who would be the most impacted by your research (i.e. by industry, job title, consumer category). Our work impacts machine-learning researchers and optimization researchers, in industry or academia.

Minimizing Cost-Plus-Dissatisfaction in Online EV Charging Under Real-Time Pricing

We consider an increasingly popular demand-response scenario where a home user schedules the flexible electric vehicle (EV) charging load in response to real-time electricity prices. The objective is to minimize the total charging cost with user dissatisfaction taken into account. We focus on the online setting where neither accurate prediction nor distribution of future real-time prices is available to the user when making irrevocable charging decisions in each time slot. The emphasis on considering user dissatisfaction and achieving optimal competitive ratio differentiates our work from existing ones and makes our study uniquely challenging. Our key contribution is two simple online algorithms with the optimal competitive ratio among all deterministic algorithms. The optimal competitive ratio is upper-bounded by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\min \left \{{ \sqrt {\alpha /p_{\min }},p_{\max }/p_{\min }}\right \} $ </tex-math></inline-formula> and the bound is asymptotically tight with respect to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p_{\max }$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p_{\min }$ </tex-math></inline-formula> are the upper and lower bounds of real-time prices and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha \geq p_{\min }$ </tex-math></inline-formula> captures the consideration of user dissatisfaction. The bounds with respect to small and large values of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> suggest the fundamental difference of the problems with and without considering user dissatisfaction. We also extend the algorithms to take minimum charging requirement and short-term prediction into account. Simulation results based on real-world traces corroborate our theoretical findings and show that the empirical performance of our algorithms can be substantially better than the theoretical worst-case guarantees. Our algorithms also achieve notable performance gains under diverse settings as compared to conceivable alternatives.

Solving sparse principal component analysis with global support

Sparse principal component analysis with global support (SPCAgs), is the problem of finding the top-r leading principal components such that all these principal components are linear combinations of a common subset of at most k variables. SPCAgs is a popular dimension reduction tool in statistics that enhances interpretability compared to regular principal component analysis (PCA). Methods for solving SPCAgs in the literature are either greedy heuristics (in the special case of $$r = 1$$ ) with guarantees under restrictive statistical models or algorithms with stationary point convergence for some regularized reformulation of SPCAgs. Crucially, none of the existing computational methods can efficiently guarantee the quality of the solutions obtained by comparing them against dual bounds. In this work, we first propose a convex relaxation based on operator norms that provably approximates the feasible region of SPCAgs within a $$c_1 + c_2 \sqrt{\log r} = O(\sqrt{\log r})$$ factor for some constants $$c_1, c_2$$ . To prove this result, we use a novel random sparsification procedure that uses the Pietsch-Grothendieck factorization theorem and may be of independent interest. We also propose a simpler relaxation that is second-order cone representable and gives a $$(2\sqrt{r})$$ -approximation for the feasible region. Using these relaxations, we then propose a convex integer program that provides a dual bound for the optimal value of SPCAgs. Moreover, it also has worst-case guarantees: it is within a multiplicative/additive factor of the original optimal value, and the multiplicative factor is $$O(\log r)$$ or O(r) depending on the relaxation used. Finally, we conduct computational experiments that show that our convex integer program provides, within a reasonable time, good upper bounds that are typically significantly better than the natural baselines.

Lossless Prioritized Embeddings

Given metric spaces (X, d) and (Y, ρ) and an ordering x1, x2,...,xn of (X, d), an embedding f : X → Y is said to have a prioritized distortion α(·), for a function α(·), if for any pair xj, x' of distinct points in X, the distortion provided by f for this pair is at most α(j). If Y is a normed space, the embedding is said to have prioritized dimension β(·), if f(xj) may have at most β(j) nonzero coordinates. The notion of prioritized embedding was introduced by Filtser and the current authors in [EFN18], where a rather general methodology for constructing such em-beddings was developed. Though this methodology enabled [EFN18] to come up with many prioritized embed-dings, it typically incurs some loss in the distortion. In other words, in the worst-case, prioritized embeddings obtained via this methodology incur distortion which is at least a constant factor off, compared to the distortion of the classical counterparts of these embeddings. This constant loss is problematic for isometric embeddings. It is also troublesome for Matousek's embedding of general metrics into l∞, which for a parameter k = 1, 2,..., provides distortion 2k−1 and dimension O(k log n·n1/k). In this paper we devise two lossless prioritized embeddings. The first one is an isometric prioritized embedding of tree metrics into l∞ with dimension O(log j), matching the worst-case guarantee of O(log n) of the classical embedding of Linial et al. [LLR95]. The second one is a prioritized Matousek's embedding of general metrics into l∞, which for a parameter k = 1, 2,..., provides prioritized distortion [MATH HERE] and dimension O(k log n · n1/k), again matching the worst-case guarantee 2k − 1 in the distortion of the classical Matousek's embedding. We also provide a dimension-prioritized variant of Matousek's embedding. Finally, we devise prioritized embeddings of general metrics into (single) ultra-metric and of general graphs into (single) spanning tree with asymptotically optimal distortion.