On the competitive ratio of the work function algorithm for the k-server problem

We consider a model for online computation in which the online algorithm receives, together with each request, some information regarding the future, referred to as advice . The advice provided to the online algorithm may allow an improvement in its performance, compared to the classical model of complete lack of information regarding the future. We are interested in the impact of such advice on the competitive ratio, and in particular, in the relation between the size b of the advice, measured in terms of bits of information per request, and the (improved) competitive ratio. Since b = 0 corresponds to the classical online model, and $ b = \lceil \log |\mathcal{A}| \rceil $, where $\mathcal{A}$ is the algorithm's action space, corresponds to the optimal (offline) one, our model spans a spectrum of settings ranging from classical online algorithms to offline ones. In this paper we propose the above model and illustrate its applicability by considering two of the most extensively studied online problems, namely, metrical task systems (MTS) and the k-server problem. For MTS we establish tight (up to constant factors) upper and lower bounds on the competitive ratio of deterministic and randomized online algorithms with advice for any choice of 1 ≤ b ≤ *** (logn ) , where n is the number of states in the system: we prove that any randomized online algorithm for MTS has competitive ratio *** ( log(n ) / b ) and we present a deterministic online algorithm for MTS with competitive ratio O (log(n ) / b ) . For the k -server problem we construct a deterministic online algorithm for general metric spaces with competitive ratio k O (1 / b ) for any choice of *** (1) ≤ b ≤ logk .

We study the k -server problem in the resource augmentation setting, i.e., when the performance of the online algorithm with k servers is compared to the offline optimal solution with h ≤ k servers. The problem is very poorly understood beyond uniform metrics. For this special case, the classic k -server algorithms are roughly (1+1/ϵ)-competitive when k =(1+ϵ) h , for any ϵ > 0. Surprisingly, however, no o ( h )-competitive algorithm is known even for HSTs of depth 2 and even when k / h is arbitrarily large. We obtain several new results for the problem. First, we show that the known k -server algorithms do not work even on very simple metrics. In particular, the Double Coverage algorithm has competitive ratio Ω ( h ) irrespective of the value of k , even for depth-2 HSTs. Similarly, the Work Function Algorithm, which is believed to be optimal for all metric spaces when k = h , has competitive ratio Ω ( h ) on depth-3 HSTs even if k =2 h . Our main result is a new algorithm that is O (1)-competitive for constant depth trees, whenever k =(1+ϵ) h for any ϵ > 0. Finally, we give a general lower bound that any deterministic online algorithm has competitive ratio at least 2.4 even for depth-2 HSTs and when k / h is arbitrarily large. This gives a surprising qualitative separation between uniform metrics and depth-2 HSTs for the ( h , k )-server problem.

We provide simple but surprisingly useful direct product theorems for proving lower bounds on online algorithms with a limited amount of advice about the future. Intuitively, our direct product theorems say that if b bits of advice are needed to ensure a cost of at most t for some problem, then r*b bits of advice are needed to ensure a total cost of at most r*t when solving r independent instances of the problem. Using our direct product theorems, we are able to translate decades of research on randomized online algorithms to the advice complexity model. Doing so improves significantly on the previous best advice complexity lower bounds for many online problems, or provides the first known lower bounds. For example, we show that - A paging algorithm needs Omega(n) bits of advice to achieve a competitive ratio better than H_k = Omega(log k), where k is the cache size. Previously, it was only known that Omega(n) bits of advice were necessary to achieve a constant competitive ratio smaller than 5/4. - Every O(n^{1-epsilon})-competitive vertex coloring algorithm must use Omega(n log n) bits of advice. Previously, it was only known that Omega(n log n) bits of advice were necessary to be optimal. For certain online problems, including the MTS, k-server, metric matching, paging, list update, and dynamic binary search tree problem, we prove that randomization and sublinear advice are equally powerful (if the underlying metric space or node set is finite). This means that several long-standing open questions regarding randomized online algorithms can be equivalently stated as questions regarding online algorithms with sublinear advice. For example, we show that there exists a deterministic O(log k)-competitive k-server algorithm with sublinear advice if and only if there exists a randomized O(log k)-competitive k-server algorithm without advice. Technically, our main direct product theorem is obtained by extending an information theoretical lower bound technique due to Emek, Fraigniaud, Korman, and Rosen [ICALP'09].

In this paper, we study a stochastic variant of the celebrated k-server problem. In the k-server problem, we are required to minimize the total movement of k servers that are serving an online sequence of t requests in a metric. In the stochastic setting we are given t independent distributions <P_1, P_2,..., P_t> in advance, and at every time step i a request is drawn from Pi. Designing the optimal online algorithm in such setting is NP-hard, therefore the emphasis of our work is on designing an approximately optimal online algorithm. We first show a structural characterization for a certain class of non-adaptive online algorithms. We prove that in general metrics, the best of such algorithms has a cost of no worse than three times that of the optimal online algorithm. Next, we present an integer program that finds the optimal algorithm of this class for any arbitrary metric. Finally, by rounding the solution of the linear relaxation of this program, we present an online algorithm for the stochastic k-server problem with the approximation factor of 3 in the line and circle metrics and O(log n) in a general metric of size n. Moreover, we define the Uber problem, in which each demand consists of two endpoints, a source and a destination. We show that given an a-approximation algorithm for the k-server problem, we can obtain an (a+2)-approximation algorithm for the Uber problem. Motivated by the fact that demands are usually highly correlated with the time we study the stochastic Uber problem. Furthermore, we extend our results to the correlated setting where the probability of a request arriving at a certain point depends not only on the time step but also on the previously arrived requests.

The study of competitive algorithms has concentrated on competitiveness — comparing online algorithms to optimal off-line algorithms on sequences of operations. Published algorithms proven (or suggested) to be competitive invariably have pessimal response time i.e. their worst-case single operation time is as bad as possible. We consider whether or not such algorithms can be adapted to improve the response time without sacrificing competitiveness. We consider lists, off-line static search trees, dynamic search trees, and the k-server problem on a line segment of length L. For lists, pessimal response time is unavoidable. For off-line static search trees our algorithm is 2-competitive and has response time 2 log n. For dynamic search trees our algorithm has logarithmic amortized time and is statically optimal (like splay trees), but the response time is O(√n log n) and Ω(log2 n). For the k-server problem we prove that any algorithm with O(optimal) response time has a competitive ratio of at least Ω(L/k). This is achieved by a simple on-line algorithm. We also show that even a weak limit on response time for the k-server problem (e.g. response time less than half pessimal) yields an Ω(L/k) separation between on-line and off-line algorithms. Our results apply to high-performance multi-head disk drives where response time is critical.

In this article, we study the online Euclidean spanners problem for points in \(\mathbb{R}^{d}\) . Given a set \(S\) of \(n\) points in \(\mathbb{R}^{d}\) , a \(t\) -spanner on \(S\) is a subgraph of the underlying complete graph \(G=(S,\binom{S}{2})\) , that preserves the pairwise Euclidean distances between points in \(S\) to within a factor of \(t\) , that is the stretch factor . Suppose we are given a sequence of \(n\) points \((s_{1},s_{2},\ldots,s_{n})\) in \(\mathbb{R}^{d}\) , where point \(s_{i}\) is presented in step \(i\) for \(i=1,\ldots,n\) . The objective of an online algorithm is to maintain a geometric \(t\) -spanner on \(S_{i}=\{s_{1},\ldots,s_{i}\}\) for each step \(i\) . The algorithm is allowed to add new edges to the spanner when a new point is presented but cannot remove any edge from the spanner. The performance of an online algorithm is measured by its competitive ratio, which is the supremum, over all sequences of points, of the ratio between the weight of the spanner constructed by the algorithm and the weight of an optimum spanner. Here, the weight of a spanner is the sum of all edge weights. First, we establish a lower bound of \(\Omega(\varepsilon^{-1}\log n/\log\varepsilon^{-1})\) for the competitive ratio of any online \((1+\varepsilon)\) -spanner algorithm, for a sequence of \(n\) points in 1-dimension. We show that this bound is tight, and there is an online algorithm that can maintain a \((1+\varepsilon)\) -spanner with competitive ratio \(O(\varepsilon^{-1}\log n/\log\varepsilon^{-1})\) . Next, we design online algorithms for sequences of points in \(\mathbb{R}^{d}\) , for any constant \(d\geq 2\) , under the \(L_{2}\) norm. We show that previously known incremental algorithms achieve a competitive ratio \(O(\varepsilon^{-(d+1)}\log n)\) . However, if the algorithm is allowed to use additional points (Steiner points), then it is possible to substantially improve the competitive ratio in terms of \(\varepsilon\) . We describe an online Steiner \((1+\varepsilon)\) -spanner algorithm with competitive ratio \(O(\varepsilon^{(1-d)/2}\log n)\) . As a counterpart, we show that the dependence on \(n\) cannot be eliminated in dimensions \(d\geq 2\) . In particular, we prove that any online spanner algorithm for a sequence of \(n\) points in \(\mathbb{R}^{d}\) under the \(L_{2}\) norm has competitive ratio \(\Omega(f(n))\) , where \(\lim_{n\rightarrow\infty}f(n)=\infty\) . Finally, we provide improved lower bounds under the \(L_{1}\) norm: \(\Omega(\varepsilon^{-2}/\log\varepsilon^{-1})\) in the plane and \(\Omega(\varepsilon^{-d})\) in \(\mathbb{R}^{d}\) for \(d\geq 3\) .

In this paper, the authors consider an on-line scheduling problem of m (m ≥ 3) identical machines with common maintenance time interval and nonresumable availability. For the case that the length of maintenance time interval is larger than the largest processing time of jobs, the authors prove that any on-line algorithm has not a constant competitive ratio. For the case that the length of maintenance time interval is less than or equal to the largest processing time of jobs, the authors prove a lower bound of 3 on the competitive ratio. The authors give an on-line algorithm with competitive ratio \(4 - \tfrac{1} {m} \). In particular, for the case of m = 3, the authors prove the competitive ratio of the on-line algorithm is \(\tfrac{{10}} {3} \).

We extend the Mobile Server problem introduced in Feldkord and Meyer auf der Heide (TOPC 6(3), 14:1–14:17 2019) to a model where k identical mobile resources, here named servers, answer requests appearing at points in the Euclidean space. To reduce communication costs, the positions of the servers can be adapted by a limited distance ms per round for each server. The costs are measured similarly to the classical Page Migration problem: i.e., answering a request induces costs proportional to the distance to the nearest server, and moving a server induces costs proportional to the distance multiplied with a weight D. We show that, in our model, no online algorithm can have a constant competitive ratio: i.e., one which is independent of the input length n, even if an augmented moving distance of (1 + δ)ms is allowed for the online algorithm. Therefore we investigate a restriction of the power of the adversary dictating the sequence of requests: We demand locality of requests: i.e., that consecutive requests come from points in the Euclidean space with distance bounded by some constant mc. We show constant lower bounds on the competitiveness in this setting (independent of n, but dependent on k, ms and mc). On the positive side, we present a deterministic online algorithm with bounded competitiveness when an augmented moving distance and locality of requests is assumed. Our algorithm simulates any given algorithm for the classical k-Page Migration problem as guidance for its servers and extends it by a greedy move of one server in every round. The resulting competitive ratio is polynomial in the number of servers k, the ratio between mc and ms, the inverse of the augmentation factor 1/δ and the competitive ratio of the simulated k-Page Migration algorithm. We also show how to directly adapt the Double Coverage algorithm (Chrobak et al. SIAM J. Discrete Math. 4(2), 172–181 11) for the k-Server problem to receive an algorithm with improved competitiveness on the line.

Online algorithms are a hallmark of worst case optimization under uncertainty. On the other hand, in practice, the input is often far from worst case, and has some predictable characteristics. A recent line of work has shown how to use machine learned predictions to circumvent strong lower bounds on competitive ratios in classic online problems such as ski rental and caching. We study how predictive techniques can be used to break through worst case barriers in online scheduling. The makespan minimization problem with restricted assignments is a classic problem in online scheduling theory. Worst case analysis of this problem gives Ω(log m) lower bounds on the competitive ratio in the online setting. We identify a robust quantity that can be predicted and then used to guide online algorithms to achieve better performance. Our predictions are compact in size, having dimension linear in the number of machines, and can be learned using standard off the shelf methods. The performance guarantees of our algorithms depend on the accuracy of the predictions, given predictions with error η, we show how to construct O(log η) competitive fractional assignments. We then give an online algorithm that rounds any fractional assignment into an integral schedule. Our algorithm is O((log log m)3)-competitive and we give a nearly matching Ω(log log m) lower bound for online rounding algorithms.1 Altogether, we give algorithms that, equipped with predictions with error η, achieve O(log η (log log m)3) competitive ratios, breaking the Ω(log m) lower bound even for moderately accurate predictions.

Parallel disks provide a cost effective way of speeding up I/Os in applications that work with large amounts of data. The main challenge is to achieve as much parallelism as possible, using prefetching to avoid bottlenecks in disk access. Efficient algorithms have been developed for some particular patterns of accessing the disk blocks. In this paper, we consider general request sequences. When the request sequence consists of unique block requests, the problem is called prefetching and is a well-solved problem for arbitrary request sequences. When the reference sequence can have repeated references to the same block, we need to devise an effective caching policy as well. While optimum offline algorithms have been recently designed for the problem, in the online case, no effective algorithm was previously known. Our main contribution is a deterministic online algorithm threshold-LRU which achieves O((MD/L)2/3) competitive ratio and a randomized online algorithm threshold-MARK which achieves O(√(MD/L) log (MD/L)) competitive ratio for the caching/prefetching problem on the parallel disk model (PDM), where D is the number of disks, M is the size of fast memory buffer, and M+L is the amount of lookahead available in the request sequence. The best-known lower bound on the competitive ratio is Ω(≾MD/L) for lookahead L ≥ M in both models. We also show that if the deterministic online algorithm is allowed to have twice the memory of the offline then a tight competitive ratio of Θ(≾MD/L) can be achieved. This problem generalizes the well-known paging problem on a single disk to the parallel disk model.

In this paper, we augment online algorithms for the knapsack problem using the total weight information. The conventional optimal online algorithm achieves the ln(U/L)+1 competitive ratio where L and U are the upper and lower bounds of the value-to-weight ratio. However, it does not consider that decision makers can know the total weight information or obtain it through machine-learned predictions. To fill this gap, we first propose the Known Weight Algorithm (KWA) which uses the exact total weight information to achieve a competitive ratio of W((U-L)/(eL))+1, where W denotes the Lambert-W function. We prove that it is optimal and tight. After that, we extend KWA to the Predicted Weight Algorithm (PWA), a learning-augmented online algorithm that uses predicted total weight. We show the consistency and robustness of PWA, and prove that its competitive ratio degrades gracefully as the prediction error grows. Finally, we introduce the Limited Volume Algorithm (LWA), which achieves a better competitive ratio than ln(U/L)+1 when the total weight is less than twice the capacity.

This paper studies the online machine minimization problem, where the jobs have real release times, uniform processing times and a common deadline. We investigate how the lookahead ability improves the performance of online algorithms. Two lookahead models are studied, that is, the additive lookahead and the multiplicative lookahead. At any time t, the online algorithm knows all the jobs to be released before time $$t+L$$ (or $$\beta \cdot t$$ ) in the additive (or multiplicative) lookahead model. We propose a $$\frac{e}{\alpha (e-1)+1}$$ -competitive online algorithm with the additive lookahead, where $$\alpha = \frac{L}{T} \le 1$$ and T is the common deadline of the jobs. For the multiplicative lookahead, we provide an online algorithm with a competitive ratio of $$\frac{\beta e}{(\beta -1) e +1}$$ , where $$\beta \ge 1$$ . Lower bounds are also provided for both of the two models, which show that our algorithms are optimal for two extreme cases, that is, $$\alpha = 0$$ (or $$\beta = 1$$ ) and $$\alpha = 1$$ (or $$\beta \rightarrow \infty $$ ), and remain a small gap for the cases in between. Particularly, for $$\alpha = 0$$ (or $$\beta = 1$$ ), the competitive ratio is e, which corresponds to the problem without lookahead. For $$\alpha = 1$$ (or $$\beta \rightarrow \infty $$ ), the competitive ratio is 1, which corresponds to the offline version (with full information).

Online algorithms are an important branch in algorithm design. Designing online algorithms with a bounded competitive ratio (in terms of worst-case performance) can be hard and usually relies on problem-specific assumptions. Inspired by adversarial training from Generative Adversarial Net and the fact that the competitive ratio of an online algorithm is based on worst-case input, we adopt deep neural networks (NNs) to learn an online algorithm for a resource allocation and pricing problem from scratch, with the goal that the performance gap between offline optimum and the learned online algorithm can be minimized for worst-case input. Specifically, we leverage two NNs as the algorithm and the adversary, respectively, and let them play a zero sum game, with the adversary being responsible for generating worst-case input while the algorithm learns the best strategy based on the input provided by the adversary. To ensure better convergence of the algorithm network (to the desired online algorithm), we propose a novel per-round update method to handle sequential decision making to break complex dependency among different rounds so that update can be done for every possible action instead of only sampled actions. To the best of our knowledge, our work is the first using deep NNs to design an online algorithm from the perspective of worst-case performance guarantee. Empirical studies show that our updating methods ensure convergence to Nash equilibrium and the learned algorithm outperforms state-of-the-art online algorithms under various settings.

Problem definition: We study a fundamental online resource allocation problem in service operations in which a heterogeneous stream of arrivals that varies in service times and rewards makes service requests from a finite number of servers/providers. This is an online adversarial setting in which nothing more is known about the arrival process of customers. Each server has a finite regular capacity but can be expanded at the expense of overtime cost. Upon arrival of each customer, the system chooses both a server and a time for service over a scheduling horizon subject to capacity constraints. The system seeks easy-to-implement online policies that admit a competitive ratio (CR), guaranteeing the worst-case relative performance. Academic/practical relevance: On the academic side, we propose online algorithms with theoretical CRs for the problem described above. On the practical side, we investigate the real-world applicability of our methods and models on appointment-scheduling data from a partner health system. Methodology: We develop new online primal-dual approaches for making not only a server-date allocation decision for each arriving customer, but also an overtime decision for each server on each day within a horizon. We also derive a competitive analysis to prove a theoretical performance guarantee. Results: Our online policies are (i) robust to future information, (ii) easy-to-implement and extremely efficient to compute, and (iii) admitting a theoretical CR. Comparing our online policy with the optimal offline policy, we obtain a CR that guarantees the worst-case performance of our online policy. Managerial implications: We evaluate the performance of our online algorithms by using real appointment scheduling data from a partner health system. Our results show that the proposed online policies perform much better than their theoretical CR, and outperform the pervasive First-Come-First-Served (FCFS) and nested threshold policies (NTPO) by a large margin.

In this paper, we present tight bounds for the k-server problem with delays in the uniform metric space. The problem is defined on n+k nodes in the uniform metric space which can issue requests over time. These requests can be served directly or with some delay using k servers, by moving a server to the corresponding node with an open request. The task is to find an online algorithm that can serve the requests while minimizing the total moving and delay costs. We first provide a lower bound by showing that the competitive ratio of any deterministic online algorithm cannot be better than (2k+1) in the clairvoyant setting. We will then show that conservative algorithms (without delay) can be equipped with an accumulative delay function such that all such algorithms become (2k+1)-competitive in the non-clairvoyant setting. Together, the two bounds establish a tight result for both, the clairvoyant and the non-clairvoyant settings.