A Randomized Algorithm for Two Servers in Cross Polytope Spaces

A Randomized Algorithm for Two Servers on the Line

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 develop a new approach for online network design and obtain improved competitive ratios for several problems. Our approach gives natural deterministic algorithms and simple analyses. At the heart of our work is a novel application of embeddings into hierarchically well-separated trees (HSTs) to the analysis of online network design algorithms --- we charge the cost of the algorithm to the cost of the optimal solution on any HST embedding of the terminals. This analysis technique is widely applicable to many problems and gives a unified framework for online network design. In a sense, our work brings together two of the main approaches to online network design. The first uses greedy-like algorithms and analyzes them using dual-fitting. The second uses tree embeddings and results in randomized $O(\log n)$-competitive algorithms, where $n$ is the total number of vertices in the graph. Our approach uses deterministic greedy-like algorithms but analyzes them via HST embeddings of the terminals. Our proofs are simpler as we do not need to carefully construct dual solutions and we get $O(\log k)$ competitive ratios, where $k$ is the number of terminals. In this paper, we apply our approach to obtain deterministic $O(\log k)$-competitive online algorithms for the following problems. - Steiner network with edge duplication. Previously, only a randomized $O(\log n)$-competitive algorithm was known. - Rent-or-buy. Previously, only deterministic $O(\log^2 k)$-competitive and randomized $O(\log k)$-competitive algorithms by Awerbuch, Azar and Bartal (2004) were known. - Connected facility location. Previously, only a randomized $O(\log^2 k)$-competitive algorithm by San Felice, Williamson and Lee (2014) was known. - Prize-collecting Steiner forest. We match the competitive ratio first achieved by Qian and Williamson (2011) and give a simpler analysis.

The peak-demand charge motivates large-load customers to flatten their demand curves, while their self-owned renewable generations aggravate demand fluctuations. Thus, it is attractive to utilize energy storage for shaping real-time loads and reducing electricity bills. In this paper, we propose the first peak-aware competitive online algorithm for leveraging stored energy (e.g., in fuel cells) to minimize peak-demand charges. Our algorithm decides the discharging quantity slot by slot to maintain the optimal worst-case performance guarantee (namely, competitive ratio) among all deterministic online algorithms. Interestingly, we show that the best competitive ratio can be computed by solving a linear number of linear-fractional problems. We can also extend our competitive algorithm and analysis to improve the average-case performance and consider short-term prediction.

We consider the following scheduling problem. The input is a set of jobs with equal processing times, where each job is specified by its release time and deadline. The goal is to determine a single-processor nonpreemptive schedule that maximizes the number of completed jobs. In the online version, each job arrives at its release time. We give two online algorithms with competitive ratios below $2$ and show several lower bounds on the competitive ratios. First, we give a barely random $5/3$-competitive algorithm that uses only one random bit. We also show a lower bound of $3/2$ on the competitive ratio of barely random algorithms that randomly choose one of two deterministic algorithms. If the two algorithms are selected with equal probability, we can further improve the bound to $8/5$. Second, we give a deterministic $3/2$-competitive algorithm in the model that allows restarts, and we show that in this model the ratio $3/2$ is optimal. For randomized algorithms with restarts we show a lower bound of $6/5$.

Cloud providers such as Amazon EC2 and Google GCE typically price their idle compute resources at a significant discount in the form of transient servers. Unlike regular servers, cloud transient servers have no reliability guarantee and can be revoked anytime. In order to enjoy the price premium of transient servers while mitigating the impact of indefinite server revocations, cloud users continuously checkpoint current computational states to stable storage. To determine the optimal checkpointing interval, prior works have largely built upon predicting future revocations. However, we argue that existing prediction models may not work well if providers employ different server revocation algorithms in the future, and the required historical information can be unavailable in practice. In this work, we propose two online algorithms, one competitive and the other heuristic, that determine the checkpointing intervals without predicting future revocation based on history. Our competitive online algorithm achieves the best possible competitive ratio against the optimal, yet impractical, offline solution. Our heuristic algorithm adaptively reacts to server revocations, achieving better average performance than the competitive algorithm, especially for short-running tasks. Extensive trace-driven simulations confirm our analytical results and demonstrate the efficacy of the two online algorithms in the context of Amazon EC2 Spot Instances.

On-Line Load Balancing of Temporary Tasks

The authors deal with a generalization of the k-server problem, in which the servers are unequal. In the weighted server model each of the servers is assigned a positive weight. The cost associated with moving a server equals the product of the distance traversed and the server weight. A weighted k-server algorithm is called competitive if the competitive ratio depends only upon the number of servers. (i.e., the competitive ratio is independent of the weights associated with the servers and the number of points in the metric space). For the uniform metric space, they give super exponential competitive algorithms for any set of weights. If the servers have one of two possible weights, they give deterministic exponential competitive algorithms and randomized polynomial competitive algorithms. They use the MIN operator for both algorithms. One can model the problem of storage management for RAM and E/sup 2/PROM type memories as a weighted server problem with two weights on the uniform metric space.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We study the advice complexity and the random bit complexity of the online knapsack problem: Given a knapsack of unit capacity, and n items that arrive in successive time steps, an online algorithm has to decide for every item whether it gets packed into the knapsack or not. The goal is to maximize the value of the items in the knapsack without exceeding its capacity. In the model of advice complexity of online problems, one asks how many bits of advice about the unknown parts of the input are both necessary and sufficient to achieve a specific competitive ratio. It is well-known that even the unweighted online knapsack problem does not admit any competitive deterministic online algorithm. We show that a single bit of advice helps a deterministic algorithm to become 2-competitive, but that Ω(log n) advice bits are necessary to further improve the deterministic competitive ratio. This is the first time that such a phase transition for the number of advice bits has been observed for any problem. We also show that, surprisingly, instead of an advice bit, a single random bit allows for a competitive ratio of 2, and any further amount of randomness does not improve this. Moreover, we prove that, in a resource augmentation model, i.e., when allowing a little overpacking of the knapsack, a constant number of advice bits suffices to achieve a near-optimal competitive ratio. We also study the weighted version of the problem proving that, with O(log n) bits of advice, we can get arbitrarily close to an optimal solution and, using asymptotically fewer bits, we are not competitive.

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].

This paper studies the online Orthogonal Variable Spreading Factor (OVSF) code assignment problem with resource augmentation introduced by Erlebach et al. (in STACS 2004. LNCS, vol. 2996, pp. 270---281, 2004). We propose a (1+1/?)-competitive algorithm with help of (1+???)lg? h trees for the height h of the OVSF code tree and any ??1. In other words, it is a (1+?)-competitive algorithm with help of (1+?1/??)lg? h trees for any constant 0<?≤1. In the case of ?=1 (or ?=1), we obtain a 2-competitive algorithm with 2lg? h trees, which substantially improves the previous resource of 3h/8+2 trees shown by Chan et al. (COCOON 2009. LNCS, vol. 5609, pp. 358---367, 2009). In another aspect, if it is not necessary to bound the incurred cost for individual requests to a constant, an amortized (4/3+?)-competitive algorithm with (11/4+4/(3?)) trees for any 0<?≤4/3 is also designed in Chan et al. (COCOON 2009. LNCS, vol. 5609, pp. 358---367, 2009). The algorithm in this paper gives us a new trade-off between the competitive ratio and the resource augmentation when ??3 (or ?≤1/3), although the incurred cost for individual requests is bounded to a constant.

This paper investigates a zero-sum game played on a weighted connected graph G between two players, the tree player and the edge player. At each play, the tree player chooses a spanning tree T and the edge player chooses an edge e. The payoff to the edge player is $\textit{cost} (T, e)$, defined as follows: If e lies in the tree T then $\textit{cost}(T, e) = 0$; if e does not lie in the tree then $\textit{cost}(T, e) = cycle(T, e)/w(e)$, where $w(e)$ is the weight of edge e and $\textit{cycle}(T, e)$ is the weight of the unique cycle formed when edge e is added to the tree T. The main result is that the value of the game on any n-vertex graph is bounded above by $\exp(O(\sqrt{\log n \log \log n}))$. It is conjectured that the value of the game is $O(\log n)$. The game arises in connection with the k-server problem on a road network; i.e., a metric space that can be represented as a multigraph G in which each edge e represents a road of length $w(e)$. It is shown that, if the value of the game on G is $\textit{Val}(G, w)$, then there is a randomized strategy that achieves a competitive ratio of $k(1 + \textit{Val}(G, w))$ against any oblivious adversary. Thus, on any n-vertex road network, there is a randomized algorithm for the k-server problem that is $k \cdot \exp(O(\sqrt{\log n \log \log n}))$ competitive against oblivious adversaries. At the heart of the analysis of the game is an algorithm that provides an approximate solution for the simple network design problem. Specifically, for any n-vertex weighted, connected multigraph, the algorithm constructs a spanning tree T such that the average, over all edges e, of $\textit{cost}(T, e)$ is less than or equal to $\exp(O(\sqrt{\log n \log \log n}))$. This result has potential application to the design of communication networks. It also improves substantially known estimates concerning the existence of a sparse basis for the cycle space of a graph.

Online decision-making in the presence of uncertain future information is abundant in many problem domains. In the critical problem of energy generation scheduling for microgrids, one needs to decide when to switch energy supply between a cheaper local generator with startup cost and the costlier on-demand external grid, considering intermittent renewable generation and fluctuating demands. Without knowledge of future input, competitive online algorithms are appealing as they provide optimality guarantees against the optimal offline solution. In practice, however, future input, e.g., wind generation, is often predictable within a limited time window, and can be exploited to further improve the competitiveness of online algorithms. In this paper, we exploit the structure of information in the prediction window to design a novel prediction-aware online algorithm for energy generation scheduling in microgrids. Our algorithm achieves the best competitive ratio to date for this important problem, which is at most $3-2/(1+\mathcal{O}(\frac{1}{w})),$ where $w$ is the prediction window size. We also characterize a non-trivial lower bound of the competitive ratio and show that the competitive ratio of our algorithm is only $9\%$ away from the lower bound, when a few hours of prediction is available. Simulation results based on real-world traces corroborate our theoretical analysis and highlight the advantage of our new prediction-aware design.

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).