Online bin packing with resource augmentation

In competitive analysis, we usually do not put any restrictions on the computational complexity of online algorithms, although efficient algorithms are preferred. Thus if such an algorithm were given the entire input in advance, it could give an optimal solution (in exponential time). Instead of giving the algorithm more knowledge about the input, in this paper we consider the effects of giving an online bin packing algorithm larger bins than the offline algorithm it is compared to. We give new algorithms for this problem that combine items in bins in an unusual way and give bounds on their performance which improve upon the best possible bounded space algorithm. We also give general lower bounds for this problem which are nearly matching for bin sizes b ?

Approximation Algorithms for Energy Minimization in Cloud Service Allocation under Reliability Constraints

We consider allocation problems that arise in the context of service allocation in Clouds. More specifically, we assume on the one part that each computing resource is associated to a capacity constraint, that can be chosen using Dynamic Voltage and Frequency Scaling (DVFS) method, and to a probability of failure. On the other hand, we assume that the service runs as a set of independent instances of identical Virtual Machines. Moreover, there exists a Service Level Agreement (SLA) between the Cloud provider and the client that can be expressed as follows: the client comes with a minimal number of service instances which must be alive at the end of the day, and the Cloud provider offers a list of pairs (price,compensation), this compensation being paid by the Cloud provider if it fails to keep alive the required number of services. On the Cloud provider side, each pair corresponds actually to a guaranteed success probability of fulfilling the constraint on the minimal number of instances. In this context, given a minimal number of instances and a probability of success, the question for the Cloud provider is to find the number of necessary resources, their clock frequency and an allocation of the instances (possibly using replication) onto machines. This solution should satisfy all types of constraints during a given time period while minimizing the energy consumption of used resources. We consider two energy consumption models based on DVFS techniques, where the clock frequency of physical resources can be changed. For each allocation problem and each energy model, we prove deterministic approximation ratios on the consumed energy for algorithms that provide guaranteed probability failures, as well as an efficient heuristic, whose energy ratio is not guaranteed

Tight results for Next Fit and Worst Fit with resource augmentation

AbstractIt is well known that the two simple algorithms for the classic bin packing problem, NF and WF both have an approximation ratio of 2. However, WF seems to be a more reasonable algorithm, since it never opens a new bin if an existing bin can still be used.Using resource augmented analysis, where the output of an approximation algorithm, which can use bins of size b>1, is compared to an optimal packing into bins of size 1, we give a complete analysis of the asymptotic approximation ratio of WF and of NF, and use it to show that WF is strictly better than NF for any 1<b<2, while they have the same asymptotic performance guarantee for all b≥2, and for b=1

Run Generation Revisited: What Goes Up May or May Not Come Down

In this paper, we revisit the classic problem of run generation. Run generation is the first phase of external-memory sorting, where the objective is to scan through the data, reorder elements using a small buffer of size M , and output runs (contiguously sorted chunks of elements) that are as long as possible. We develop algorithms for minimizing the total number of runs (or equivalently, maximizing the average run length) when the runs are allowed to be sorted or reverse sorted. We study the problem in the online setting, both with and without resource augmentation, and in the offline setting. (1) We analyze alternating-up-down replacement selection (runs alternate between sorted and reverse sorted), which was studied by Knuth as far back as 1963. We show that this simple policy is asymptotically optimal. Specifically, we show that alternating-up-down replacement selection is 2-competitive and no deterministic online algorithm can perform better. (2) We give online algorithms having smaller competitive ratios with resource augmentation. Specifically, we exhibit a deterministic algorithm that, when given a buffer of size 4M , is able to match or beat any optimal algorithm having a buffer of size M . Furthermore, we present a randomized online algorithm which is 7/4-competitive when given a buffer twice that of the optimal. (3) We demonstrate that performance can also be improved with a small amount of foresight. We give an algorithm, which is 3/2-competitive, with foreknowledge of the next 3M elements of the input stream. For the extreme case where all future elements are known, we design a PTAS for computing the optimal strategy a run generation algorithm must follow. (4) Finally, we present algorithms tailored for nearly sorted inputs which are guaranteed to have optimal solutions with sufficiently long runs

Competitive Analysis of Task Scheduling Algorithms on a Fault-Prone Machine and the Impact of Resource Augmentation

Abstract Reliable task execution in machines that are prone to unpredictable crashes and restarts is challenging and of high importance. However, not much work exists on the worst case analysis of such systems. In this paper, we analyze the fault-tolerant properties of four popular scheduling algorithms: Longest In System (LIS), Shortest In System (SIS), Largest Processing Time (LPT) and Shortest Processing Time (SPT), under worst case scenarios on a fault-prone machine. We use three metrics for the evaluation and comparison of their competitive performance, namely, completed time, pending time and latency. We also investigate the effect of resource augmentation in their performance, by increasing the speed of the machine. To do so, we compare the behavior of the algorithms for different speed intervals and show that between LIS, SIS and SPT there is no clear winner with respect to all the three considered metrics, while LPT is not better than SPT

Modeling and Practical Evaluation of a Service Location Problem in Large Scale Networks

International audienceWe consider a generalization of a classical optimization problem related to server and replica location problems in networks. More precisely, we suppose that a set of users distributed over a network wish to have access to a particular service proposed by a set of providers. The aim is then to distinguish a set of service providers able to offer a sufficient amount of resources in order to satisfy the requests of the clients. Moreover, a quality of service following some requirements in terms of latencies is desirable. A smart repartition of the servers in the network may also ensure good fault tolerance properties. We model this problem as a variant of Bin Packing, namely Bin Packing under Distance Constraint (BPDC) where the goal is to build a minimal number of bins (i.e. to choose a minimal number of servers) so that (i) each client is associated to exactly one server, (ii) the capacity of the server is large enough to satisfy the requests of its clients and (iii) the distance between two clients associated to the same server is minimized. We prove that this problem is hard to approximate even when using resource augmentation techniques : we compare the number of obtained bins when using polynomial time algorithms allowed to build bins of diameter at most b*dmax, for b>1, to the optimal number of bins of diameter at most dmax. On the one hand, we prove that (i) if b=(2-e), BPDC is hard to approximate within any constant approximation ratio, for any e>0; and that (ii) BPDC is hard to approximate at a ratio lower than 3/2 even if resource augmentation is used. On the other hand, if b=2, we propose a polynomial time approximation algorithm for BPDC with approximation ratio 7/3 in the general case. We show how to turn an approximation algorithm for BPDC into an approximation algorithm for the non-uniform capacitated K-center problem and vice-versa. Then, we present a comparison of the quality of results for BPDC in the context of several Internet latency embedding tools such as Sequoia and Vivaldi, using datasets based on PlanetLab latency measurements.Nous considérons une généralisation d'un problème d'optimisation classique lié au placement de serveurs et de réplicats dans les réseaux. Plus précisément, nous supposons qu'un ensemble d'utilisateurs au sein d'un réseau souhaite accéder à un service particulier proposé par un ensemble de fournisseurs de ce service. L'objectif est alors d'identifier un ensemble de fournisseurs de service capable d'offrir suffisamment de ressources pour répondre aux requêtes des clients. Par ailleurs, une certaine qualité de service relativement aux temps de communications est désirable. Une répartition judicieuse des serveurs dans le réseau offrirait également de bonnes propriétés de tolérance aux pannes. Nous modélisons ce problème comme une variante de Bin Packing, le Bin Packing avec Contrainte de Distance (BPDC en anglais) où le but est de construire un minimum de groupes (i.e. de choisir un nombre minimal de serveurs) de telle sorte que (i) chaque client est associé à exactement un serveur, (ii) la capacité dudit serveur est suffisante pour répondre aux requêtes des clients qui lui sont associés et (iii) la distance entre deux clients associés au même serveur est minimisée. Nous prouvons que ce problème est inapproximable même en utilisant des techniques d'augmentation de ressources : le nombre de groupes obtenus en utilisant des algorithmes s'exécutant en temps polynomial et autorisés à construire des groupes de diamètre au plus b*dmax, avec b>1, est comparé au nombre de groupes d'une solution optimale construisant des groupes de diamètre au plus dmax. D'un côté, nous prouvons que (i) si b=(2-e), BPDC est inapproximable à facteur constant, pour tout e>0; et que (ii) BPDC est inapproximableà un facteur inférieur à 3/2 même en utilisant de l'augmentation de ressources. D'un autre côté, si b=2, nous proposons un algorithme s'exécutant en temps polynomial pour BPDC assurant un facteur d'approximation de 7/3 dans le cas général. Nous montrons également comment transformer un algorithme d'approximation pour BPDC en un algorithme d'approximation pour le K-centre non uniforme avec capacités, et vice-versa. Enfin, nous présentons une comparaison qualitative de nos résultats pour BPDC en utilisant plusieurs outils de plongement de l'espace des latences d'Internet, comme Sequoia et Vivaldi

Reliable Service Allocation in Clouds

International audienceWe consider several reliability problems that arise when allocating applications to processing resources in a Cloud computing platform. More specifically, we assume on the one hand that each computing resource is associated to a capacity constraint and to a probability of failure. On the other hand, we assume that each service runs as a set of independent instances of identical Virtual Machines, and that the Service Level Agreement between the Cloud provider and the client states that a minimal number of instances of the service should run with a given probability. In this context, given the capacity and failure probabilities of the machines, and the capacity and reliability demands of the services, the question for the cloud provider is to find an allocation of the instances of the services (possibly using replication) onto machines satisfying all types of constraints during a given time period. In this paper, our goal is to assess the impact of the reliability constraint on the complexity of resource allocation problems. We consider several variants of this problem, depending on the number of services and whether their reliability demand is individual or global. We prove several fundamental complexity results ($\#$P' and NP-completeness results) and we provide several optimal and approximation algorithms. In particular, we prove that a basic randomized allocation algorithm, that is easy to implement, provides optimal or quasi-optimal results in several contexts, and we show through simulations that it also achieves very good results in more general settings

Online dynamic bin packing

In this thesis we study online algorithms for dynamic bin packing. An online algorithm is presented with input throughout time and must make irrevocable decisions without knowledge of future input. The classical bin packing problem is a combinatorial optimization problem in which a set of items must be packed into a minimum number of uniform-sized bins without exceeding their capacities. The problem has been studied since the early 1970s and many variants continue to attract researchers’ attention today. The dynamic version of the bin packing problem was introduced by Coffman, Garey and Johnson in 1983. The problem is a generalization of the bin packing problem in which items may arrive and depart dynamically. In this setting, an online algorithm for bin packing is presented with one item at a time, without knowledge of its departure time, nor arrival and departure times of future items, and must decide in which bin the item should be packed. Migration of items between bins is not allowed, however rearrangement of items within a bin is permitted. The objective of problem is to minimize the maximum number of bins used over all time. In multi-dimensional generalizations of the problem, multi-dimensional items must be packed without overlap in multi-dimensional bins of uniform size in each dimension. In this work, we study the setting where items are oriented and cannot be rotated. We first consider online one-dimensional dynamic bin packing and present a lower bound of 8/3 ∼ 2.666 on the achievable competitive ratio of any deterministic online algorithm, improving the best known 2.5-lower bound. Since the introduction of the problem by Coffman, Garey and Johnson, the progress on the problem has focused on improving the original lower bound of 2.388 to 2.428, and to the best known 2.5-lower bound. Our improvement from 2.5 to 8/3 ∼ 2.666 makes a big step forward in closing the gap between the lower bound and the upper bound, which currently stands at 2.788. Secondly we study the online two- and three-dimensional dynamic bin packing problem by designing and analyzing algorithms for special types of input. Bar-Noy et al. initiated the study of the one-dimensional unit fraction bin packing problem, a restricted version where all sizes of items are of the form 1/k, for some integer k > 0. Another related problem is for power fraction items, where sizes are of the form 1/2k, for some integer k ≥ 0. We initiate the study of online multi-dimensional dynamic bin packing of unit fraction items and power fraction items, where items have lengths unit fraction and power fraction in each dimension, respectively. While algorithms for general input are suitable for unit fraction and power fraction items, their worst-case performance guarantees are the same for special types of input. For unit fraction and power fraction items, we design and analyze online algorithms that achieve better worst-case performance guarantees compared to their classical counterparts. Our algorithms give careful consideration to unit and power fraction items, which allows us to reduce the competitive ratios for these types of inputs. Lastly we focus on obtaining lower bounds on the performance of the family of Any- Fit algorithms (Any-Fit, Best-Fit, First-Fit, Worst-Fit) for online multi-dimensional dynamic bin packing. Any-Fit algorithms are classical online algorithms initially studied for the one-dimensional version of the bin packing problem. The common rule that the algorithms use is to never pack a new item to a new bin if the item can be packed in any of the existing bins. While the family of Any-Fit algorithms is always O(1)-competitive for one-dimensional dynamic bin packing, we show that this is no longer the case for multi-dimensional dynamic bin packing when using Best-Fit and Worst-Fit, even if the input consists of power fraction items or unit fraction items. For these restricted inputs, we prove that Best-Fit and Worst-Fit have unbounded competitive ratios, while for First-Fit we provide lower bounds that are higher than the lower bounds for any online algorithm. Furthermore, for general input we show that all classical Any-Fit algorithms are not competitive for online multi-dimensional dynamic bin packing

Online scheduling in fault-prone systems: performance optimization and energy efficiency

Mención Internacional en el título de doctorEveryone is familiar with the problem of online scheduling (even if they are not aware of it), from the way we prioritize our everyday decisions to the way a delivery service must decide on the route to follow in order to cover the ongoing requests. In computer science, this is a problem of even greater importance. This thesis considers two main families of online scheduling problems in computer science, and aims to provide an extended clear framework for their analysis, presenting at the same time some common characteristics that connect these problems. The first and main family of online scheduling problems considered, is task scheduling in fault-prone computing systems. As the number of clients and the possibilities offered by the rapid development of computing systems, grow with time, the increase of demands of computationally intensive tasks is inevitable. Uniprocessors are no longer capable of coping with the escalation of these demands, which among others, has led to the development of multicore-based parallel machines, Internet-based computing platforms and co-operational distributed systems. Nonetheless, the challenges of these systems, even of the simplest ones, are numerous: They have to deal with continuous dynamic requests from the clients, which are probably not of the same nature (require different amount of computational resources). The processing elements (i.e., machines) may suffer from unpredictable failures, either malicious or due to overload. Furthermore, depending on the size of these systems and the exact processing units, their power consumption may be of significant amount; even equal to the electricity needed for a small town. Hence, limiting their power consumption is another challenge. To analyze such a system one must consider the online nature of the problem; the dynamic task arrivals (client requests) of different sizes (computational demands), and the unpredictable machine crashes and restarts (failures). It is important to give guarantees for the performance of the algorithms used in these systems, thus the thesis conducts worst-case competitive analysis and covers a significant level of the three dimensions of the problem. More precisely, it studies the effects of the number of machines, the number of different task sizes and the speed of the machines – which as will be explained through the thesis, affects the power consumption of the system – on the efficiency of online scheduling algorithms. As performance measures, this thesis uses the completed load, the pending load and the latency competitiveness of the algorithms. In some cases, it considers the long-term competitiveness versions of these measures as well. One of the most important results shown, is that resource augmentation in the form of increasing the machine speedup, is necessary in order to achieve some competitiveness, or to reach optimal competitiveness. The sufficient amount of speedup is found, and online algorithms that achieve the desired competitiveness are proposed and analyzed. Apart from the algorithms designed, some of the most widely used algorithms in scheduling are also analyzed in the model considered for the first time; namely, Longest In System (LIS), Shortest In System (SIS), Largest Processing Time (LPT), and Smallest Processing Time (SPT). Nonetheless, deciding on the best algorithm between them, is not easy. Each algorithm behaves better with respect to a different evaluation metric and under different model parameters. The second family of problems considered, is packet scheduling over an unreliable wireless communication link. As claimed, these problems have a strong connection to the task scheduling problem, especially when considering one machine and no speedup, hence some of the results can be shared. A setting with a single pair of nodes is considered, connected through an unreliable wireless channel. The sending station transmits packets to a receiving station over the channel, which can be jammed and hence corrupt the packet being transmitted. First, worst-case scenarios are assumed for the channel jams, modeled by a malicious adversarial entity. The packet arrivals however, follow a stochastic distribution and competitive analysis of scheduling algorithms is pursued giving matching bounds for the most pessimistic scenarios of channel jams. The aim of the algorithms is to find the schedule (or order or transmission of the arriving packets) in order to maximize the asymptotic throughout, which corresponds to the long-term competitive ratio of total length of successfully transmitted packets. Then, a slightly different problem is considered, assuming infinite amount of data to be transmitted over the same unreliable communication link. This time however, an adversarial entity with constrained power is assumed for the channel jams. The constrained power is modeled by an Adversarial Queueing Theory (AQT) approach, defined with two main parameters; "the error availability rate", and, the maximum batch of errors available to the adversary at any time. This is the first time AQT is used to model channel jams; it has been mostly used to model the packet arrivals in networking problems. In this problem, the scheduling algorithms must decide on the length of the packets to be transmitted, with the objective of maximizing the goodput rate; the rate of successfully transmitted load. It is seen, that even for the simplest settings, the analysis and results are not trivial.This work has been supported by IMDEA Networks InstitutePrograma Oficial de Doctorado en Ingeniería TelemáticaPresidente: María Serna Iglesias.- Secretario: Vincenzo Mancuso.- Vocal: Leszek Antoni Gasieni