Performance Bounds of Algorithms for Scheduling Advertisements on a Web Page

Consider a set of n advertisements (hereafter called ads) A ={A₁,...,A_n} competing to be placed in a planning horizon which is divided into N time intervals called slots. An ad A _i is specified by its size s _i and frequency w _i. The size s _i represents the amount of space the ad occupies in a slot. Ad A _i is said to be scheduled if exactly w _i copies of A _i are placed in the slots subject to the restriction that a slot contains at most one copy of an ad. In this paper, we consider two problems. The MINSPACE problem minimizes the maximum fullness among all slots in a feasible schedule where the fullness of a slot is the sum of the sizes of ads assigned to the slot. For the MAXSPACE problem, in addition, we are given a common maximum fullness S for all slots. The total size of the ads placed in a slot cannot exceed S. The objective is to find a feasible schedule of ads such that the total occupied slot space is maximized. We examine the complexity status of both problems and provide heuristics with performance guarantees.     

New Algorithms for Disk Scheduling

Abstract. Processor speed and memory capacity are increasing several times faster than disk speed. This disparity suggests that disk I/ O performance could become an important bottleneck. Methods are needed for using disks more efficiently. Past analysis of disk scheduling algorithms has largely been experimental and little attempt has been made to develop algorithms with provable performance guarantees. We consider the following disk scheduling problem. Given a set of requests on a computer disk and a convex reachability function that determines how fast the disk head travels between tracks, our goal is to schedule the disk head so that it services all the requests in the shortest time possible. We present a 3/2 -approximation algorithm (with a constant additive term). For the special case in which the reachability function is linear we present an optimal polynomial-time solution. The disk scheduling problem is related to the special case of the Asymmetric Traveling Salesman Problem with the triangle inequality (ATSP-Δ ) in which all distances are either 0 or some constant α . We show how to find the optimal tour in polynomial time and describe how this gives another approximation algorithm for the disk scheduling problem. Finally we consider the on-line version of the problem in which uniformly distributed requests arrive over time. We present an algorithm related to the above ATSP-Δ .     

New Algorithms for Disk Scheduling

Processor speed and memory capacity are increasing several times faster than disk speed. This disparity suggests that disk I/ O performance could become an important bottleneck. Methods are needed for using disks more efficiently. Past analysis of disk scheduling algorithms has largely been experimental and little attempt has been made to develop algorithms with provable performance guarantees. We consider the following disk scheduling problem. Given a set of requests on a computer disk and a convex reachability function that determines how fast the disk head travels between tracks, our goal is to schedule the disk head so that it services all the requests in the shortest time possible. We present a 3/2 -approximation algorithm (with a constant additive term). For the special case in which the reachability function is linear we present an optimal polynomial-time solution. The disk scheduling problem is related to the special case of the Asymmetric Traveling Salesman Problem with the triangle inequality (ATSP-Δ ) in which all distances are either 0 or some constant α . We show how to find the optimal tour in polynomial time and describe how this gives another approximation algorithm for the disk scheduling problem. Finally we consider the on-line version of the problem in which uniformly distributed requests arrive over time. We present an algorithm related to the above ATSP-Δ .     

Robust Algorithms for Preemptive Scheduling

Preemptive scheduling problems on parallel machines are classic problems. Given the goal of minimizing the makespan, they are polynomially solvable even for the most general model of unrelated machines. In these problems, a set of jobs is to be assigned to run on a set of m machines. A job can be split into parts arbitrarily and these parts are to be assigned to time slots on the machines without parallelism, that is, for every job, at most one of its parts can be processed at each time. Motivated by sensitivity analysis and online algorithms, we investigate the problem of designing robust algorithms for constructing preemptive schedules. Robust algorithms receive one piece of input at a time. They may change a small portion of the solution as an additional part of the input is revealed. The capacity of change is based on the size of the new piece of input. For scheduling problems, the supremum ratio between the total size of the jobs (or parts of jobs) which may be re-scheduled upon the arrival of a new job j, and the size of j, is called migration factor. We design a strongly optimal algorithm with the migration factor $1-\frac{1}{m}$ for identical machines. Strongly optimal algorithms avoid idle time and create solutions where the (non-increasingly) sorted vector of completion times of the machines is lexicographically minimal. In the case of identical machines this results not only in makespan minimization, but the created solution is also optimal with respect to any ? _p norm (for p>1). We show that an algorithm of a smaller migration factor cannot be optimal with respect to makespan or any other ? _p norm, thus the result is best possible in this sense as well. We further show that neither uniformly related machines nor identical machines with restricted assignment admit an optimal algorithm with a constant migration factor. This lower bound holds both for makespan minimization and for any ? _p norm. Finally, we analyze the case of two machines and show that in this case it is still possible to maintain an optimal schedule with a small migration factor in the cases of two uniformly related machines and two identical machines with restricted assignment.     

无线调度算法

为满足无线应用中的Qos要求,调度算法是很重要的。由于无线网络变化的链路错误率和容量,因此为其设计调度算法非常有挑战性。近来提出了多种适合无线网络的调度算法。该文分析了3种基于TDD的调度算法。讨论了各种算法的实现过程和优缺点,最后对它们的性能进行比较．得出了有意义的结论。     

Approximation Algorithms for Average Stretch Scheduling

We study the basic problem of preemptive scheduling of a stream of jobs on a single processor. Consider an on-line stream of jobs, and let the ith job arrive at time r(i) and have processing time p(i). If C(i) is the completion time of job i, then the flow time of i is C(i) − r(i) and the stretch of i is the ratio of its flow time to its processing time; that is, . Flow time measures the time that a job is in the system regardless of the service it requests; the stretch measure relies on the intuition that a job that requires a long service time must be prepared to wait longer than jobs that require small service times. We present the improved algorithmic results for the average stretch metric in preemptive uniprocessor scheduling. Our first result is an off-line polynomial-time approximation scheme (PTAS) for average stretch scheduling. This improves upon the 2-approximation achieved by the on-line algorithm srpt that always schedules a job with the shortest remaining processing time. In a recent work, Chekuri and Khanna (Proc. 34th Ann. Symp. Theory Comput., 297–305, 2002) have presented approximation algorithms for weighted flow time, which is a more general metric than average stretch; their result also yields a PTAS for average stretch. Our second set of results considers the impact of incomplete knowledge of job sizes on the performance of on-line scheduling algorithms. We show that a constant-factor competitive ratio for average stretch is achievable even if the processing times (or remaining processing times) of jobs are known only to within a constant factor of accuracy.     

Barely Random Algorithms for Multiprocessor Scheduling

We consider randomized algorithms for on-line scheduling on identical machines. For two machines, a randomized algorithm achieving a competitive ratio of was found by Bartal et al. (1995). Seiden has presented a randomized algorithm which achieves competitive ratios of 1.55665, 1.65888, 1.73376, 1.78295, and 1.81681, for m=3,4,5,6,7, respectively (Seiden, 2000). A barely random algorithm is one which is a distribution over a constant number of deterministic strategies. The algorithms of Bartal et al. and Seiden are not barely random–in fact, these algorithms potentially make a random choice for each job scheduled. We present the first barely random on-line scheduling algorithms. In addition, our algorithms use less space and time than the previous algorithms, asymptotically.     

Approximation Algorithms for Time Constrained Scheduling

In this paper we consider the following time constrained scheduling problem. Given a set of jobsJwith execution timese(j)(0, 1] and an undirected graphG=(J, E), we consider the problem to find a schedule for the jobs such that adjacent jobs (j, j′)Eare assigned to different machines and that the total execution time for each machine is at most 1. The goal is to find a minimum number of machines to execute all jobs under this time constraint. This scheduling problem is a natural generalization of the classical bin-packing problem. We propose and analyse several approximation algorithms with constant absolute worst case ratio for graphs that can be colored in polynomial time.     

Optimal Scheduling Algorithms for Tertiary Storage

The ever growing needs of large multimedia systems cannot be met by magnetic disks due to their high cost and low storage density. Consequently, cheaper and denser tertiary storage systems are being integrated into the storage hierarchies of these applications. Although tertiary storage is cheaper, the access latency is very high due to the need to load and unload media on the drives. This high latency and the bursty nature of I/O traffic result in the accumulation of I/O requests for tertiary storage. We study the problem of scheduling these requests to improve performance. In particular we address the issues of scheduling across multiple tapes or disks as opposed to most other studies which consider only one or two media. We focus on algorithms that minimize the number of switches and show through simulation that these result in near-optimal schedules. For single drive libraries an efficient algorithm that produces optimal schedules is developed. Formultiple drives the problem is shown to be NP-Complete. Efficient and effective heuristics are presented for both single and multiple drives. The scheduling policies developed achieve significant performance gains over naive policies. The algorithms are simple to implement and are not restrictive. The study encompasses all types of storage libraries handling removable media, such as tapes and optical disks.     

Competitive Algorithms for Due Date Scheduling

We consider several online scheduling problems that arise when customers request make-to-order products from a company. At the time of the order, the company must quote a due date to the customer. To satisfy the customer, the company must produce the good by the due date. The company must have an online algorithm with two components: The first component sets the due dates, and the second component schedules the resulting jobs with the goal of meeting the due dates.     

基于启发式规则的混合遗传算法及其在生产计划优化中的应用

以最优或近似最优的作业顺序编制满足关键资源约束的生产计划优化问题一直是企业生产管理中重要的研究课题之一。文章提出了一种基于传统启发式规则的混合遗传算法。该算法将染色体分为两段,前段表示资源安排策略,后段表示为优先分配规则序列,并设计了一种新的交叉算子。最后,介绍了根据此算法编制的一个制造企业生产控制的软件系统。     

Semi-Online Algorithms for Parallel Machine Scheduling Problems

This paper considers two semi-online versions of scheduling problem P2||C_max where one type of partial information is available and one type of additional algorithmic extension is allowed simultaneously. For the semi-online version where a buffer of length 1 is available and the total size of all jobs is known in advance, we present an optimal algorithm with competitive ratio 5/4. We also show that it does not help that the buffer length is greater than 1. For the semi-online version where two parallel processors are available and the total size of all jobs is known in advance, we present an optimal algorithm with competitive ratio 6/5.The second author is supported by TRAPOYT of China, National Natural Science Foundation of China (10271110). Corresponding author: Y. He.     

Approximation Algorithms for Multiprocessor Scheduling under Uncertainty

Motivated by applications in grid computing and project management, we study multiprocessor scheduling in scenarios where there is uncertainty in the successful execution of jobs when assigned to processors. We consider the problem of multiprocessor scheduling under uncertainty, in which we are given n unit-time jobs and m machines, a directed acyclic graph C giving the dependencies among the jobs, and for every job j and machine i, the probability p _ij of the successful completion of job j when scheduled on machine i in any given particular step. The goal of the problem is to find a schedule that minimizes the expected makespan, that is, the expected time at which all of the jobs are completed.     

Multicycle Polling Scheduling Algorithms for FieldBus Networks

The paper deals with the scheduling of periodic information flow in a FieldBus environment. The scheduling problem is defined from an analytical point of view, giving a brief survey of the most well-known solutions. One of these is called multicycle polling scheduling, which is based on the hypothesis that all the production periods of the periodic processes to be scheduled are harmonic. Although in some process control or manufacturing scenarios, this hypothesis may be acceptable, there are many real industrial processes to which it cannot be applied. The aim of the paper is to make a contribution towards solving the scheduling problem. It essentially concerns extension of the theory on which multicycle polling scheduling is based to a much more realistic and general scenario, where the periods of all the processes to be scheduled have arbitrary values. The authors present a new formulation of multicycle polling scheduling, called extended multicycle polling scheduling, and demonstrate that it comprises the scenario currently considered in the literature. Two algorithmic solutions for extended multicycle polling scheduling are then proposed, giving a computational complexity analysis which will highlight the capability of the algorithmic scheduling solutions to be performed on-line. The paper concludes by comparing the multicycle polling scheduling approach known in literature and the one presented in the paper. Comparison is performed by evaluating the use of available bandwidth to serve both periodic and asynchronous traffic in the two approaches.     

Algorithms for Divisible Load Scheduling of Data-intensive Applications

In this paper we introduce the Divisible Load Scheduling (DLS) family of algorithms for data-intensive applications. The polynomial time algorithms partition the input data and generate optimal mappings to collection of autonomous and heterogeneous computational systems. We prove the optimality of the solution and report a simulation study of the algorithms.     

模糊动态抢占调度算法

针对不确定任务特征,提出应用模糊理论进行动态抢占调度,用语言模糊集来描述任务的不确定特征和不同的优先级等级,利用最大隶属度原理确定任务的优先级等级,采用优先调度高优先级等级任务的调度策略提高重要任务的调度成功率,实现具有不确定任务特征的抢占调度,与传统的EDF和LSF算法相比较,仿真表明,所提算法能够提高重要任务的调度成功率,并降低重要任务的截止期错失率;同时,任务间的平均切换次数大大小于LSF的平均切换次数,而与EDF保持相当,该方法可应用于计算机控制系统的控制任务调度,并借鉴于其它具有不确定任务特征或具有有限优先级等级的实时调度问题研究中。     

Tabu Search Algorithms for Cyclic Machine Scheduling Problems

Tabu search algorithms are developed for solving a large class of cyclic machine scheduling problems with the objective to minimize the cycle time. Neighborhoods are derived which generalize the block-approach based neighborhoods which have been successfully applied to noncyclic job-shop problems. For a variant of this neighborhood opt-connectivity is proved.The tabu-search procedure is applied to cyclic job-shop scheduling problems. Computational results are presented.Supported by INTAS, Project 00-217.Supported by Cusanuswerk.     

Fair Scheduling Algorithms in Grids

In this paper, we propose a new algorithm for fair scheduling, and we compare it to other scheduling schemes such as the earliest deadline first (EDF) and the first come first served (FCFS) schemes. Our algorithm uses a max-min fair sharing approach for providing fair access to users. When there is no shortage of resources, the algorithm assigns to each task enough computational power for it to finish within its deadline. When there is congestion, the main idea is to fairly reduce the CPU rates assigned to the tasks so that the share of resources that each user gets is proportional to the users weight. The weight of a user may be defined as the users contribution to the infrastructure or the price he is willing to pay for services or any other socioeconomic consideration. In our algorithms, all tasks whose requirements are lower than their fair share CPU rate are served at their demanded CPU rates. However, the CPU rates of tasks whose requirements are larger than their fair share CPU rate are reduced to fit the total available computational capacity in a fair manner. Three different versions of fair scheduling are adopted in this paper: the simple fair task order (SFTO), which schedules the tasks according to their respective fair completion times, the adjusted fair task order (AFTO), which refines the SFTO policy by ordering the tasks using the adjusted fair completion time, and the max-min fair share (MMFS) scheduling policy, which simultaneously addresses the problem of finding a fair task order and assigning a processor to each task based on a max-min fair sharing policy. Experimental results and comparisons with traditional scheduling schemes such as the EDF and the FCFS are presented using three different error criteria. Validation of the simulations using real experiments of tasks generated from 3D image- rendering processes is also provided. The three proposed scheduling schemes can be integrated into existing grid computing architectures.     

Normalization-Based Task Scheduling Algorithms for Heterogeneous Multi-Cloud Environment

Cloud computing is one of the most successful technologies that offer on-demand services through the Internet. However, datacenters of the clouds may not have unlimited capacity which can fulfill the demanded services in peak hours. Therefore, scheduling workloads across multiple clouds in a federated manner has gained a significant attention in the recent years. In this paper, we present four task scheduling algorithms, called CZSN, CDSN, CDN and CNRSN for heterogeneous multi-cloud environment. The first two algorithms are based on traditional normalization techniques, namely z-score and decimal scaling respectively which are hired from data mining. The next two algorithms are based on two newly proposed normalization techniques, called distribution scaling and nearest radix scaling respectively. All the proposed algorithms are shown to work on-line. We perform rigorous experiments on the proposed algorithms using various synthetic as well as benchmark datasets. Their performances are evaluated through simulation run by measuring two performance metrics, namely makespan and average cloud utilization. The experimental results are compared with that of existing algorithms to show the efficacy of the proposed algorithms.     

并行设计子任务调度的遗传算法原理与实现方法

建立了设计子任务调度的目标模型,提出了一种针对并行设计子任务调度的遗传算法．应用结果表明,在满足子任务间偏序关系条件下,文中算法能够得到设计子任务的最优调度方案．