I/O scheduling for digital continuous media

A growing set of applications require access to digital video and audio. In order to provide playback of such continuous media (CM), scheduling strategies for CM data servers (CMS) are necessary. In some domains, particularly defense and industrial process control, the timing requirements of these applications are strict and essential to their correct operation. In this paper we develop a scheduling strategy for multiple access to a CMS such that the timing guarantees are maintained at all times. First, we develop a scheduling strategy for the steady state, i.e., when there are no changes in playback rate or operation. We derive an optimal Batched SCAN (BSCAN) algorithm that requires minimum buffer space to schedule concurrent accesses. The scheduling strategy incorporates two key constraints: (1) data fetches from the storage system are assumed to be in integral multiples of the block size, and (2) playback guarantees are ensured for frame-oriented streams when each frame can span multiple blocks. We discuss modifications to the scheduling strategy to handle compressed data like motion-JPEG and MPEG. Second, we develop techniques to handle dynamic changes brought about by VCR-like operations executed by applications. We define a suite of primitive VCR-like operations that can be executed. We show that an unregulated change in the BSCAN schedule, in response to VCR-like operations, will affect playback guarantees. We develop two general techniques to ensure playback guarantees while responding to VCR-like operations: passive and active accumulation. Using user response time as a metric we show that active accumulation algorithms outperform passive accumulation algorithms. An optimal response-time algorithm in a class of active accumulation strategies is derived. The results presented here are validated by extensive simulation studies.     

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 .     

Optimizing multiple dimensional queries simultaneously in multidimensional databases

Some significant progress related to multidimensional data analysis has been achieved in the past few years, including the design of fast algorithms for computing datacubes, selecting some precomputed group-bys to materialize, and designing efficient storage structures for multidimensional data. However, little work has been carried out on multidimensional query optimization issues. Particularly the response time (or evaluation cost) for answering several related dimensional queries simultaneously is crucial to the OLAP applications. Recently, Zhao et al. first exploited this problem by presenting three heuristic algorithms. In this paper we first consider in detail two cases of the problem in which all the queries are either hash-based star joins or index-based star joins only. In the case of the hash-based star join, we devise a polynomial approximation algorithm which delivers a plan whose evaluation cost is $ O(n^{\epsilon }$) times the optimal, where n is the number of queries and is a fixed constant with . We also present an exponential algorithm which delivers a plan with the optimal evaluation cost. In the case of the index-based star join, we present a heuristic algorithm which delivers a plan whose evaluation cost is n times the optimal, and an exponential algorithm which delivers a plan with the optimal evaluation cost. We then consider a general case in which both hash-based star-join and index-based star-join queries are included. For this case, we give a possible improvement on the work of Zhao et al., based on an analysis of their solutions. We also develop another heuristic and an exact algorithm for the problem. We finally conduct a performance study by implementing our algorithms. The experimental results demonstrate that the solutions delivered for the restricted cases are always within two times of the optimal, which confirms our theoretical upper bounds. Actually these experiments produce much better results than our theoretical estimates. To the best of our knowledge, this is the only development of polynomial algorithms for the first two cases which are able to deliver plans with deterministic performance guarantees in terms of the qualities of the plans generated. The previous approaches including that of [ZDNS98] may generate a feasible plan for the problem in these two cases, but they do not provide any performance guarantee, i.e., the plans generated by their algorithms can be arbitrarily far from the optimal one. Received: July 21, 1998 / Accepted: August 26, 1999     

Single-machine scheduling with periodic maintenance to minimize makespan

We consider a single-machine scheduling problem with periodic maintenance activities. Although the scheduling problem with maintenance has attracted researchers’ attention, most of past studies considered only one maintenance period. In this research several maintenance periods are considered where each maintenance activity is scheduled after a periodic time interval. The objective is to find a schedule that minimizes the makespan, subject to periodic maintenance and nonresumable jobs. We first prove that the worst-case ratio of the classical LPT   algorithm is 2. Then we show that there is no polynomial time approximation algorithm with a worst-case ratio less than 2 unless P=NP$P=\text{<italic>NP</italic>}$, which implies that the LPT algorithm is the best possible.     

Fast techniques for the optimal smoothing of stored video

Work-ahead smoothing is a technique whereby a server, transmitting stored compressed video to a client, utilizes client buffer space to reduce the rate variability of the transmitted stream. The technique requires the server to compute a schedule of transfer under the constraints that the client buffer neither overflows nor underflows. Recent work established an optimal off-line algorithm (which minimizes peak, variance and rate variability of the transmitted stream) under the assumptions of fixed client buffer size, known worst case network jitter, and strict playback of the client video. In this paper, we examine the practical considerations of heterogeneous and dynamically variable client buffer sizes, variable worst case network jitter estimates, and client interactivity. These conditions require on-line computation of the optimal transfer schedule. We focus on techniques for reducing on-line computation time. Specifically, (i) we present an algorithm for precomputing and storing the optimal schedules for all possible client buffer sizes in a compact manner; (ii) we show that it is theoretically possible to precompute and store compactly the optimal schedules for all possible estimates of worst case network jitter; (iii) in the context of playback resumption after client interactivity, we show convergence of the recomputed schedule with the original schedule, implying greatly reduced on-line computation time; and (iv) we propose and empirically evaluate an “approximation scheme” that produces a schedule close to optimal but takes much less computation time.     

Multiprocessor Scheduling with Machine Allotment and Parallelism Constraints

Abstract. Modern computer systems distribute computation among several machines to speed up the execution of programs. Yet, setup and communication costs, as well as parallelism constraints, bound the number of machines that can share the execution of a given application, and the number of machines by which it can be processed simultaneously . We study the resulting scheduling problem, stated as follows. Given a set of n jobs and m uniform machines, assign the jobs to the machines subject to parallelism and machine allotment constraints, such that the overall completion time of the schedule (or makespan ) is minimized. Indeed, the multiprocessor scheduling problem (where each job can be processed by a single machine) is a special case of our problem; thus, our problem is strongly NP-hard. We present a (1+ α) -approximation algorithm for this problem, where α ∈ (0,1] depends on the minimal number of machine allotments and the minimal parallelism allowed for any job. Also, we show that when the maximal number of machines that can share the execution of a job is some fixed constant, our problem has a polynomial time approximation scheme ; for other special cases we give optimal polynomial time algorithms. Finally, through the relation of our problem to the classic preemptive scheduling problem on multiple machines, we shed some fresh light on what is known in scheduling folklore as the power of preemption.     

Scheduling Independent Multiprocessor Tasks

Abstract. We study the problem of scheduling a set of n independent multiprocessor tasks with prespecified processor allocations on a fixed number of processors. We propose a linear time algorithm that finds a schedule of minimum makespan in the preemptive model, and a linear time approximation algorithm that finds a schedule of makespan within a factor of (1+\eps) of optimal in the non-preemptive model. We extend our results by obtaining a polynomial time approximation scheme for the parallel processors variant of the multiprocessor task model.     

Guarding the walls of an art gallery

P so that each point of P is seen by at least one guard. We introduce and explore the edge-covering problem; the guards are required to observe the edges of P; metaphorically the paintings on the walls of the art gallery, and not necessarily every interior point. We compare minimum edge and interior covers for a given polygon and analyze the bounds and complexity for the edge-covering problem. We also introduce and analyze a restricted edge covering problem, where full visibility of each edge from at least one guard is required. For this problem we present an algorithm that computes a set of regions where a minimum set of guards must be located. The algorithm can also deal with the external visibility of a set of polygons.     

Minimizing Makespan and Preemption Costs on a System of Uniform Machines

It is well known that for preemptive scheduling on uniform machines there exist polynomial time exact algorithms, whereas for non-preemptive scheduling there are probably no such algorithms. However, it is not clear how many preemptions (in total, or per job) suffice in order to guarantee an optimal polynomial time algorithm. In this paper we investigate exactly this hardness gap, formalized as two variants of the classic preemptive scheduling problem. In generalized multiprocessor scheduling (GMS) we have a job-wise or total bound on the number of preemptions throughout a feasible schedule. We need to find a schedule that satisfies the preemption constraints, such that the maximum job completion time is minimized. In minimum preemptions scheduling (MPS) the only feasible schedules are preemptive schedules with the smallest possible makespan. The goal is to find a feasible schedule that minimizes the overall number of preemptions. Both problems are NP-hard, even for two machines and zero preemptions. For GMS, we develop polynomial time approximation schemes, distinguishing between the cases where the number of machines is fixed, or given as part of the input. Our scheme for a fixed number of machines has linear running time, and can be applied also for instances where jobs have release dates, and for instances with arbitrary preemption costs. For MPS, we derive matching lower and upper bounds on the number of preemptions required by any optimal schedule. Our results for MPS hold for any instance in which a job, J_j, can be processed simultaneously by ρ_j machines, for some ρ_j ≥ 1.     

Two-machine flow-shop scheduling with rejection

We study a scheduling problem with rejection on a set of two machines in a flow-shop scheduling system. We evaluate the quality of a solution by two criteria: the first is the makespan and the second is the total rejection cost. We show that the problem of minimizing the makespan plus total rejection cost is NP-hard and for its solution we provide two different approximation algorithms, a pseudo-polynomial time optimization algorithm and a fully polynomial time approximation scheme (FPTAS). We also study the problem of finding the entire set of Pareto-optimal points (this problem is NP-hard due to the NP-hardness of the same problem variation on a single machine [20]). We show that this problem can be solved in pseudo-polynomial time. Moreover, we show how we can provide an FPTAS that, given that there exists a Pareto optimal schedule with a total rejection cost of at most R and a makespan of at most K, finds a solution with a total rejection cost of at most (1+?)R and a makespan value of at most (1+?)K. This is done by defining a set of auxiliary problems and providing an FPTAS algorithm to each one of them.     

Providing QOS guarantees for disk I/O

In this paper, we address the problem of providing different levels of performance guarantees or quality of service for disk I/O. We classify disk requests into three categories based on the provided level of service. We propose an integrated scheme that provides different levels of performance guarantees in a single system. We propose and evaluate a mechanism for providing deterministic service for variable-bit-rate streams at the disk. We will show that, through proper admission control and bandwidth allocation, requests in different categories can be ensured of performance guarantees without getting impacted by requests in other categories. We evaluate the impact of scheduling policy decisions on the provided service. We also quantify the improvements in stream throughput possible by using statistical guarantees instead of deterministic guarantees in the context of the proposed approach.     

Scheduling Independent Multiprocessor Tasks

We study the problem of scheduling a set of n independent multiprocessor tasks with prespecified processor allocations on a fixed number of processors. We propose a linear time algorithm that finds a schedule of minimum makespan in the preemptive model, and a linear time approximation algorithm that finds a schedule of makespan within a factor of (1+\eps) of optimal in the non-preemptive model. We extend our results by obtaining a polynomial time approximation scheme for the parallel processors variant of the multiprocessor task model.     

A buffer-inventory-based dynamic scheduling algorithm for multimedia-on-demand servers

We present a producer-consumer model of multimedia-on-demand (MOD) servers. The producer retrieves media data from a disk and places it into a set of buffers, while the consumer sends out the data in the buffers to the users. We develop for the producer a buffer-inventory-based dynamic scheduling (BIDS) algorithm that guarantees non-zero inventory and non-overflow of data in the buffers to meet the continuity requirement and no-loss of data for each media stream. The algorithm can deal with heterogeneous me dia streams as well as the transient circumstances upon service completions and arrivals of new requests. To smooth out the impact of bursty data of variable-bit-rate media streams and therefore increase the maximum admissible load of requests, we also introduce into the scheduling scheme a time-scale-dependent peak consumption rate and a virtual cycle time. Based on BIDS, an effective admission control mechanism can be easily established by checking two simple conditions respectively on the overall system load and buffer size. Our algorithm is very easy to implement. Experiments carried out with an actual disk system and real video stream data verify that it is more robust compared to static scheduling algorithms previously proposed in the literature, especially when handling variable-bit-rate media streams.     

Converging to periodic schedules for cyclic scheduling problems with resources and deadlines

Cyclic scheduling has been widely studied because of the importance of applications in manufacturing systems and in computer science. For this class of problems, a finite set of tasks with precedence relations and resource constraints must be executed repetitively while maximizing the throughput. Many applications also require that execution schedules be periodic i.e. the execution of each task is repeated with a fixed global period w.The present paper develops a new method to build periodic schedules with cumulative resource constraints, periodic release dates and deadlines. The main idea is to fix the period w, to unwind the cyclic scheduling problem for some number of iterations, and to add precedence relations so that the minimum time lag between two successive executions of any task equals w. Then, using any usual (not cyclic) scheduling algorithm to compute task starting times for the unwound problem, we prove that either the method converges to a periodic schedule of period w or it fails to compute a schedule. A non-polynomial upper bound on the number of iterations to unwind in order to guarantee that cyclic precedence relations and resource constraints are fulfilled is also provided. This method is successfully applied to a real-life problem, namely the software pipelining of inner loops on an embedded VLIW processor core by using a Graham list scheduling algorithm.     

Preemptive scheduling of independent jobs on identical parallel machines subject to migration delays

We present hardness and approximation results for the problem of preemptive scheduling of n independent jobs on m identical parallel machines subject to a migration delay d with the objective to minimize the makespan. We give a sharp threshold on the value of d for which the complexity of the problem changes from polynomial time solvable to NP-hard. Next, we give initial results supporting a conjecture that there always exists an optimal schedule with at most m − 1 job migrations. Finally, we provide a O(n) time (1 + 1/log₂ n)-approximation algorithm for m = 2.     

Adaptive general perfectly periodic scheduling

We propose an adaptive algorithm Adaptmin to create perfectly periodic schedules. A perfectly periodic schedule schedules a client regularly after a predefined amount of time known as the period of the client. The periodicity of such schedules can be used to save battery life of nodes in a wireless network. The quality of a perfectly periodic schedule is a function of the ratio between the granted and requested periods. We find a worst case performance bound on the quality of schedules produced by Adaptmin. We compare our algorithm to previously proposed algorithm A in [Z. Brakerski, A. Nisgav, B. Patt-Shamir, General perfectly periodic scheduling, in: Proc. 21st Annual Symp. on Principles of Distributed Computing, 2002, pp. 163-172], and show families of input instances where either Adaptmin does no worse than A, or always outperforms A. The better performance of the proposed algorithm is also confirmed by simulations results for randomly generated input instances. Adaptmin produces 25% more efficient schedules as compared to A in our experiments. We also propose a variant of Adaptmin which is computationally much less demanding compared to A, but is very close to Adaptmin in terms of efficiency. Finally, we compare our algorithms to exponential-time optimal scheduling. Our simulation results indicate that the performance of the proposed algorithms is close to that of optimal scheduling.     

Scheduling of a single crane in batch annealing process

This paper studies a single crane scheduling problem motivated by batch annealing process in the iron and steel industry. Each coil stack placed on fixed base needs to go through two-stage processing: heating and cooling. During each stage, limited special machines (furnace and cooler) must be operated by crane, respectively. Our problem is to assign the shared machines and schedule a single crane for minimizing the last coil stack completion time (makespan). A mixed integer linear programming (MILP) model is formulated by considering both machine and crane positions. We show that the problem is NP-hard in the strong sense. Some optimal properties on the problem are derived. A two-phase algorithm is constructed for the problem. In the first phase, a fully polynomial time approximation scheme (FPTAS) is developed for the assignment subproblem. In the second phase, a heuristics is proposed for the scheduling subproblem. From an absolute performance point of view, we analyze the quality of the two-phase algorithm. We also consider special cases where some properties or algorithms are developed. In order to further verify the performance of the two-phase algorithm, we develop a lower bound on the optimal objective function. Computational experiments on the randomly generated problem instances show that the algorithm is close to the lower bound within a reasonable computation time.     

Parallel-machine scheduling of simple linear deteriorating jobs

We consider parallel-machine scheduling problems in which the processing time of a job is a simple linear increasing function of its starting time. The objectives are to minimize the makespan, total machine load, and total completion time. We show that all the problems are strongly NP-hard with an arbitrary number of machines and NP-hard in the ordinary sense with a fixed number of machines. For the former two problems, we prove that there exists no polynomial time approximation algorithm with a constant worst-case bound when the number of machines is arbitrary unless P=NP. When the number of machines is fixed, we propose two similar fully polynomial-time approximation schemes for the former two problems.     

On periodic resource scheduling for continuous-media databases

The Enhanced Pay-Per-View (EPPV) model for providing continuous-media services associates with each continuous-media clip a display frequency that depends on the clip's popularity. The aim is to increase the number of clients that can be serviced concurrently beyond the capacity limitations of available resources, while guaranteeing a constraint on the response time. This is achieved by sharing periodic continuous-media streams among multiple clients. The EPPV model offers a number of advantages over other data-sharing schemes (e.g., batching), which make it more attractive to large-scale service providers. In this paper, we provide a comprehensive study of the resource-scheduling problems associated with supporting EPPV for continuous-media clips with (possibly) different display rates, frequencies, and lengths. Our main objective is to maximize the amount of disk bandwidth that is effectively scheduled under the given data layout and storage constraints. Our formulation gives rise to -hard combinatorial optimization problems that fall within the realm of hard real-time scheduling theory. Given the intractability of the problems, we propose novel heuristic solutions with polynomial-time complexity. We also present preliminary experimental results for the average case behavior of the proposed scheduling schemes and examine how they compare to each other under different workloads. A major contribution of our work is the introduction of a robust scheduling framework that, we believe, can provide solutions for a variety of realistic EPPV resource-scheduling scenarios, as well as any scheduling problem involving regular, periodic use of a shared resource. Based on this framework, we propose various interesting research directions for extending the results presented in this paper. Received June 9, 1998 / Accepted October 13, 1998     

An efficient distributed algorithm for constructing small dominating sets

The dominating set problem asks for a small subset D of nodes in a graph such that every node is either in D or adjacent to a node in D. This problem arises in a number of distributed network applications, where it is important to locate a small number of centers in the network such that every node is nearby at least one center. Finding a dominating set of minimum size is NP-complete, and the best known approximation is logarithmic in the maximum degree of the graph and is provided by the same simple greedy approach that gives the well-known logarithmic approximation result for the closely related set cover problem. We describe and analyze new randomized distributed algorithms for the dominating set problem that run in polylogarithmic time, independent of the diameter of the network, and that return a dominating set of size within a logarithmic factor from optimal, with high probability. In particular, our best algorithm runs in rounds with high probability, where n is the number of nodes, is one plus the maximum degree of any node, and each round involves a constant number of message exchanges among any two neighbors; the size of the dominating set obtained is within of the optimal in expectation and within of the optimal with high probability. We also describe generalizations to the weighted case and the case of multiple covering requirements. Received: January 2002 / Accepted: August 2002 RID="*" ID="*" Supported by NSF CAREER award NSF CCR-9983901 RID="*" ID="*" Supported by NSF CAREER award NSF CCR-9983901