Hadas Shachnai

Disk load balancing for video-on-demand systems

Abstract. For a video-on-demand computer system, we propose a scheme which balances the load on the disks, thereby helping to solve a performance problem crucial to achieving maximal video throughput. Our load-balancing scheme consists of two components. The static component determines good assignments of videos to groups of striped disks. The dynamic component uses these assignments, and features a “DASD dancing” algorithm which performs real-time disk scheduling in an effective manner. Our scheme works synergistically with disk striping. We examine the performance of the proposed algorithm via simulation experiments.

On Two Class-Constrained Versions of the Multiple Knapsack Problem

Abstract. We study two variants of the classic knapsack problem, in which we need to place items of different types in multiple knapsacks; each knapsack has a limited capacity, and a bound on the number of different types of items it can hold: in the class-constrained multiple knapsack problem (CMKP) we wish to maximize the total number of packed items; in the fair placement problem (FPP) our goal is to place the same (large) portion from each set. We look for a perfect placement, in which both problems are solved optimally. We first show that the two problems are NP-hard; we then consider some special cases, where a perfect placement exists and can be found in polynomial time. For other cases, we give approximate solutions. Finally, we give a nearly optimal solution for the CMKP. Our results for the CMKP and the FPP are shown to provide efficient solutions for two fundamental problems arising in multimedia storage subsystems.

DASD dancing: a disk load balancing optimization scheme for video-on-demand computer systems

For a video-on-demand computer system we propose a scheme which balances the load on the disks, thereby helping to solve a performance problem crucial to achieving maximal video throughput. Our load balancing scheme consists of two stages. The static stage determines good assignments of videos to groups of striped disks. The dynamic phase uses these assignments, and features a DASD dancing algorithm which performs real-time disk scheduling in an effective manner. Our scheme works synergistically with disk striping. We examine the performance of the DASD dancing algorithm via simulation experiments.

Design and analysis of a look-ahead scheduling scheme to support pause-resume for video-on-demand applications

In a video-on-demand (VOD) system, subscribers can choose both the movie they wish to view and the time they wish to view it. In such an environment there are invariably “hot” videos which are requested by many viewers. The requirement that each viewer be able to independently pause the video at any instant and later resume the viewing with little delay can cause difficulties in batching viewers for each showing. Under batching, a single video stream is shared by multiple concurrent viewers and a resume request has to wait for additional stream capacity to become available before actual resumption can occur. The conventional approach to the support of on-demand pause-resume provides one video access stream to disks for each video request. This can greatly increase the disk arm requirements of a VOD system. In this paper, we propose a more efficient mechanism to support the pause-resume feature usinglook-ahead scheduling withlook-aside buffering. The idea is to use buffering to increase the number of concurrent viewers supportable. The concept of look-ahead scheduling is not to back up each viewer with a real stream capacity so he can pause and resume at any time, but rather with a (look-ahead) stream that is currently being used for another showing which is close to completion. Before the look-ahead stream becomes available, the pause and resume features have to be supported by the original stream through (look-aside) buffering of the missed content. It is shown via simulations that the proposed scheme can provide a substantially greater throughput than the approach without batching. Furthermore, for a given amount of buffer, the improvement in throughput grows more than linearly with the stream capacity of the server. In other words, the look-ahead stream scheduling scheme operates with good economy of scale because it is easier to form look-ahead streams for video servers with larger stream capacity.

Sum Multicoloring of Graphs

Scheduling dependent jobs on multiple machines is modeled by the graph multicoloring problem. In this paper we consider the problem of minimizing the average completion time of all jobs. This is formalized as the sum multicoloring problem: Given a graph and the number of colors required by each vertex, find a multicoloring which minimizes the sum of the largest colors assigned to the vertices. It reduces to the known sum coloring problem when each vertex requires exactly one color.This paper reports a comprehensive study of the sum multicoloring problem, treating three models: with and without preemptions, as well as co-scheduling where jobs cannot start while others are running. We establish a linear relation between the approximability of the maximum independent set problem and the approximability of the sum multicoloring problem in all three models, via a link to the sum coloring problem. Thus, for classes of graphs for which the independent set problem is ?-approximable, we obtain O(?)-approximations for preemptive and co-scheduling sum multicoloring and O(?logn)-approximation for nonpreemptive sum multicoloring. In addition, we give constant ratio approximations for a number of fundamental classes of graphs, including bipartite, line, bounded degree, and planar graphs.

Exploring wait tolerance in effective batching for video-on-demand scheduling

Abstract. In a video-on-demand (VOD) environment, batching requests for the same video to share a common video stream can lead to significant improvement in throughput. Using the wait tolerance characteristic that is commonly observed in viewers behavior, we introduce a new paradigm for scheduling in VOD systems. We propose and analyze two classes of scheduling schemes: the Max_Batch and Min_Idle schemes that provide two alternative ways for using a given stream capacity for effective batching. In making a video selection, the proposed schemes take into consideration the next stream completion time, as well as the viewer wait tolerance. We compared the proposed schemes with the two previously studied schemes: (1) first-come-first-served (FCFS) that schedules the video with the longest waiting request and (2) the maximum queue length (MQL) scheme that selects the video with the maximum number of waiting requests. We show through simulations that the proposed schemes substantially outperform FCFS and MQL in reducing the viewer turn-away probability, while maintaining a small average response time. In terms of system resources, we show that, by exploiting the viewers wait tolerance, the proposed schemes can significantly reduce the server capacity required for achieving a given level of throughput and turn-away probability as compared to the FCFS and MQL. Furthermore, our study shows that an aggressive use of the viewer wait tolerance for batching may not yield the best strategy, and that other factors, such as the resulting response time, fairness, and loss of viewers, should be taken into account.

Sum Coloring Interval and k-Claw Free Graphs with Application to Scheduling Dependent Jobs

Abstract We consider the sum coloring and sum multicoloring problems on several fundamental classes of graphs, including the classes of interval and k-claw free graphs. We give an algorithm that approximates sum coloring within a factor of 1.796, for any graph in which the maximum k-colorable subgraph problem is polynomially solvable. In particular, this improves on the previous best known ratio of 2 for interval graphs. We introduce a new measure of coloring, robust throughput}, that indicates how “quickly” the graph is colored, and show that our algorithm approximates this measure within a factor of 1.4575. In addition, we study the contiguous (or non-preemptive) sum multicoloring problem on k-claw free graphs. This models, for example, the scheduling of dependent jobs on multiple dedicated machines, where each job requires the exclusive use of at most k machines. Assuming that k is a fixed constant, we obtain the first constant factor approximation for the problem.

Approximation Schemes for Packing with Item Fragmentation

Abstract We consider two variants of the classical bin packing problem in which items may be fragmented. This can potentially reduce the total number of bins needed for packing the instance. However, since fragmentation incurs overhead, we attempt to avoid it as much as possible. In bin packing with size increasing fragmentation (BP-SIF), fragmenting an item increases the input size (due to a header/footer of fixed size that is added to each fragment). In bin packing with size preserving fragmentation (BP-SPF), there is a bound on the total number of fragmented items. These two variants of bin packing capture many practical scenarios, including message transmission in community TV networks, VLSI circuit design and preemptive scheduling on parallel machines with setup times/setup costs. While both BP-SPF and BP-SIF do not belong to the class of problems that admit a polynomial time approximation scheme (PTAS), we show in this paper that both problems admit a dual PTAS and an asymptotic PTAS. We also develop for each of the problems a dual asymptotic fully polynomial time approximation scheme (AFPTAS). Our AFPTASs are based on a non-standard transformation of the mixed packing and covering linear program formulations of our problems into pure covering programs, which enables to efficiently solve these programs.

Approximation schemes for generalized two-dimensional vector packing with application to data placement

Given is a set of items and a set of devices, each possessing two limited resources. Each item requires some amounts of these resources. Further, each item is associated with a profit and a color, and items of the same color can share the use of one resource. The goal is to allocate the resources to the most profitable (feasible) subset of items. In alternative formulation, the goal is to pack the most profitable subset of items in a set of two-dimensional bins (knapsacks), in which the capacity in one dimension is sharable. Indeed, the special case where there is a single item in each color is the well-known two-dimensional vector packing (2DVP) problem. Thus, unless P = NP, the problem that we study does not admit a fully polynomial time approximation scheme (FPTAS) for a single bin, and is MAX-SNP hard for multiple bins. Our problem has several important applications, including data placement on disks in media-on-demand systems. We present approximation algorithms as well as optimal solutions for some instances. In some cases, our results are similar to the best known results for 2DVP. Specifically, for a single bin, we show that the problem is solvable in pseudo-polynomial time and develop a polynomial time approximation scheme (PTAS) for general instances. For a natural subclass of instances we obtain a simpler scheme. This yields the first combinatorial PTAS for a non-trivial subclass of instances for 2DVP. For multiple bins, we develop a PTAS for a subclass of instances arising in the data placement problem. Finally, we show that when the number of distinct colors in the instance is fixed, our problem admits a PTAS, even if the items have arbitrary sizes and profits, and the bins are arbitrary.

There is no EPTAS for two-dimensional knapsack

In the d-dimensional (vector) knapsack problem given is a set of items, each having a d-dimensional size vector and a profit, and a d-dimensional bin. The goal is to select a subset of the items of maximum total profit such that the sum of all vectors is bounded by the bin capacity in each dimension. It is well known that, unless P=NP, there is no fully polynomial-time approximation scheme for d-dimensional knapsack, already for d=2. The best known result is a polynomial-time approximation scheme (PTAS) due to Frieze and Clarke [A.M. Frieze, M. Clarke, Approximation algorithms for the m-dimensional 0-1 knapsack problem: worst-case and probabilistic analyses, European J. Operat. Res. 15 (1) (1984) 100-109] for the case where d>=2 is some fixed constant. A fundamental open question is whether the problem admits an efficient PTAS (EPTAS). In this paper we resolve this question by showing that there is no EPTAS for d-dimensional knapsack, already for d=2, unless W[1]=FPT. Furthermore, we show that unless all problems in SNP are solvable in sub-exponential time, there is no approximation scheme for two-dimensional knapsack whose running time is f(1/@e)|I|^o^(^1^/^@e^), for any function f. Together, the two results suggest that a significant improvement over the running time of the scheme of Frieze and Clarke is unlikely to exist.