Sungjin Im

A tutorial on amortized local competitiveness in online scheduling

Recently the use of potential functions to analyze online scheduling algorithms has become popular [19, 7,29, 13, 31, 4, 30, 3, 21, 15, 14, 28, 12, 2, 5, 6, 9, 11, 23, 33, 24, 8, 17, 16, 25, 1, 20, 26, 22, 18]. Thesepotential functions are used to show that a particular online algorithm is locally competitive in an amortizedsense. Algorithm analyses using potential functions are sometimes criticized as seeming to be black magicas the formal proofs do not require, and commonly do not contain, any discussion of the intuition behindthe design of the potential function. Sometimes, as in the case for the first couple uses of potential functionsin the online scheduling literature, this is because the authors arrived at the potential function by trial anderror, and there was not a cohesive underlying intuitionguidingthe development. However, now that tens ofonline scheduling papers have used potential functions, one can see that a “standard” potential function hasemerged that seems to be applicable to a wide range of problems. The use of this standard potential functionto prove amortized local competitiveness can no longer be considered to be magical, and is a learnabletechnique. Our main goal here is to give a tutorialteaching this technique to readers with some modest priorknowledge of scheduling, online problems, and the concept of worst-case performance ratios.Online Scheduling: We consider online schedulingproblems where jobs/tasksarrive at a server (e.g. a webserver, a database server, an operating system, etc.) over time. Throughoutthis paper N willdenote the totalnumber of jobs and jobs are indexed J

Scheduling jobs with varying parallelizability to reduce variance

We give a (2+ε)-speed <i>O</i>(1)-competitive algorithm for scheduling jobs with arbitrary speed-up curves for the l₂ norm of flow. We give a similar result for the broadcast setting with varying page sizes.

Competitive algorithms from competitive equilibria: non-clairvoyant scheduling under polyhedral constraints

We introduce and study a general scheduling problem that we term the Packing Scheduling problem (PSP). In this problem, jobs can have different arrival times and sizes; a scheduler can process job j at rate xj, subject to arbitrary packing constraints over the set of rates (x) of the outstanding jobs. The PSP framework captures a variety of scheduling problems, including the classical problems of unrelated machines scheduling, broadcast scheduling, and scheduling jobs of different parallelizability. It also captures scheduling constraints arising in diverse modern environments ranging from individual computer architectures to data centers. More concretely, PSP models multidimensional resource requirements and parallelizability, as well as network bandwidth requirements found in data center scheduling. In this paper, we design non-clairvoyant online algorithms for PSP and its special cases -- in this setting, the scheduler is unaware of the sizes of jobs. Our results are summarized as follows. • For minimizing total weighted completion time, we show a O(1)-competitive algorithm. Surprisingly, we achieve this result by applying the well-known Proportional Fairness algorithm (PF) to perform allocations each time instant. Though PF has been extensively studied in the context of maximizing fairness in resource allocation, we present the first analysis in adversarial and general settings for optimizing job latency. Our result is also the first O(1)-competitive algorithm for weighted completion time for several classical non-clairvoyant scheduling problems. •For minimizing total weighted flow time, for any constant ε > 0, any O(n1---ε)-competitive algorithm requires extra speed (resource augmentation) compared to the offline optimum. We show that PF is a O(log n)-speed O(log n)-competitive non-clairvoyant algorithm, where n is the total number of jobs. We further show that there is an instance of PSP for which no non-clairvoyant algorithm can be O(n1---ε)-competitive with o(√log n) speed. •For the classical problem of minimizing total flow time for unrelated machines in the non-clairvoyant setting, we present the first online algorithm which is scalable ((1 + ε)-speed O(1)-competitive for any constant ε > 0). No non-trivial results were known for this setting, and the previous scalable algorithm could handle only related machines. We develop new algorithmic techniques to handle the unrelated machines setting that build on a new single machine scheduling policy. Since unrelated machine scheduling is a special case of PSP, when contrasted with the lower bound for PSP, our result also shows that PSP is significantly harder than perhaps the most general classical scheduling settings. Our results for PSP show that instantaneous fair scheduling algorithms can also be effective tools for minimizing the overall job latency, even when the scheduling decisions are non-clairvoyant and constrained by general packing constraints.

SelfishMigrate: A Scalable Algorithm for Non-clairvoyantly Scheduling Heterogeneous Processors

We consider the classical problem of minimizing the total weighted flow-time for unrelated machines in the online non-clairvoyant setting. In this problem, a set of jobs J arrive over time to be scheduled on a set of M machines. Each job j has processing length pj, weight wj, and is processed at a rate of ℓij when scheduled on machine i. The online scheduler knows the values of wj and ℓij upon arrival of the job, but is not aware of the quantity pj. We present the first online algorithm that is scalable ((1 + ϵ)-speed O(1/2)-competitive for any constant ϵ > 0) for the total weighted flow-time objective. No non-trivial results were known for this setting, except for the most basic case of identical machines. Our result resolves a major open problem in online scheduling theory. Moreover, we also show that no job needs more than a logarithmic number of migrations. We further extend our result and give a scalable algorithm for the objective of minimizing total weighted flow-time plus energy cost for the case of unrelated machines. In this problem, each machine can be sped up by a factor of f4-1 i (P) when consuming power P, wherefi is an arbitrary strictly convex power function. In particular, we get an O(γ2)-competitive algorithm when all power functions are of form sγ. These are the first non-trivial non-clairvoyant results in any setting with heterogeneous machines. The key algorithmic idea is to let jobs migrate selfishly until they converge to an equilibrium. Towards this end, we define a game where each jobs utility which is closely tied to the instantaneous increase in the objective the job is responsible for, and each machine declares a policy that assigns priorities to jobs based on when they migrate to it, and the execution speeds. This has a spirit similar to coordination mechanisms that attempt to achieve near optimum welfare in the presence of selfish agents (jobs). To the best our knowledge, this is the first work that demonstrates the usefulness of ideas from coordination mechanisms and Nash equilibria for designing and analyzing online algorithms.

Coordination mechanisms from (almost) all scheduling policies

We study the price of anarchy of coordination mechanisms for a scheduling problem where each job j has a weight wj, processing time pij, assignment cost hij, and communication delay (or release date) rij, on machine i. Each machine is free to declare its own scheduling policy. Each job is a selfish agent and selects a machine that minimizes its own disutility, which is equal to its weighted completion time plus its assignment cost. The goal is to minimize the total disutility incurred by all the jobs. Our model is general enough to capture scheduling jobs in a distributed environment with heterogeneous machines (or data centers) that are situated across different locations. Our main result is a characterization of scheduling policies that give a small (robust) Price of Anarchy. More precisely, we show that whenever each machine independently declares any scheduling policy that satisfies a certain bounded stretch condition introduced in this paper, the game induced between the jobs has a small Price of Anarchy. Our characterization is powerful enough to test almost all popular scheduling policies. On the technical side, to derive our results, we use a potential function whose derivative leads to an instantaneous smoothness condition, and linear programming and dual fitting. To the best of our knowledge, this is a novel application of these techniques in the context of coordination mechanisms, and we believe these tools will find more applications in analyzing PoA of games. We also extend our results to the lk-norms and l∞ norm (makespan) objectives.

Longest wait first for broadcast scheduling [extended abstract]

We consider online algorithms for broadcast scheduling. In the pull-based broadcast model there are n unit-sized pages of information at a server and requests arrive online for pages. When the server transmits a page p, all outstanding requests for that page are satisfied. There is a lower bound of Ω(n) on the competitiveness of online algorithms to minimize average flow-time; therefore we consider resource augmentation analysis in which the online algorithm is given extra speed over the adversary. The longest-wait-first (LWF) algorithm is a natural algorithm that has been shown to have good empirical performance [2]. Edmonds and Pruhs showed that LWF is 6-speed O(1)-competitive using a novel yet complex analysis; they also showed that LWF is not O(1)-competitive with less than 1.618-speed. In this paper we make two main contributions to the analysis of LWF and broadcast scheduling. We give an intuitive and easy to understand analysis of LWF which shows that it is O(1/e2)-competitive for average flow-time with (4+e) speed. We show that a natural extension of LWF is O(1)-speed O(1)-competitive for more general objective functions such as average delay-factor and Lk norms of delay-factor (for fixed k). These metrics generalize average flow-time and Lk norms of flow-time respectively and ours are the first non-trivial results for these objective functions in broadcast scheduling.

Optimized scheduling of multi-IMA partitions with exclusive region for synchronized real-time multi-core systems

Integrated Modular Avionics (IMA) architecture has been widely adopted by the avionics industry due to its strong temporal and spatial isolation capability for safety-critical real-time systems. The fundamental challenge to integrating an existing set of single-core IMA partitions into a multi-core system is to ensure that the isolation of the partitions will be maintained without incurring huge redevelopment and recertification costs. To address this challenge, we developed an optimized partition scheduling algorithm which considers exclusive regions to achieve the synchronization between partitions across cores. We show that the problem of finding the optimal partition schedule is NP-complete and present a Constraint Programming formulation. In addition, we relax this problem to find the minimum number of cores needed to schedule a given set of partitions and propose an approximation algorithm which is guaranteed to find a feasible schedule of partitions if there exists a feasible schedule of exclusive regions.

Tight Bounds for Online Vector Scheduling

Modern data centers face a key challenge of effectively serving user requests that arrive online. Such requests are inherently multi-dimensional and characterized by demand vectors over multiple resources such as processor cycles, storage space, and network bandwidth. Typically, different resources require different objectives to be optimized, and Lr norms of loads are among the most popular objectives considered. Furthermore, the server clusters are also often heterogeneous making the scheduling problem more challenging. To address these problems, we consider the online vector scheduling problem in this paper. Introduced by Chekuri and Khanna (SIAM J. of Comp. 2006), vector scheduling is a generalization of classical load balancing, where every job has a vector load instead of a scalar load. The scalar problem, introduced by Graham in 1966, and its many variants (identical and unrelated machines, makespan and Lr-norm optimization, offline and online jobs, etc.) have been extensively studied over the last 50 years. In this paper, we resolve the online complexity of the vector scheduling problem and its important generalizations - for all Lr norms and in both the identical and unrelated machines settings. Our main results are: · For identical machines, we show that the optimal competitive ratio is Θ(log d/ log log d) by giving an online lower bound and an algorithm with an asymptotically matching competitive ratio. The lower bound is technically challenging, and is obtained via an online lower bound for the minimum mono-chromatic clique problem using a novel online coloring game and randomized coding scheme. Our techniques also extend to asymptotically tight upper and lower bounds for general Lr norms. · For unrelated machines, we show that the optimal competitive ratio is Θ(log m + log d) by giving an online lower bound that matches a previously known upper bound. Unlike identical machines, however, extending these results, particularly the upper bound, to general Lr norms requires new ideas. In particular, we use a carefully constructed potential function that balances the individual Lr objectives with the overall (convexified) min-max objective to guide the online algorithm and track the changes in potential to bound the competitive ratio.

Minimizing Maximum Response Time and Delay Factor in Broadcast Scheduling

We consider online algorithms for pull-based broadcast scheduling. In this setting there are n pages of information at a server and requests for pages arrive online. When the server serves (broadcasts) a page p, all outstanding requests for that page are satisfied. We study two related metrics, namely maximum response time (waiting time) and maximum delay-factor and their weighted versions. We obtain the following results in the worst-case online competitive model. We show that FIFO (first-in first-out) is 2-competitive even when the page sizes are different. Previously this was known only for unit-sized pages [10] via a delicate argument. Our proof differs from [10] and is perhaps more intuitive. We give an online algorithm for maximum delay-factor that is O(1/e 2)-competitive with (1 + e)-speed for unit-sized pages and with (2 + e)-speed for different sized pages. This improves on the algorithm in [13] which required (2 + e)-speed and (4 + e)-speed respectively. In addition we show that the algorithm and analysis can be extended to obtain the same results for maximum weighted response time and delay factor. We show that a natural greedy algorithm modeled after LWF (Longest-Wait-First) is not O(1)-competitive for maximum delay factor with any constant speed even in the setting of standard scheduling with unit-sized jobs. This complements our upper bound and demonstrates the importance of the tradeoff made in our algorithm.

Online Scheduling with General Cost Functions

We consider a general online scheduling problem on a single machine with the objective of minimizing Σjwjg(Fj), where wj is the weight/importance of job Jj, Fj is the flow time of the job in the schedule, and g is an arbitrary non-decreasing cost function. Numerous natural scheduling objectives are special cases of this general objective. We show that the scheduling algorithm Highest Density First (HDF) is (2+e)-speed O(1)-competitive for all cost functions g simultaneously. We give lower bounds that show the HDF algorithm and this analysis are essentially optimal. Finally, we show scalable algorithms are achievable in some special cases.