A note on the complexity of scheduling of communication-aware

BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 66, No. 2, 2018 DOI: 10.24425/122099

A note on the complexity of scheduling of communication-aware directed acyclic graph J. MUSIAL1*, M. GUZEK 2, P. BOUVRY 2, and J. BLAZEWICZ 1, 3 1

2

Institute of Computing Science, Poznan University of Technology, ul. Piotrowo 2, 60-965 Poznan, Poland Comp. Sci. and Commun. Res. Unit, University of Luxembourg, 6 Avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg 3 Institute of Bioorganic Chemistry, Polish Academy of Sciences, Z. Noskowskiego 12/14, 61-704 Poznan, Poland

Abstract. The most recent incarnation of distributed paradigm is cloud computing. It can be seen as the first widely accepted business model of mass consumption of the distributed computing resources. Despite the differences in business models and technical details regarding cloud platforms, the distributed computing underlies cloud. Communications in cloud systems include transmissions of the results of cloud applications, users interactions, and exchange of data between different services that compose applications. The latter becomes more critical as applications become richer as well as more complex, and may consist of services operated by various providers. The effective communication between components of cloud systems is thus critical to the end user satisfaction and to the success of cloud services. We will discuss different cloud computing models (communication aware and unaware). Main focus will be placed on communication-aware directed acyclic graph (CA-DAG), which extends the classical DAG model by explicitly modeling communication tasks. Moreover, we will analyze and consult computational complexity of this innovative distributed computation model inspired by the characteristics of cloud computing. Providing a proof of strong NP-hardness of the problem allows for a future implementation and evolution of the communication-aware DAG models. Key words: computational complexity, models in cloud computing, communication-aware cloud computing, directed acyclic graphs, NP-hardness in cloud computing.

1. Introduction Scheduling is undoubtedly one of the most important fields of operations research. The scheduling problems can be generally described as the allocation of resources over time to execute a group of tasks, which are a part of the processes [1]. Each task needs a particular resource to be completed. The nature of resources, tasks, connections and relations between them can be defined in many different ways. Predominantly, the scheduling is motivated by applications from industry and service operations management like: transportation planning, factory production lines, events planning, timetabling for general transportation purposes, vehicle routing, duty roster, and many more. Actually, if we perceive scheduling problems in general, they can be observed almost everywhere in real world situations. If one runs many computers in parallel and the goal is to solve a problem computationally (which is very often complicated and intensive), it can be described as a parallel processing/ computing [2]. The main idea is to provide the results in a time that no single computer could deliver. The idea is not new, however, its nature is experimental and therefore is being still strongly developed and studied. There are many new concepts that constantly try to increase speed of computations. Neverthe*e-mail: [email protected]

Manuscript submitted 2017-06-20, revised 2017-10-14, initially accepted for publication 2017-10-17, published in April 2018.

Bull. Pol. Ac.: Tech. 66(2) 2018

less, it has to be noted that the parallel processing is strongly connected with a networking and hardware issues. If we consider a parallel application as a set of communicating processes then it can be easily modeled using a directed acyclic graph (DAG) [1, 3]. Most of the general scheduling theory assumptions can be applied. However, one should notice that the task execution in parallel distributed systems is connected with some communication delays. This communication delay cost occurs when two consecutive tasks are performed (on different machines, on the same machine, on the same processor, or even on the same processor’s core). The most recent incarnation of distributed paradigm is cloud computing [4]. It can be seen as the first widely accepted business model of mass consumption of distributed computing resources. The resources are offered as services (or their components) which means that they are no longer addressed to the IT professionals, but can be also directly offered to the end users. The users can use the resources seamlessly, as the necessary knowledge about a service set up and the underlying hardware [5]. The exact predictions of cloud market growth vary [6], but all of them recognize its large potential, justified by the strong growth in the past. Despite differences in business models and technical details regarding cloud platforms, the distributed computing underlies cloud. Communications in cloud systems include transmissions of the results of cloud applications, users interactions, and exchange of data between different services that compose applications. The latter becomes more critical as applications become richer and therefore, more complex. They may consist of services operated by various providers. The ef-

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J. Musial, M. Guzek, P. Bouvry, and J. Blazewicz

fective communication between components of cloud systems is thus critical to the end user satisfaction and to the success of cloud services. In this paper, we discuss the complexity of an innovative distributed computation model inspired by these characteristics of cloud computing. The model, communication-aware directed acyclic graph (CA-DAG), extends the classical DAG model by explicitly modeling communication tasks. It enables to model multicast communications or gather and scatter patterns, and explicitly schedule communications on various network topologies. Because of its relevance, the model has been already studied experimentally [7] however, rapid practical developments left the theoretical analysis behind. This paper is devoted to filling that gap.

2. Resource allocation in cloud computing Basic requirements of cloud computing (such as managing multiple machines, providing services that work on different systems) have already been addressed and initially solved with some first ideas, analysis and computations with specific algorithms [8]. Further improvements require more holistic approaches, i.e. simultaneous taking into account different aspects and layers of the system to enhance efficiency and performance of cloud computing systems. New research opportunities and challenges reside also in linking the existing cloud computing issues [9] with other known problems, sometimes from different or neighboring fields [10], e.g. problems that tackle shopping optimization of products available over the Internet [11, 12] and the analysis of possible usage of their algorithms [13]. As communications are crucial for cloud services, a network becomes one of the most important resources. The current approach to resource optimization and scheduling is far from optimal. One of the reasons is that cloud computing is a highly dynamic online environment. Therefore, management of resources must react and adapt to a large amount of state changes in communication. Traditional approaches to resource optimization are not sufficient [14]. At this point it is worth mentioning some general information regarding task scheduling and the scheduling area (cf. [1] or [15] for a general treatment of the topic). Let us remind a basic presentation of the scheduling problem with a set J = { J1, J2, …, Jn} of jobs/tasks which need to be scheduled on a set M = { M1, M2, …, Mm} of non-identical, parallel machines. Additional parameters that describe the problem are: ● pij – processing time of job Ji on the machine Mj, ● tj – setup (warm-up) time for machine Mj. The goal is to find a non-preemptive schedule that minimizes total time (including set-ups) the machines used to process the jobs. Workflows can be modeled by directed acyclic graphs (DAG) Gj = (Vj, Ej). Vj is the set tasks. Ej = (Tu , Tv) is the set of directed arcs between tasks where Tu , Tv 2 Vj and u 6= v. There should be no cycles Tu  → Tv → Tu . Precedence constraints are represented by arcs (Tu , Tv), where task Tu has to be completed

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Fig. 1. CU-DAG model

before execution of Tv. It is worth noting that arc can be additionally connected with communication costs. Communication delays are not taken into account if two following tasks are executed by the same processor [16]. Task scheduling with communication costs is a recognized and extensively studied problem in the scheduling community and it was investigated for different systems and applications, e.g. complete and homogeneous network [17], network communication model with shared medium [18], or more detailed way of communication delay descriptions based on LogP model [19]. Classification of scheduling problems with communication delays was widely described by Drozdowski [2]. Recently Kliazovich et al. [7] proposed an innovative cloud computing model where communications between tasks are crucial – the Communication-Aware DAG model (CA-DAG, Fig. 3). They showed many advantages of the model over already known CU-DAG [20] (Communication-Unaware DAG, Fig. 1) and EB-DAG [18, 21] (Edges-Based DAG, Fig. 2), and discussed the positive impact of the CA-DAG model on existing scheduling algorithms. The computational experiments showed that the usage of CA-DAG model enables more efficient scheduling in distributed systems. Since the great part of cloud computing applications are highly dependent on communication requirements (cost, deadlines, stability, etc.) the CA-DAG is a complex model that focuses on sophisticated communication-aware issues. A new type of task is introduced: the computing tasks Vc correspond to tasks in the classical DAG model, while all communications are modeled as a new type of task: communication tasks Vcomm. Each communication task is additionally described by parameters S and Dcomm. Dcomm determines deadline for transmitting S bits of information. Moreover with each node vi there is associated cost of communication vicomm. Bull. Pol. Ac.: Tech. 66(2) 2018

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S bits S of bits information. Moreover of with Moreover Moreover node with v with each node node v v to tasks in the to tasks the DAG classical model, DAG while model, all communications while communications polynomial polynomial transformation polynomial transformati polynomial from transfo str S and and . D determines deadline for i i i hardness of stability, lines, stability, etc.) stability, etc.) the CA-DAG etc.) the CA-DAG the CA-DAG is a complex is a complex is a model complex model that model that that comm comm eters S D . D determines deadline for transmitting P(m),N(c)∣prec, p ∣C (1) T T T T HEOREM 1. HEOREM Problem HEOREM 1. HEOREM P(1),N Proble 1. first problem first is defined problem first problem as is follows. defined is defi strongly NP-hard. comm Moreover comm j max Proof. We m. Thus, by using a slightly modifie systems. there is asSeters bits of information. with each node v comm comm comm P(m),N(c)∣prec, P(m),N(c)∣pr P(m),N( ptocaja Each communication Each communication task communication Each is additionally communication task Each istask additionally described communication isrequirements task additionally isaby additionally described paramtask described is by additionally described paramby described by paramitransmitting P(1),N(1)∣c focuses onof sophisticated issues. A new tasks in the classical DAG highly model, while all communications Stobits and D .D determines deadline for hardness of problem P(1),N(1)∣chain, pimizing =for 1∣C . .the for cloud for cloud computing the cloud compu highly dependent highly dependent dependent on communication on communication on communication requirements (cost, (cost, dead(cost, deaddeadimizing imizing CparamClem C comm communication-aware j by max max max max Proof. 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We will prove this theorem by using a pseudoscheduling problems [22], we can den Since great part of computing applications are . sociated cost of communication v T HEOREM Proof. We will prove this theorem by using pseudoeters S.tasks and D eters Seters and SDCA-DAG eters D S.CA-DAG and DD eters and . tasks Dthe determines D deadline determines .Scommunication determines for deadline determines transmitting .model deadline D for deadline transmitting determines for transmitting for deadline transmitting for transmitting first problem comm ofoftask iscost introduced: the computing V correspond .the are modeled as acommunication new type ofwith communication Vand there isis asStype bitssociated information. Moreover each node vintroduced: i task: comm comm comm comm comm comm comm comm comm Using some ideas [23], we can prove strong NPlines, lines, stability, lines, etc.) stability, etc.) etc.) the CA-DAG is.the acomm complex is aD complex is aDmodel complex that model that that ciof comm polynomial transformation strongly NP-complete probEach communication Each task is additionally is additionally described by described paramby paramP(1),N(1)∣chain, P(1),N(1)∣chain, P(1),N(1)∣chain, p=aideas P(1),N(1)∣c pfrom =P(m),N(c)∣pr 1∣C ,intege deno =∑ 1ip of communication type type type ofstability, istask is task introduced: introduced: the computing the computing computing tasks V tasks tasks Vfrom Vfrom correspond correspond correspond Proof. We will prove this theorem by using athe pseudojfrom max j[23] Using Using some Using some ideas some ideas from [23], we .task sociated cost of vvcomm c (cost, cProblem c task i of ,a ,...,a and ,a ,...,a integer ,a ,...,a and b with and a a a T HEOREM 1. P(1),N(1)∣chain, p 1∣C is P(m),N(c)∣prec, P(m),N(c p lem partiti comm 1 2 3t 1 2 1 2 3t 3t polynomial transformation from strongly NP-complete probj max i aj for cloud computing imizing C highly dependent on communication requirements deadstrongly NP polynomial transformation from NP-complete probmax there isA asthere there is asis there asas-vitransmitting there isproblem asSwhile offocuses information. S bits Sofsophisticated bits information. Moreover of S bits information. with information. Moreover S each bits Moreover of node with information. Moreover veach with node each with Moreover vnode each vstrongly with v1∣C each node partition: G to tasks in the DAG model, all communications Each communication task is described by paramsociated cost ofclassical communication vi additionally . bits idetermines ideadline inode ideadline hardness ofwhile problem P(1),N(1)∣chain, phardness =NP-complete . is focuses focuses on sophisticated on sophisticated on communication-aware communication-aware communication-aware issues. issues. Acommunications new issues. new A first new jproblem max lem partition [24] to the decision version of problem eters Sofand D eters S and D . D . D determines for transmitting for first is defined problem first problem as first is follows. defined is defi Proof. We Proof. will Proof. prove We will Proof. We this prove will theore We pa to tasks to in tasks to the tasks in classical the in classical the DAG classical DAG model, DAG model, while model, all communications while all communications all comm comm comm comm polynomial transformation from strongly probhardness of hardness problem of problem of P(1),N(1)∣chain problem P(1),N(1)∣ P(1),N tioned into t tioned disjoint tioned into 3-element t into disjoint t disjoin subs 3-e strongly NP-hard. comm comm comm comm comm P(1),N(1)∣c lem partition [24] to the decision version of problem lines, stability, etc.) the CA-DAG is a complex model that lem partition [24] the decision version of problem .bits .vof .to .each . there sociated cost sociated offor communication sociated cost sociated cost communication ofvcost sociated of communication vby apartition: ,a .ofbits are modeled as aD new type of task: communication Vnew eters S and . Dcomm determines deadline 1from 2a,...,a 3 type of type type of is task introduced: is task is the computing the computing the computing tasks tasks V tasks Vvwith correspond correspond correspond comm comm icommunication iV i Using Using Using some ideas some from ideas [23], from [23] we cptasks ccommunication ci tasks P(1),N(1)∣chain, 1∣C ,V denoted hereafter. The is1∣C asthere is asStype of information. Stask: ofv1.icost information. Moreover Moreover with vsome node vfor ,a partition: Given partition: partition: ai =ideas set Given Sitransformati {a Given set P(m),N(c)∣prec, pSaG polynomial polynomial transformation polynomial polynomial from str j = max .decision .version are modeled aretask modeled are as modeled aoftasks new asintroduced: atransmitting type asintroduced: aof new task: of type communication task: of communication communication tasks V.denoted i1each i1,...,t? lem partition [24] to of problem 1transfo 2 ,.. comm comm b,node for iΠ =by 1,...,t? b, for b, 1,...,t? T HEOREM Problem P(1),N(1)∣chain, pT is==The first problem Proof. We P(1),N(1)∣chain, pjV =the 1∣C by Π hereafter. The jHEOREM max focuses on sophisticated communication-aware issues. A=comm new jcommunications max,, denoted 1= P(1),N(1)∣chain, p 1∣C Π hereafter. tioned into t comm comm EachScommunication task isMoreover additionally described by paramthere is asbits of information. with each node v T T HEOREM HEOREM 1. Problem 1. Problem 1. P(1),N(1 Problem P(1) max 1 to tasks to tasks in to the tasks in classical the in classical the DAG classical DAG model, DAG model, while model, while all communications while all communications all i hardness hardness of hardness problem of problem of P(1),N(1)∣chain, problem P(1),N(1)∣c P(1),N first problem is defined as follows. . paramcost of communication cost of communication vi paramvthis ,...,a ,athis ,...,a integer ,...,a and bthe ,awith ,...,a intege and alem alem alem partition partition lem [24] partiti decis [2 tow ∑ Each Each communication Each communication communication task sociated istask additionally is task additionally is sociated additionally described described by described paramby by Proof. We theorem athis pseudoP(1),N(1)∣chain, p jcorrespond = prove 1∣C denoted by Πproblem The 1 ,a 3t 1and 2[24] 1 ,a 2to 3ta 1problem 23t a3i i2 partition max 1.byhereafter. 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D determines determines deadline deadline for deadline transmitting for transmitting for transmitting polynomial transformation probfirst problem is defined follows. j max j comm comm comm comm comm comm bHEOREM < aTj3t