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Minimizing Average Flow-Time Naveen Garg IIT Delhi Joint work with Amit Kumar, Jivi Chadha,V Muralidhara, S. Anand

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Problem Definition Given : A set of M machines A set of jobs A matrix of processing times of job i on machine j. Each job specifies a release date

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Problem Definition Conditions : Pre-emption allowed Migration not allowed

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Problem Definition Flow-time of j, Goal : Find a schedule which minimizes the average flow-time

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Special Cases Parallel : all machines identical Related : machines have different speeds Subset Parallel : parallel except that a job can only go on a subset of machines Subset Related All of these are NP-hard.

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Previous work The problem is well studied when the machines are identical. For a single machine the Shortest- remaining-processing-time (SRPT) rule is optimum. [Leonardi-Raz 97] argued that for parallel machines SRPT is (min (log n/m, log P)) competitive, where P is max/min processing time. They also show a lower bound of (log P) on competitive ratio

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Preemptive, unweighted Flow time OnlineOffline Parallel machines O(log P), (log P) [LR97] (log 1-ε P) [GK07] Related machines O(log P) [GK06] Subset parallel Unbounded [GK07] O(log P) (log P/loglogP) [GK07] Unrelated machines O(k) [S09]

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Fractional flow-time p j (t ) = remaining processing of job j at time t remaining fraction at time t = Fractional flow-time of j = t r j p j (t )/p j Recall, flow-time of j =

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Fractional flow-time 1512 Fractional flow-time = 1*2 + 2/3*3 + 1/3*7 Fractional flow-time can be much smaller than (integral) flow-time 20

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Integer Program Define 0-1 variables : x(i,j,t) : 1 iff job j processed on i during [t,t+1] Write constraints and objective in terms of these variables. Fractional flow-time of j = t r j ( t-r j ) x(i,j,t)/ p ij

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LP Relaxation One Caveat …

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Fractional flow-time A job can be done simultaneously on many machines : flow-time is almost 0

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LP Relaxation Add a term for processing time

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Class of a job : The processing time of j rounded up to nearest power of 2 If, we say k is the class of job j Number of different classes = O(log P)

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Modified Linear Program

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Modified LP LP value changes by a constant factor only. But : rearranging jobs of the same class does not change objective value.

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From fractional to integral The solution to the LP is not feasible for our (integral) problem since it schedules the same job on multiple m/cs. We now show how to get a feasible, non- migratory schedule.

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Rounding the LP solution Consider jobs of one class, say blue. Find the optimum solution to the LP.

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Rounding the LP solution (contd.) Rearrange blue jobs in the space occupied by the blue jobs so that each job is scheduled on only one m/c. If additional space is needed it is created at the end of the schedule Additional space

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Preemptive, unweighted Flow time OnlineOffline Parallel machines O(log P), (log P) (log 1-ε P) Related machines O(log P) Subset parallelUnboundedO(log P) (log P/loglogP) Unrelated machines O(k)

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Assignment as flow Fix a class k : arrange the jobs in ascending order of release dates. r1r1 0 0 i v(i,k,j) s Flow = ? r7r7 r6r6 r5r5 r4r4 r3r3 r2r2

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Unsplittable Flow Problem s d1 d2 d3

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Unsplittable Flow Problem s d1 d2 d3 Flow can be converted to an unsplittable flow such that excess flow on any edge is at most the max demand [Dinitz,Garg, Goemans]

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Back to scheduling... Fix a class k : find unsplittable flow i v(i,j,k) s Gives assignment of jobs to machines

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Back to scheduling... J(i,k) : jobs assigned to machine i i v(i,j,k) s Can we complete J(i,k) on class k slots in I ?

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Building the Schedule Flow increases by at most max processing time = 2 k So all but at most 2 jobs in J(i,k) can be packed into these slots Extra slots are created at the end to accommodate this spillover

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Increase in Flow-time How well does capture the flow-time of j ? Charge to the processing time of other classes

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Finally... Since there are only log P classes… Can get OPT + O(log P).processing time flow-time for subset parallel case.

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Preemptive, unweighted Flow time OnlineOffline Parallel machines O(log P), (log P) (log 1-ε P) Related machines O(log P) Subset parallelUnboundedO(log P) (log P/loglogP) Unrelated machines O(k)

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Integrality Gap for our LP(identical m/c) 0 m k-1 2m k-1 mkmk +m k-1 mkmk m k-1 m T 1 Phase0 1k-1k

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Integrality Gap for our LP(identical m/c) For sufficiently large T, flow time mT(1+k/2) 0 m k-1 2m k-1 mkmk +m k-1 T Phase0 1k-1k Blue jobs can be scheduled only in this area of volume (m k +m k-1 )m/2 At least m/2 blue jobs left At least mk/2 jobs left

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Integrality Gap for our LP(identical m/c) Optimum fractional solution is roughly mT 0 m k-1 2m k-1 mkmk +m k-1 mkmk m k-1 m T 1 Phase0 1k-1k

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Integrality gap Optimum flow time is at least mT(1+k/2) Optimum LP solution has value roughly mT So integrality gap is (k). Largest job has size P = m k. For k = m c, c>1, we get an integrality gap of (log P/loglogP)

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Hardness results We use the reduction from 3-dimensional matching to makespan minimization on unrelated machines [lenstra,shmoys,tardos] to create a hard instance for subset-parallel. Each phase of the integrality gap example would have an instance created by the above reduction. To create a hard instance for parallel machines we do a reduction from 3- partition.

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Preemptive, unweighted Flow time OnlineOffline Parallel machines O(log P), (log P) (log 1-ε P) Related machines O(log P) Subset parallelUnboundedO(log P) (log P/loglogP) Unrelated machines O(k)

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A bad example 0 T B x A x A+B=T A> T/2 T+L A x Flow time is at least AxL > T L/2 OPT flow time is O(T 2 +L) Ω(T) lower bound on any online algorithm

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Other Models What if we allow the algorithm extra resources ? In particular, suppose the algorithm can process (1+ε) units in 1 time-unit. [first proposed by Kalyanasundaram,Pruhs95] Resource Augmentation Model

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Resource Augmentation For a single machine, many natural scheduling algorithms are O(1/ O(1) )- competitive with respect to any L p norm [Bansal Pruhs 03] Parallel machines : randomly assign each job to a machine – O(1/ O(1) ) competitive [Chekuri, Goel, Khanna, Kumar 04] Unrelated Machines : O(1/ 2 )-competitive, even for weighted case. [Chadha, Garg, Kumar, Muralidhara 09]

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Our Algorithm When a job arrives, we dispatch it to one of the machines. Each machine just follows the optimal policy : Shortest Remaining Processing Time (SRPT) What is the dispatch policy ? GREEDY

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p j 1 (t ) The dispatch policy When a job j arrives, compute for each machine i the increase in flow-time if we dispatch j to i. j1j1 j2j2 jrjr j r+1 jsjs j arrives at time t : p ij 1 (t) p ij 2 (t) … p ij r (t) < p ij < p ij r+1 (t) j Increase in flow-time = p j 1 (t ) + … + p j r (t ) + p ij + p ij (s-r)

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Our Algorithm When a job j arrives, compute for each machine i the increase in flow-time if we dispatch j to i. Dispatch j to the machine for which increase in fractional flow-time is minimum.

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Analyzing our algorithm Primal LPDual LP LP opt. value Algorithms value Construct a dual solution Show that the dual solution value and algorithms flow-time are close to each other.

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Dual LP αjαj β it

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Dual LP

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p j 1 (t ) α j = p j 1 (t ) + … + p j r (t ) + p ij + p ij (s-r) Setting the Dual Values When a job j arrives, set α j to the increase in flow- time when j is dispatched greedily. j1j1 j2j2 jrjr j r+1 jsjs j arrives at time t : p ij 1 (t) p ij 2 (t) … p ij r (t) < p ij < p ij r+1 (t) Thus j α j is equal to the total flow- time.

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β it = s Setting the Dual Values j1j1 j2j2 jrjr j r+1 jsjs p j 1 (t ) Set β it to be the number of jobs waiting at time t for machine i. Thus i,t β it is equal to the total flow- time.

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Need to verify Dual Feasibility j1j1 j2j2 jljl j l+1 jsjs p j 1 (t ) Fix a machine i, a job j and time t. Suppose p ij l (t) < p ij < p ij l+1 (t)

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Dual Feasibility j1j1 j2j2 jljl j l+1 jsjs p j 1 (t ) What happens when t = t ?

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Dual Feasibility j1j1 j2j2 jljl j l+1 jsjs δ What happens when t = t + δ? Suppose at time t job j k is being processed Case 1: k l

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Dual Feasibility j2j2 jrjr j r+1 jsjs Case 2: k > l ¢

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Dual Feasibility So, α j, β it are dual feasible But i,t β it and j α j both equal the total flow time and hence the dual objective value is Hence, for any machine i, time t and job j

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Incorporating machine speed-up So the values α j, β it /(1+ ε) are dual feasible for an instance with processing times larger by a factor (1+ ε) For any machine i, time t and job j Equivalently, schedule given instance on machines of speed (1+ ε) to determine α j, β it. The values α j, β it /(1+ ε) are dual feasible.

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Dual Objective Value The dual value is less than the optimum fractional flow time. Since i,t β it = j α j the value of the dual is Hence, the flow time of our solution, j α j, is at most (1+1/ ε) times the optimum fractional flow time.

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Extensions Can extend this analysis to the L p -norm of the flow time to get a similar result. Analysis also extends to the case of minimizing sum of flow time and energy on unrelated machines.

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Open Problems Single Machine : Constant factor approximation algorithm for weighted flow-time. loglog n approx [Bansal Pruhs 10] 2+ε quasi polynomial time algorithm [Chekuri Khanna Zhu 01]

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Open Problems Parallel machines : Constant factor approximation algorithm if we allow migration of a job from one machine to another. The (log 1-ε P) hardness is for non- migratory schedules

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Open Problems Unrelated Machines : poly-log approximation algorithm (LP integrality gap ?) O(k) approximation [Sitters 08] is known, where k is the number of different processing times.

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Thank You

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