Garg Slides

7/30/2019 Garg Slides

1/58

Minimizing Average Flow-Time

Naveen Garg

IIT Delhi

Joint work with Amit Kumar, Jivi Chadha ,V Muralidhara, S. Anand


2/58

Problem Definition

Given : A set of M machinesA set of jobs

A matrix of processing times of

job ion machine j.Each job specifies a release date

pij

rj

rj


3/58

Problem Definition

Conditions :

rj

rj Pre-emption allowed

Migration not allowed


4/58

Problem Definition

rj Cj

Flow-time of j,

Goal : Find a schedule which minimizes the average flow-time

Fj= Cj rj


5/58

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

p i j= p j

p i j= p j/si

pij pj

All of these are NP-hard.


6/58

Previous work

The problem is well studied when themachines are identical.

For a single machine the Shortest-

remaining-processing-time (SRPT) rule isoptimum.

[Leonardi-Raz 97] argued that for parallel

machines SRPT is O(min (log n/m, log P))competitive, where P is max/minprocessing time. They also show a lowerbound of(log P) on competitive ratio


7/58

Preemptive, unweighted Flow

time

Online Offline

Parallel

machines

O(log P),

(log P) [LR97]

(log1- P)

[GK07]

Related

machines

O(log P) [GK06]

Subset

parallel

Unbounded

[GK07]

O(log P)

(log P/loglogP)

[GK07]

Unrelated

machines

O(k) [S09]


8/58

Fractional flow-time

pj(t) = remaining processing of job j at time t

remaining fraction at time t =

timeprocessingtotal

(t)pj

Fractional flow-time of j = trj

pj(t)/pj

Recall, flow-time of j =

j

j

C

rt

1


9/58


1 5 12

Fractional flow-time = 1*2 + 2/3*3 + 1/3*7

Fractional flow-time can be much smaller than(integral) flow-time

20


10/58

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 rj (t-rj) x(i,j,t)/pij


11/58

LP Relaxation

0t)j,x(i,

ti,allfor1t)j,x(i,

jallfor1p

t)j,x(i,

)r(tp

t)j,x(i,Min

j

ti, ij

jtj,i, ij

One Caveat


12/58


A job can be done simultaneously on many

machines : flow-time is almost 0


13/58

LP Relaxation

0t)j,x(i,

ti,allfor1t)j,x(i,

jallfor1p

t)j,x(i,

t)j,x(i,)r(tp

t)j,x(i,Min

j

ti, ij

tj,i,j

tj,i, ij

Add a term for processing time


14/58

Class of a job

pj

: The processing time of j rounded up

to nearest power of 2

pj

2k

If , we say k is the class of job j

Number of different classes = O(log P)


15/58

Modified Linear Program

0t)j,x(i,

ti,allfor1t)j,x(i,

jallfor1p

t)j,x(i,

t)j,x(i,)r(tp

t)j,x(i,Min

j

ti, ij

tj,i,j

tj,i, ij


16/58

ModifiedLP

LP value changes by a constant factor only.

But : rearranging jobs of the same class does

not change objective value.


17/58

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.


18/58

Rounding the LP solution

Consider jobs of one class, say blue.

Find the optimum solution to the LP.


19/58

Rounding the LP solution

(contd.)

Rearrange blue jobs in the space occupied

by the blue jobs so that each job isscheduled on only one m/c.

If additional space is needed it is createdat the end of the schedule

Additional

space


20/58


time

Online Offline

Parallel

machines

O(log P),

(log P) (log1- P)

Related

machines

O(log P)

Subset parallel Unbounded O(log P)

(log P/loglogP)

Unrelated

machines

O(k)


21/58

Assignment as flow

Fix a class k : arrange the jobs in ascending order ofrelease dates.

r10

iv(i,k,j)

z i , j t

x i , j , t

p j

s

Flow = ?

r7

r6r5r4r3r2


22/58

Unsplittable Flow Problem

s

d1 d2

d3


23/58

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]


24/58

Back to scheduling...

Fix a class k : find unsplittable flow

10 2 3 4 5 6 7

iv(i,j,k) p j

s

Gives assignment of jobs to machines


25/58

Back to scheduling...

J(i,k) : jobs assigned to machine i

10 2 3 4 5 6 7

iv(i,j,k) p j

s

Can we complete J(i,k) on class k slots in I ?


26/58

Building the Schedule

Flow increases by at most

max processing time = 2k

So all but at most 2 jobs in J(i,k) can be packedinto these slots

Extra slots are created at the end to accommodate

this spillover


27/58

Increase in Flow-time

How well does

capture the flow-time of j ?

Charge to the processing time of other classes

1ijj

ti, p)r(tt)j,x(i,


28/58

Finally...

Since there are only log P classes

Can get OPT + O(log P).processing time

flow-time for subset parallel case.


29/58


time

Online Offline

Parallel

machines

O(log P),

(log P) (log1- P)

Related

machines

O(log P)


(log P/loglogP)

Unrelated

machines

O(k)


30/58

Integrality Gap for our

LP(identical m/c)

0 mk-1 2mk-1 mk +mk-1

mk

mk-1

m

T

1

Phase 0 1 k-1 k


31/58


LP(identical m/c)

For sufficiently large T, flow time mT(1+k/2)

0 mk-1 2mk-1 mk +mk-1T

Phase 0 1 k-1 k

Blue jobs can be scheduled only in this area of volume (mk+mk-1)m/2

At least m/2 blue jobs left At least mk/2 jobs left


32/58


LP(identical m/c)

Optimum fractional solution is roughly mT

0 mk-1 2mk-1 mk +mk-1

mk

mk-1

m

T

1

Phase 0 1 k-1 k


33/58

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 = mk.

For k = mc, c>1, we get an integrality gap of

(log P/loglogP)


34/58

Hardness results

We use the reduction from 3-dimensionalmatching to makespan minimization onunrelated machines [lenstra,shmoys,tardos]to create a hard instance for subset-parallel.

Each phase of the integrality gap examplewould have an instance created by theabove reduction.

To create a hard instance for parallelmachines we do a reduction from 3-partition.


35/58


time

Online Offline

Parallel

machines

O(log P),

(log P) (log1- P)

Related

machines

O(log P)


(log P/loglogP)

Unrelated

machines

O(k)


36/58

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(T2+L)

(T) lower bound on any online

algorithm


37/58

Other Models

What if we allow the algorithm extra

resources ?

In particular, suppose the algorithm can

process (1+) units in 1 time-unit. [first

proposed by Kalyanasundaram,Pruhs95]

Resource Augmentation Model


38/58

Resource Augmentation

For a single machine, many natural

scheduling algorithms are O(1/eO(1))-

competitive with respect to any Lp norm[Bansal Pruhs 03]

Parallel machines : randomly assign each

job to a machine O(1/eO(1)) competitive[Chekuri, Goel, Khanna, Kumar 04]

Unrelated Machines : O(1/e2)-competitive, evenfor weighted case. [Chadha, Garg, Kumar, Muralidhara 09]


39/58

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


40/58

pj1(t)

The dispatch policy

When a job j arrives, compute for each machine ithe increasein flow-time if we dispatch j to i.

j1

j2

jr

jr+1

js

j arrives at time t : pij1(t) pij2(t)

pijr(t) < pij < pijr+1(t)

j

Increase in flow-time =

pj1(t) + + pjr(t) + pij+ pij(s-r)


41/58

Our Algorithm

When a job j arrives, compute for each machine ithe increasein flow-time if we dispatch j to i.

Dispatch j to the machine for which

increase in fractional flow-time is minimum.


42/58

Analyzing our algorithm

Primal LPDual LP

LP opt. valueAlgorithms value

Construct a dual

solution

Show that the dual solution value and algorithms

flow-time are close to each other.


43/58

Dual LP

0t)j,x(i,

ti,allfor1t)j,x(i,

jallfor1p

t)j,x(i,

t)j,x(i,)r(tp

t)j,x(i,Min

j

ti, ij

tj,i,j

tj,i, ij

j

it


44/58

Dual LP

0,

rtt,j,i,allfor1prt

p

Max

itj

jij

jit

ij

j

ti,it

jj


45/58

pj1(t)

j=

pj1(t) + +

pjr(t) +

pij+

pij(s-r)

Setting the Dual Values

When a job j arrives, set j to the increase in flow-time when j is dispatched greedily.

j1

j2

jr

jr+1

js

j arrives at time t : pij1

(t) pij2

(t)

pijr(t) < pij < pijr+1(t)

Thus jj is equal to the total flow-

time.


46/58

it = s

Setting the Dual Values

j1

j2

jr

jr+1

js

pj1(t)

Set it to be the number of jobs waiting at time t for

machine i.

Thus i,tit is equal to the total flow-time.


47/58

Need to verify

Dual Feasibility

j1

j2

jl

jl+1

js

pj1(t)

)( ls ji'ji'jjj pp(t)p...(t)p l1

Fix a machine i, a job j and time t.

1p tt'p ji't'i'

ji'

j

Suppose pijl(t) < pij < pijl+1(t)


48/58

Dual Feasibility

j1

j2

jl

jl+1

js

pj1(t) lsji'ji'jjj

pp(t)p...(t)pl1

11p

(t)p...(t)p

p ti'ji'

jj

ji'

j l1

sls

1

What happens when t = t ?


49/58

Dual Feasibility

j1

j2

jl

jl+1

js

t'i'ji'ji'

ji'

jj

ji'

jjjj

ji'

j

1p

t)(t'1)k-(s1

p

t)(t'

1

p

(t)p...(t)pt)(t'

1p

(t)p...(t)p(t)p...(t)p

p

lk

lk1-k1

ls

ls

What happens when t = t + ?Suppose at time t job jk is being processed

Case 1: k l


50/58

Dual Feasibility

j2

jr

jr+1

js

t'i'ji'ji'

ji'

jjjj

ji'

ji'ji'jj

ji'

j

1p

t)(t'1)k-(s1

p

t)(t'

1

p

(t)p...(t)p(t)p...(t)p

p

pp(t)p...(t)p

p

1-k1lr1

l1

ks

ls

1

Case 2: k > l


51/58

Dual Feasibility

So, j, it are dual feasible

But i,tit and jj both equal the total flow time

and hence the dual objective value is

1p

t)(t'

p ji't'i'

ji'

j

Hence, for any machine i, time t and job j

0,

, ti

ti

j

j


52/58

Incorporating machine speed-up

So the values j, it/(1+) are dual feasible for an

instance with processing times larger by a factor (1+)

1

p

t)(t'

p ji'

t'i'

ji'

j

eee 111

For any machine i, time t and job j

Equivalently, schedule given instance on machines ofspeed (1+) to determine j, it. The values j, it/(1+)

are dual feasible.


53/58

Dual Objective Value

The dual value is less than the optimum fractional

flow time.

j

j

ti

ti

j

j e

e

e

11,

,

Since i,tit = jj the value of the dual is

Hence, the flow time of our solution, jj, is at most(1+1/) times the optimum fractional flow time.


54/58

Extensions

Can extend this analysis to the Lp-norm of

the flow time to get a similar result.

Analysis also extends to the case of

minimizing sum of flow time and energy onunrelated machines.


55/58

Open Problems

Single Machine :

Constant factor approximation algorithm for

weighted flow-time.

loglog n approx [Bansal Pruhs 10]

2+ quasi polynomial time algorithm [Chekuri Khanna

Zhu 01]


56/58

Open Problems

Parallel machines :

Constant factor approximation algorithm if

we allow migration of a job from one

machine to another.

The (log1- P) hardness is for non-

migratory schedules


57/58

Open Problems

Unrelated Machines :

poly-log approximation algorithm

(LP integrality gap ?)

O(k) approximation [Sitters 08] is known,

where k is the number of differentprocessing times.


58/58

Thank You

Garg Slides

Documents

Transcript of Garg Slides