Scheduling Shared Continuous Resources On Many-Cores PRESENTER - - PowerPoint PPT Presentation
Scheduling Shared Continuous Resources On Many-Cores PRESENTER PRESENTER PRESENTER PRESENTER: LIOR BELINSKY AUTHORS AUTHORS: ANDRE BRINKMANN - PETER KLING EQ - TIM SUB AUTHORS AUTHORS UUUJJJJJIIILARS NAGEL UUUUU - SORE RIECHERSIU
PRESENTER PRESENTER PRESENTER PRESENTER: LIOR BELINSKY AUTHORS AUTHORS AUTHORS AUTHORS: ANDRE BRINKMANN - PETER KLING EQ - TIM SUB UUUJJJJJIIILARS NAGEL UUUUU - SORE RIECHERSIU
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First Part: Review the process scheduling problem [~30 minutes] Second Part: Algorithms and Approximations [~20 minutes]
Introduction To the CRSHARING Problem Motivation & Usage Hyper Graph Representation Complexity Of The Problem (NPC)
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We consider the problem of scheduling a number of jobs on identical processors sharing a continuously divisible resource. Time is considered discrete and separated by time steps. At every time step the scheduler distributes the resource among the processors. Each processor assigned a share of the resource it can use at time step . For each processor there is a sequence of jobs to process in the given order. the -th job on processor will be denoted as
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Consider a job whose processing started at time step The job arrives with resource requirement
and a process volume (size)
The job is granted with a share of the resource, and thus
Therefore, after time step finishes, the remaining processing volume is:
"
#$ % &$' ! %( %)%*
Goal
finding a resource assignment to processors that minimize the makespan, i.e. the +,-
. +/0
;; Feasible Solution
A schedule consist of resource assignment functions < = that specify the resource’s distributions among the processors for all time steps, without overusing the resource.
In other Words
Our goal is to find a feasible schedule (solution) having a minimal makespan.
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System Resource limit System Resource limit System Resource limit System Resource limit -
K
L )
Per job Per job Per job Per job Resource limit Resource limit Resource limit Resource limit -
job is capped by
" "
) L )
time steps to finish a given set of jobs.
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Consider a job whose processing started at time step The job arrives with resource requirement
and a process volume
The job is granted with a share of the resource, and thus
Therefore, after time step finishes, the remained processing volume is:
"
#$ % &$' ! %( %)%*
Exceed computational performance
Devices or energy consumption are not the only bottleneck of a computation. Distribution of the bandwidth (resource) shared by processors can speed up the computation.
Usage Examples
Many-core systems - chip’s cores share a single data bus to the outside UUUUUUUUUUUUUUIworld. Virtual systems – different virtual machines share a single divisible U UUUUUUUUU resource of a given host system.
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The problem is kind of similar to running machines at the gym
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Meaning Term/Notation
The -th job on processor , ) The share of resource granted to processor at time step The number of jobs that will be processed by processor
() Job (, ) is active in time step if − = − 1 Active job Processoris active in time step if > 0 Active processor The set of all processors having at least j jobs to process S
≔ {| ≥ }
HyperGraph V = (W X consist of a finite set Wof vertices, and edges X which iiis a non-empty subset of V. For example: W = Y YZ [ Y\ X 8 8Z 8] 8^
8 Y YZ Y] 8Z YZ Y] 8] Y] Y_ Y` 8^ .Y^;
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Given a problem instance of CRSHARING with unit size jobs and corresponding iischedule a, we define a weighted HyperGraph V4 = W X named the iischeduling graph of a: W = b ; X 8 8Z [ 84 J a the edge 8% c d is defined as follows: 8% T b ; J W e8fg
W= Jobs X= Time steps 8% Active jobs at time
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Processor 1 Processor 2 Processor 3 W= jobs X= time steps 8% =active jobs at time
The connected components formed by the edges of scheduling graph V4 carry iiia lot of information about the schedule
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Meaning Notation
Number of connected components h The i-th connected component (i ∈ [h]) 5j The number of edges of the i-th component #j The size of the first edge in the i-th component Component class lj
Observation 2 – consider a connected component 5 c d of V4 and two time iiisteps K Z with 8%( m 8%n c o. then for all . [ Z; we have 8% c o.
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Theorem – CRSHARING with jobs of unit size is NP-hard uuuuuuuuuu if the number of processors is part of the input. PROOF Highlight Reduction from the PARTITION problem
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processors 3 jobs on each processor elements PARTITION CRSHARING Reduction Open Question For a constant ≥ p , Does the CRSHARING remain NP-hard?
Round Robin Approximation Unique properties expected of a feasible schedule (solution) Algorithm for 2 processors A (q −
L) approximation for processors
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Let be the maximal number of jobs on a processor. The algorithm operates in phases s.t. during phase it processes the -th job on any Uremained processor.
Theorem - The RoundRobin algorithm for the CRSharing problem with unit sized jobs has a
UUUUUUUUIworst-case approximation ratio of exactly q
PROOF
The -th phase requires exactly time steps. Hence all phases last "
r
R ) K " "
R )
K tu: tu: qtu:
v
s'
v
w
Each Reasonable Schedule for the CRSHARING problem should have the following basic properties: sNon-Wasting – finishes all active jobs during every time step with "
x
L )
Progressive – among all jobs that are assigned resources, at most one job is only partially processed during any time step y more formally: J , bCz A UK Balanced – whenever a processor finishes a job at time step iany processor { with
| > does also finish a job.
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Non-Wasting – finishes all active jobs during every time step with i"
x
L )
Progressive – among all jobs that are assigned resources, at most one job is
more formally: J , bCz A UK Given an arbitrary schedule awe can transform it into a non-wasting and progressive schedule a{ with a| K ay Moreover, the resulting schedule a{ finishes at least one job per time step
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For every balanced schedule, 2 processors with ≥ Z and for all we have : -zZ A K K -zZ A -z -zZ
Balanced – whenever a processor finishes a job at time step I any processor { with
| > does also finish a job.
Lemma Consider a non-wasting, progressive, and balanced schedule. The number of vertexes and edges in a component are related via the following properties: (a) The inequality |5j| ≥ kj lj holds for all k ∈ { 1, 2, . . . ,N − 1 } (b) The last component satisfies |5}| ≥ k}
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The inequality |5j| ≥ kj lj holds for all k ∈ { 1, 2, . . . ,N − 1 }
Let i ∈ ~ − , and let 8
∈ X be the first edge in 5j, i.e.
J8%• c 5j we have K |
Thus, 8
= lj
For all the other edges in 5j, there is at least one new vertex in every edge, since every time step, at least
Therefore:
First row
lj
5j ≥ Anything but First row
+ k€
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The last component satisfies |5}| ≥ k}
Every time step at least one job must finish. Each edge represents all the active jobs in a specific time step. a component contains at least 1 edge (time step). Therefore:
5} ! k}
OptResAssignment algorithm uses a dynamic programming approach. Let • be a two-dimensional array of size ‚Z
IIIhas finished all jobs and q Z for x and Z x Zy Ss ui will be the remaining total resource requirement of and Z, i.e.
S
sF Fz( A Fzn A Iƒ„…†‡ < •
n
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Invariant
At phase ℓ, all entries on the (ℓ−1)-th diagonal (i.e., all B[ , Z] with + Z = ℓ) correspond to subschedules with minimal (and, for this , minimal ) reaching the jobs (1, ) and (2, Z)
Output
Runtime - t Z
O(n) phases – B has Z − diagonals. Each phase considers the t entries on the corresponding diagonal.
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we’ll prove the invariant’s correctness with induction on the phase number: Base: at the first phase •
n , as there are no jobs preceding
q y Step: Assume the invariant holds for the first ℓ phases, and consider an entry
that has iprocessed all jobs preceding ( and q Z y Since each processor can finish, at most, one job per time step, this subschedule must originate from one of the following subschedules:
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L Approximation
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Consider a CRSHARING instance and a feasible schedule a for it that is non-wasting, progressive and balanced. Then a is a q −
L approximation with respect to the optimal makespan
Let kˆdenote the average number of edges in a component. Our proof uses two bounds on the approximation ratio:
GreedyBalance is an example for a greedy algorithm for balanced schedule. the schedule priorities processors with a higher number of remaining jobs, and in case of a tie, prioritizing jobs with larger remaining resource requirement.
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The GreedyBallance algorithm for the CRSHARING problem has a worst-case approximation ratio of exactly q −
L
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What have we seen
New resource scheduling problem, where job processing depends on the share of the resource a job is assigned. Efficient optimal algorithm for 2 processors. Approximation Algorithm with a worst-case approximation ratio of q −
L for processors.
Authors Outlook
Believes that It is possible to find some analytical results with arbitrary job sizes.
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