uet multiprocessor scheduling problems

UET MultiprocessorScheduling Problems

Nan [email protected]

Nan Zang

Overview of the paper

Introduction

Classification

Complexity results for Pm | prec, pj = 1 | Cmax

Algorithms for Pk | prec, pj = 1 | Cmax

Future research directions and conclusions

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Scheduling

Scheduling concerns optimal allocation of scarce resources to activities.

For example Class scheduling problems

Courses Classrooms Teachers

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Problem notation (1)

A set of n jobs J = { J1, J2, ……, Jn}

The execution time of the job Jj is p(Jj).

A set of processors P = {P1 , P2, ……}

Schedule Specify which job should be executed by

which processor at what time.

Objective Optimize one or more performance criteria

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Problem Notation (2)

α describes the processor environment Number of processors, speed, …

β provides the job characteristics release time, precedence constraints, preemption,

…

γ represents the objective function to be optimized

The finishing time of the last job (makespan) The total waiting time of the jobs

3-field notation α|β|γ (Graham et al.)

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Job Characteristics

Release time (rj) - earliest time at which job Jj can start processing.

- available job

Preemption (prmp) - jobs can be interrupted during processing.

Precedence constraints (prec) - before certain jobs are allowed to start processing,

one or more jobs first have to be completed. - a ready job

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Precedence constraints (prec)

Definition Successor Predecessor Immediate successor Immediate

predecessor Transitive Reduction

Before certain jobs are allowed to start processing, one or more jobs first have to be completed.

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Precedence constraints (prec)

Definition Successor Predecessor Immediate successor Immediate

predecessor Transitive Reduction

One or more job have to be completed before another job is allowed to start processing.

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Special precedence constraints (1)

In-tree (Out-tree)

In-forest (Out-forest)

Opposing forest

Interval orders

Quasi-interval orders

Over-interval orders

Series-parallel orders

Level orders

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Special precedence constraints (2)

In-forest

Out-treeIn-tree

Out-forest

Opposing forest

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UET scheduling problem formal definition

Pm| prec, pj = 1 | Cmax (m≥1) Processor Environment

m identical processors are in the system.

Job characteristics Precedence constraints are given by a precedence

graph; Preemption is not allowed; The release time of all the jobs is 0.

Objective function Cmax : the time the last job finishes execution.

( If cj denotes the finishing time of Jj in a schedule S,

) jnjmax cmaxC 1

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Gantt Chart

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A Gantt chart indicates the time each

job spends in execution, as well as

the processor on which it executes

Time axis

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Introduction

Classification

Complexity results for Pm | prec, pj = 1 | Cmax



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Classification

Due to the number of processors

Number of processors is a variable (m) Pm | prec, pj = 1 | Cmax

Number of processors is a constant (k) Pk | prec, pj = 1 | Cmax

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Classification

Due to the number of processors

Number of processors is a variable (m) Pm| prec, pj = 1 | Cmax

Number of processors is a constant (k) Pk| prec, pj = 1 | Cmax

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SP and OF

Series-parallel orders Does NOT have a substructure isomorphic to Fig 1. Opposing-forest orders Is a disjoint union of in-tree orders and out-tree orders.

Fig 2: Opposing forest

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Conclusion on Pm| prec, pj = 1 | Cmax

3SAT is reducible to the corresponding scheduling problem.

m is a function of the number of clauses in the 3SAT problem.

Results and techniques do not hold for the case Pk | prec, pj = 1 | Cmax

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Classification

Number of processors is a variable (m) Pm | prec, pj = 1 | Cmax

Number of processors is a constant (k) Pk | prec, pj = 1 | Cmax

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Optimal Schedule for Pk | prec, pj = 1 | Cmax

The complexity of Pk | prec, pj = 1 | Cmax is open. 8th problem in Garey and Johnson’s open problems list.

(1979) One of the three problems remaining unsolved in that list

If k = 2, P2 | prec, pj = 1 | Cmax is solvable in polynomial time. Fujii, Kasami and Ninomiya (1969) Coffman and Graham (1972)

For any fixed k, when the precedence graph is restricted to certain special forms, Pk | prec, pj = 1 | Cmax turns out to be solvable in polynomial time. In-tree, Out-tree, Opposing-forest, Interval orders…

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Special precedence constraints In-tree (Out-tree)

In-forest (Out-forest)

Opposing forest

Interval orders

Quasi-interval orders

Over-interval orders

Level orders

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List scheduling policiesGraham’s list algorithmHLF algorithmMSF algorithmCG algorithm

FKN algorithm (Matching algorithm)

Merge algorithm

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List scheduling policies (1)

Set up a priority list L of jobs. When a processor is idle, assign the first ready job

to the processor and remove it from the list L.

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List scheduling policiesGraham’s list algorithmHLF algorithmMSF algorithmCG algorithm


Merge algorithm

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Graham’s list algorithm

Graham first analyzed the performance of the simplest list scheduling algorithm.

List scheduling algorithm with an arbitrary job list is called Graham’s list algorithm.

Approximation ratio for Pk | prec, pj = 1 | Cmax

δ = 2-1/k. (Tight!)Approximation ratio is δ if for each input instance, the

makespan produced by the algorithm is at most δ times of the optimal makespan.

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Algorithms for Pk| prec, pj = 1 | Cmax

List scheduling policies

Graham’s list algorithmHLF algorithmMSF algorithmCG algorithm


Merge algorithm

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HLF algorithm (1)

T. C. Hu (1961) Critical Path algorithm or Hu’s algorithm Algorithm

1. Assign a level h to each job. If job has no successors, h(j) equals 1. Otherwise, h(j) equals one plus the maximum

level of its immediate successors.2. Set up a priority list L by nonincreasing order of the

jobs’ levels.3. Execute the list scheduling policy on this level

based priority list L.

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HLF algorithm (2)

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Level 3 Level 2 Level 1

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HLF algorithm (3)

Time complexity O(|V|+|E|) (|V| is the number of jobs and |E| is the number of

edges in the precedence graph)

Theorem 4 (Hu, 1961) The HLF algorithm is optimal for Pk | pj = 1 , in-tree (out-tree) | Cmax.

Corollary 4.1 The HLF algorithm is optimal for Pk | pj = 1 , in-forest (out-forest) |

Cmax.

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HLF algorithm (4)

N.F. Chen & C.L. Liu (1975) The approximation ratio of HLF algorithm for the

problem with general precedence constraints:

If k = 2, δHLF ≤ 4/3.

If k ≥ 3, δHLF ≤ 2 – 1/(k-1).Tight!

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Merge algorithm

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MSF algorithm (1)

Algorithm:Set up a priority list L by nonincreasing order of

the jobs’ successors numbers. (i.e. the job having more successors should have a higher priority in L than the job having fewer successors)

Execute the list scheduling policy based on this priority list L.

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MSF algorithm (2)

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Special precedence constraints

Interval orders Does NOT have a substructure isomorphic to Fig 1. Quasi – interval orders

Does NOT have a substructure isomorphic to TYPE I, II or III.

Type I Type II Type III

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Merge algorithm

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CG algorithm (1)

Coffman and Graham (1972)

An optimal algorithm for P2 | prec, pj = 1 | Cmax.

Best approximation algorithm known for Pk | prec, pj = 1 | Cmax, where k ≥ 3.

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Definitions

– Let IS(Jj) denote the immediate successors set of Jj.

– A job is ready to label, if all its immediate successors are labeled and it has not been labeled yet.

– N(Jj) denotes the decreasing sequence of integers formed by ordering of the set { label (Ji) | Ji IS(Jj) }.

– Let label(Jj) be an integer label assigned to Jj.

CG algorithm (2)

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CG algorithm (3)1. Assign a label to each job:

a. Choose an arbitrary task Jk such that IS(Jk) = 0, and define label(Jk) to be 1

b. for i 2 to n do

I. R be the set of jobs that are ready to label.

II. Let J* be the task in R such that N(J*) is lexicographically smaller than N(J) for all J in R

III. Let label(J*) i

end for

2. Construct a list of jobs L = {Jn, Jn-1,…, J2, J1} according to the decreasing order of the labels of the jobs.

3. Execute the list scheduling policy on this priority list L.

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CG algorithm (4)

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CG algorithm (4)

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Special precedence constraints

Quasi – interval ordersDoes NOT have a substructure isomorphic to TYPE I, II or III.

Over – interval ordersDoes NOT have a substructure isomorphic to TYPE I or II.

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CG algorithm (6)

The performance of CG algorithm when k≥3

Lam and Sethi (1978) δCG ≤ 2 – 2/k

Braschi and Trystram (1994) Cmax(S) ≤ (2 – 2/k) Cmax(S*) – (k – 2 – odd(k))/k (tight!)

Note: S is a CG schedule. S* is an optimal schedule. If k is an odd, odd(k):=1; otherwise, odd(k):=0.

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List Scheduling Policy Conclusions


Graham’s list algorithm HLF algorithm MSF algorithm CG algorithm

Property easy to implement extended to the problem Pm | prec, pj = 1|

Cmax

Research directions:

Allow priority lists to depend on the number k of processors.

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List scheduling policy algorithm



Merge algorithm

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FKN algorithm (1)

Fujii, Kasami and Ninomiya (1969) First optimal algorithm for P2 | prec, pj = 1 | Cmax. Basic Idea

Find a minimum partition of the jobs• There are at most two jobs in each set.• The pair of jobs in the same set can be executed

together. Make a valid schedule according to a particular order

of the partition. (Some clever swap work needed!) The length of the result schedule = value of the min

partition Can be solved by some maximum matching

algorithm.

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FKN algorithm (2) Hard to extend! FKN algorithm cannot be extended to k = 3 directly.

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Minimal partition is {J1, J5, J6} {J4, J2, J3} and |P|=2.

However, The optimal Cmax corresponds partition {J1, J4} {J2,J3,J5} {J6}

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Merge algorithm

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Merge Algorithm (1)

Dolev and Marmuth (1985)

Required input:

An optimal schedule S for a high-graph H(G).

Merge algorithms show how to “ Merge” the known optimal schedule S with the remaining jobs.

Produce an optimal schedule for the whole job set G.

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Merge Algorithm (2)

Definitions height h(G) highest level of the vertices in G.

median μ(G) height of kth highest component +1

high-graph H(G) a subgraph of G, made up of all the components

which are strictly higher than the median. low-graph L(G) remaining subgraph of G, except for H(G)

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Merge Algorithm (3)

C1 C2 C3 C4

H(C1)=5 H(C2)=3 H(C3)=3 H(C4)=2

K=3 3rd highest

μ(G)=4

H(G) L(G)

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Merge Algorithm (4)

Idea of the Merge Algorithm If there is an idle period in S, then fill it

with a highest initial vertices in L(G), and remove it from L(G)

Similar to HLF Algorithm

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Merging Algorithm (5)

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Merge Algorithm (6)

Theorem 10 (Reduction theorem)

Let G be a precedence graph and S be an optimal schedule for H(G). Then, the Merge algorithm finds an optimal schedule for the whole graph G in time and space O(|V|+|E|).

Corollary 10

If H(G) is empty, then HLF is optimal for G.

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Merge Algorithm (7)

Why Merge algorithm is useful?

1. Find an optimal schedule for a subgraph H(G).

2. H(G) contains fewer than k-1 components.

Precedence Constraints

Time complexity

Level orders O(nk-1)

Opposing forest O(n2k-2logn)

Bounded height O(nh(k-1))

Dolev and Marmuth

How to use it?1. If H(G) is easy to solve.2. If every closed subgraph

of G can be classified into polynomial number of classes.

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Merge algorithm

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Main results known

LIST

Approximation ratio

K=2

K≥3

Intree OFInterval

Quasi-interval

Over-interval

Arbitrary

List 3/2 2-1/k

HLF 4/3 opt 2-1/(k-1)

MSF 4/3 opt opt≥2-1/(k+1)

CG opt opt opt opt opt ≈2-2/k

FKN opt

Merge opt

(Opt: We can get optimal solution in polynomial time. )

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Introduction

Classification

Complexity result for Pm | prec, pj = 1 | Cmax



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Future research directions (1)

Finding a new class of orders which can be solved by known algorithms or their generalizations.

over-interval (Moukrim, 2005)

Find an algorithm with a better approximation ratio for the UET scheduling problem.

CG algorithm (1972) Ranade (2003)

special precedence constraints (loosely connected task graphs )

δ = 1.875

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Future research directions (2)

Use the UET multiprocessor scheduling algorithms to solve other related scheduling problems CG algorithm is optimal for P2 | rj, pj = 1| Cmax.

HLF algorithm is optimal for P | rj, pj = 1, outtree| Cmax.

(Huo and Leung, 2005) MSF algorithm is optimal for P | pj = 1, cm=1, quasi-interval| Cmax.

(Moukrim, 2003) CG algorithm is optimal for P2| prmp, pj = 1| Cmax and ∑Cj

(Coffman, Sethuraman and Timkovsky, 2003)

The most challenging research task: Solve the famous open problem Pk |pj = 1| Cmax (k≥3).

Thank You!

uet multiprocessor scheduling problems

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