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Space-Efficient Scheduling of Stochastically Generated Tasks Tom Brzdil 1 Javier Esparza 2 Stefan Kiefer 3 Michael Luttenberger 2 1 Masaryk University, Brno (Czech Republic) 2 TU Mnchen (Germany) 3 University of Oxford (UK) ICALP ,

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SLIDE 1

Space-Efficient Scheduling of Stochastically Generated Tasks

Tomáš Brázdil1 Javier Esparza2 Stefan Kiefer3 Michael Luttenberger2

1Masaryk University, Brno (Czech Republic) 2TU München (Germany) 3University of Oxford (UK)

ICALP , Bordeaux (July 2010)

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 2

Task Systems

We study tasks that stochastically generate new tasks. The execution of a task can generate subtasks. Examples: Divide-and-conquer Algorithms: Either solve the problem directly or solve sub-instances. Multi-Threaded Programs: Either terminate, or spawn new thread, or none of that.

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 3

Task Systems

We study tasks that stochastically generate new tasks. The execution of a task can generate subtasks. Examples: Divide-and-conquer Algorithms: Either solve the problem directly or solve sub-instances. Multi-Threaded Programs: Either terminate, or spawn new thread, or none of that.

How much memory is needed?

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 4

Task Systems

Formally, task systems are like context-free grammars. X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 5

Task Systems

Formally, task systems are like context-free grammars. X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X X Y X Y Y Y X Y Y 0.2 0.2 0.3 0.8 0.1 0.6 0.3 0.8 0.6 0.6

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 6

Task Systems

Formally, task systems are like context-free grammars. X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X X Y X Y Y Y X Y Y 0.2 0.2 0.3 0.8 0.1 0.6 0.3 0.8 0.6 0.6 Probability of this tree: 0.2 · 0.2 · 0.3 · 0.8 · 0.1 · 0.6 · 0.3 · 0.8 · 0.6 · 0.6

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 7

A Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 8

A Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X X Y X Y Y Y X Y Y X

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 9

A Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X X Y X Y Y Y X Y Y X ⇒ XY

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 10

A Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y Y X Y Y X ⇒ XY ⇒ XXY XXY

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 11

A Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XXY ⇒ YY

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 12

A Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XXY ⇒ YY ⇒ XY

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 13

A Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XXY ⇒ YY ⇒ XY ⇒ Y

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 14

A Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XXY ⇒ YY ⇒ XY ⇒ Y ⇒ YY

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 15

A Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XXY ⇒ YY ⇒ XY ⇒ Y ⇒ YY ⇒ Y

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 16

A Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XXY ⇒ YY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ YY

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 17

A Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XXY ⇒ YY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ YY ⇒ Y

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 18

A Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XXY ⇒ YY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ YY ⇒ Y ⇒ ε

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 19

A Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XXY ⇒ YY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ YY ⇒ Y ⇒ ε Time: 10 (number of nodes) Space: 3

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 20

A Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XXY XXY ⇒ YY ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ YY ⇒ Y ⇒ ε Time: 10 (number of nodes) Space: 3 3

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 21

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 22

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X X Y X Y Y Y X Y Y X

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 23

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X X Y X Y Y Y X Y Y X ⇒ XY

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 24

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X X Y X Y Y Y Y X Y Y X ⇒ XY ⇒ XYY

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 25

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X X Y X Y Y Y Y X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY XYYY

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 26

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X X Y X Y Y Y Y X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY ⇒ XYY

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 27

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X X Y X Y Y Y Y X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY ⇒ XYY ⇒ XY

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 28

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY ⇒ XYY ⇒ XY ⇒ X

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 29

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY ⇒ XYY ⇒ XY ⇒ X ⇒ XY

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-30
SLIDE 30

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY ⇒ XYY ⇒ XY ⇒ X ⇒ XY ⇒ XX

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 31

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY ⇒ XYY ⇒ XY ⇒ X ⇒ XY ⇒ XX ⇒ X

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-32
SLIDE 32

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY ⇒ XYY ⇒ XY ⇒ X ⇒ XY ⇒ XX ⇒ X ⇒ ε

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-33
SLIDE 33

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY ⇒ XYY ⇒ XY ⇒ X ⇒ XY ⇒ XX ⇒ X ⇒ ε Time: 10 (number of nodes) Space: 4

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-34
SLIDE 34

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY XYYY ⇒ XYY ⇒ XY ⇒ X ⇒ XY ⇒ XX ⇒ X ⇒ ε Time: 10 (number of nodes) Space: 4 4

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-35
SLIDE 35

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY XYYY ⇒ XYY ⇒ XY ⇒ X ⇒ XY ⇒ XX ⇒ X ⇒ ε Time: 10 (number of nodes) Space: 4 4

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

Each finite tree has its Probability (does not depend on scheduler) Time (does not depend on scheduler) Space (depends on scheduler)

slide-36
SLIDE 36

Another Derivation

X ֒ − → XY X ֒ − → ε Y ֒ − → YY Y ֒ − → X Y ֒ − → ε X X Y X Y Y Y X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY XYYY ⇒ XYY ⇒ XY ⇒ X ⇒ XY ⇒ XX ⇒ X ⇒ ε Time: 10 (number of nodes) Space: 4 4

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

Each finite tree has its Probability (does not depend on scheduler) Time (does not depend on scheduler) Space (depends on scheduler) Raises questions like: What is the expected time and the expected space? How are these random variables distributed? Time has been studied before. We focus on space.

slide-37
SLIDE 37

Related Work: Branching Processes

X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X X Y X Y Y Y X Y Y

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-38
SLIDE 38

Related Work: Branching Processes

X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X X Y X Y Y Y X Y Y

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-39
SLIDE 39

Related Work: Branching Processes

X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X X Y X Y Y Y X Y Y

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-40
SLIDE 40

Related Work: Branching Processes

X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X X Y X Y Y Y X Y Y

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-41
SLIDE 41

Related Work: Branching Processes

X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X X Y X Y Y Y X Y Y

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-42
SLIDE 42

Related Work: Branching Processes

X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X X Y X Y Y Y X Y Y

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-43
SLIDE 43

Related Work: Branching Processes

X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X X Y X Y Y Y X Y Y Branching Processes have been extensively studied. They are models for biological or physical systems. But they assume an unbounded number of “processors”.

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-44
SLIDE 44

Termination Probability

X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X X Y X Y Y Y X Y Y 0.2 0.2 0.3 0.8 0.1 0.6 0.3 0.8 0.6 0.6 Each tree has its probability. The sum of these probabilities is the “termination probability”. Is it always 1?

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-45
SLIDE 45

Termination Probability and the function f

A task system induces a vector f(x). For our example: x = x y

  • and f(x) =

fx(x, y) fy(x, y)

  • with

X

0.2

֒ − → XY X

0.8

֒ − → ε

  • fx(x, y) = 0.2xy + 0.8

Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε        fy(x, y) = 0.3y2 + 0.1x + 0.6

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-46
SLIDE 46

Termination Probability and the function f

A task system induces a vector f(x). For our example: x = x y

  • and f(x) =

fx(x, y) fy(x, y)

  • with

X

0.2

֒ − → XY X

0.8

֒ − → ε

  • fx(x, y) = 0.2xy + 0.8

Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε        fy(x, y) = 0.3y2 + 0.1x + 0.6 Proposition (well-known, see [Harris]) The termination probability is the (first component of the) least fixed point of f.

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-47
SLIDE 47

The function f

The subcritical case: termination probability = 1 expected time = finite

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2

f(x) = 0.4x2 + 0.6

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X

0.4

֒ − → XX X

0.6

֒ − → ε

slide-48
SLIDE 48

The function f

The supercritical case: termination probability < 1 expected time = ∞

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2

f(x) = 0.7x2 + 0.3

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X

0.7

֒ − → XX X

0.3

֒ − → ε

slide-49
SLIDE 49

The function f

The critical case: termination probability = 1 expected time = ∞

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2

f(x) = 0.5x2 + 0.5

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X

0.5

֒ − → XX X

0.5

֒ − → ε

slide-50
SLIDE 50

The function f

The critical case: termination probability = 1 expected time = ∞

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2

f(x) = 0.5x2 + 0.5

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

Assumption We assume termination probability = 1 in the following, i.e., subcritical or critical. X

0.5

֒ − → XX X

0.5

֒ − → ε

slide-51
SLIDE 51

Optimal Scheduling

Given a tree t with two children t0, t1. What is the optimal scheduling? t t0 t1 Sop(t) = max

  • Sop(t0) + 1 , Sop(t1)
  • ,

max

  • Sop(t1) + 1 , Sop(t0)
  • Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger

Space-Efficient Scheduling of Stochastically Generated Tasks

slide-52
SLIDE 52

Optimal Scheduling

Given a tree t with two children t0, t1. What is the optimal scheduling? t t0 t1 Sop(t) = max

  • Sop(t0) + 1 , Sop(t1)
  • ,

max

  • Sop(t1) + 1 , Sop(t0)

        

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-53
SLIDE 53

Optimal Scheduling

Given a tree t with two children t0, t1. What is the optimal scheduling? t t0 t1 Sop(t) = max

  • Sop(t0) + 1 , Sop(t1)
  • ,

max

  • Sop(t1) + 1 , Sop(t0)

        

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-54
SLIDE 54

Optimal Scheduling

Given a tree t with two children t0, t1. What is the optimal scheduling? t t0 t1 Sop(t) = max

  • Sop(t0) + 1 , Sop(t1)
  • ,

max

  • Sop(t1) + 1 , Sop(t0)

        

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-55
SLIDE 55

Optimal Scheduling

Given a tree t with two children t0, t1. What is the optimal scheduling? t t0 t1 Sop(t) = max

  • Sop(t0) + 1 , Sop(t1)
  • ,

max

  • Sop(t1) + 1 , Sop(t0)

        

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-56
SLIDE 56

Optimal Scheduling

Given a tree t with two children t0, t1. What is the optimal scheduling? t t0 t1 Sop(t) = min max

  • Sop(t0) + 1 , Sop(t1)
  • ,

max

  • Sop(t1) + 1 , Sop(t0)

        

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 57

Optimal Scheduling

Given a tree t with just one child t0. What is the optimal scheduling? t t0 Sop(t) =

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 58

Optimal Scheduling

Given a tree t with just one child t0. What is the optimal scheduling? t t0 Sop(t) = Sop(t0)

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 59

Optimal Scheduling

So we can determine Sop for any tree t: Sop(t) =            min

  • max{Sop(t0) + 1, Sop(t1)},

max{Sop(t1) + 1, Sop(t0)}

  • if t has two children t0, t1

Sop(t0) if t has one child t0 1 if t has no children.

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 60

Optimal Scheduling

So we can determine Sop for any tree t: Sop(t) =            min

  • max{Sop(t0) + 1, Sop(t1)},

max{Sop(t1) + 1, Sop(t0)}

  • if t has two children t0, t1

Sop(t0) if t has one child t0 1 if t has no children. What is the distribution of Sop, if trees are randomly generated?

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 61

Newton’s Method

Let g(x) := f(x) − x and apply Newton’s method to g(x) = 0:

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2

f(x) = 0.4x2 + 0.6

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 62

Newton’s Method

Let g(x) := f(x) − x and apply Newton’s method to g(x) = 0:

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2

g(x) = f(x) − x

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 63

Newton’s Method

Let g(x) := f(x) − x and apply Newton’s method to g(x) = 0:

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2

ν(0) g(x) = f(x) − x

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 64

Newton’s Method

Let g(x) := f(x) − x and apply Newton’s method to g(x) = 0:

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2

ν(0) ν(1) g(x) = f(x) − x

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 65

Newton’s Method

Let g(x) := f(x) − x and apply Newton’s method to g(x) = 0:

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2

ν(0) ν(1) ν(2) g(x) = f(x) − x

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 66

Newton’s Method

Let g(x) := f(x) − x and apply Newton’s method to g(x) = 0:

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2

ν(0) ν(1) ν(2) g(x) = f(x) − x

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

Proposition (Etessami,Yannakakis, 2005) Newton’s method converges to the least solution.

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SLIDE 67

Newton’s Method

Let g(x) := f(x) − x and apply Newton’s method to g(x) = 0:

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2

ν(0) ν(1) ν(2) g(x) = f(x) − x

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

Proposition (Etessami,Yannakakis, 2005) Newton’s method converges to the least solution. The least solution is = 1. Why is he talking about Newton’s method??

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SLIDE 68

Newton’s Method

Let g(x) := f(x) − x and apply Newton’s method to g(x) = 0:

0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2

ν(0) ν(1) ν(2) g(x) = f(x) − x

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

Proposition (Etessami,Yannakakis, 2005) Newton’s method converges to the least solution. The least solution is = 1. Why is he talking about Newton’s method?? Theorem Pr

  • Sop ≤ k
  • = ν(k)
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SLIDE 69

Tail Bounds for the Optimal Scheduler

Theorem Pr

  • Sop ≤ k
  • = ν(k)

It follows: Pr

  • Sop ≥ k
  • = 1 − ν(k−1)

The ν(k) converge to 1, so Pr

  • Sop ≥ k
  • goes to 0.

But how fast??

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 70

Tail Bounds for the Optimal Scheduler

Theorem Pr

  • Sop ≤ k
  • = ν(k)

It follows: Pr

  • Sop ≥ k
  • = 1 − ν(k−1)

The ν(k) converge to 1, so Pr

  • Sop ≥ k
  • goes to 0.

But how fast?? Corollary (follows from KLE’07, EKL ’08) general task systems: Pr

  • Sop ≥ k
  • ∈ O(dk)

(d < 1) subcritical task systems: Pr

  • Sop ≥ k
  • ∈ O(d2k)

(d < 1)

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 71

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X X X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε

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SLIDE 72

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY X X Y X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε

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SLIDE 73

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY X X Y X Y X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε

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SLIDE 74

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY X X Y X Y X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε

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SLIDE 75

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX X X Y X Y X X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε

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SLIDE 76

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X X X Y X Y X X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε

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SLIDE 77

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε X X Y X Y X X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε

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SLIDE 78

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε The urn model is more appropriate: X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X

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SLIDE 79

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε The urn model is more appropriate: X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X

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SLIDE 80

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε The urn model is more appropriate: X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε Y X

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SLIDE 81

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε The urn model is more appropriate: X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε Y X

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SLIDE 82

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε The urn model is more appropriate: X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε Y Y X

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SLIDE 83

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε The urn model is more appropriate: X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε Y X Y

slide-84
SLIDE 84

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε The urn model is more appropriate: X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε Y X Y

slide-85
SLIDE 85

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε The urn model is more appropriate: X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X Y

slide-86
SLIDE 86

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε The urn model is more appropriate: X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X X

slide-87
SLIDE 87

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε The urn model is more appropriate: X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X X

slide-88
SLIDE 88

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε The urn model is more appropriate: X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X X

slide-89
SLIDE 89

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε The urn model is more appropriate: X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X

slide-90
SLIDE 90

Online Scheduling

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε The urn model is more appropriate: X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε X

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SLIDE 91

Weights: An Auxiliary Notion

Let v > 1 be a vector with v ≥ f(v). Choose h > 1 and for all types X a weight wX with hwX = vX for all types X. X wX Y wY Denote by W the maximum weight of a derivation. For instance: X ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε yields X X Y Y Y Y Y

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 92

Weights: An Auxiliary Notion

Let v > 1 be a vector with v ≥ f(v). Choose h > 1 and for all types X a weight wX with hwX = vX for all types X. X wX Y wY Denote by W the maximum weight of a derivation. For instance: X ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε yields X X Y Y Y Y Y W = 2 · wY

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-93
SLIDE 93

An Upper Bound for All Online Schedulers

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

(Recall: v > 1 with v ≥ f(v) and hwX = vX.) One can show by a martingale argument: Pr

  • W ≥ k
  • ≤ vX0

hk Note: Whenever S ≥ k then W ≥ k · wmin . So we obtain: Pr

  • S ≥ k
  • ≤ Pr
  • W ≥ k · wmin
  • ≤

vX0 hk·wmin = vX0 vmink

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SLIDE 94

An Upper Bound for All Online Schedulers

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

(Recall: v > 1 with v ≥ f(v) and hwX = vX.) One can show by a martingale argument: Pr

  • W ≥ k
  • ≤ vX0

hk Note: Whenever S ≥ k then W ≥ k · wmin . So we obtain: Pr

  • S ≥ k
  • ≤ Pr
  • W ≥ k · wmin
  • ≤

vX0 hk·wmin = vX0 vmink

slide-95
SLIDE 95

An Upper Bound for All Online Schedulers

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

Theorem Let v > 1 with v ≥ f(v). Then Pr

  • Sσ ≥ k
  • ≤

vX0 vmink for all online schedulers σ and all k ∈ N. (Recall: v > 1 with v ≥ f(v) and hwX = vX.) One can show by a martingale argument: Pr

  • W ≥ k
  • ≤ vX0

hk Note: Whenever S ≥ k then W ≥ k · wmin . So we obtain: Pr

  • S ≥ k
  • ≤ Pr
  • W ≥ k · wmin
  • ≤

vX0 hk·wmin = vX0 vmink

slide-96
SLIDE 96

An Upper Bound for All Online Schedulers

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

Theorem Let v > 1 with v ≥ f(v). Let u > 1 with u ≤ f(u). Then c umax k ≤ Pr

  • Sσ ≥ k
  • ≤

vX0 vmink for all online schedulers σ and all k ∈ N. (Recall: v > 1 with v ≥ f(v) and hwX = vX.) One can show by a martingale argument: Pr

  • W ≥ k
  • ≤ vX0

hk Note: Whenever S ≥ k then W ≥ k · wmin . So we obtain: Pr

  • S ≥ k
  • ≤ Pr
  • W ≥ k · wmin
  • ≤

vX0 hk·wmin = vX0 vmink

slide-97
SLIDE 97

An Upper Bound for All Online Schedulers

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

Theorem Let v > 1 with v ≥ f(v). Let u > 1 with u ≤ f(u). Then c umax k ≤ Pr

  • Sσ ≥ k
  • ≤

vX0 vmink for all online schedulers σ and all k ∈ N. Example Consider the task system from the beginning: fx(x, y) = 0.2xy + 0.8 fy(x, y) = 0.3y2 + 0.1x + 0.6 One can show: f has two fixed points: 1 1

  • and

1.4 2.2

  • .

So: c 2.2k ≤ Pr

  • Sσ ≥ k
  • ≤ 1.4

1.4k holds for all σ.

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SLIDE 98

An Upper Bound for All Online Schedulers

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

Theorem Let v > 1 with v ≥ f(v). Let u > 1 with u ≤ f(u). Then c umax k ≤ Pr

  • Sσ ≥ k
  • ≤

vX0 vmink for all online schedulers σ and all k ∈ N. Example Consider the task system from the beginning: fx(x, y) = 0.2xy + 0.8 fy(x, y) = 0.3y2 + 0.1x + 0.6 One can show: f has two fixed points: 1 1

  • and

1.4 2.2

  • .

So: c 2.2k ≤ Pr

  • Sσ ≥ k
  • ≤ 1.4

1.4k holds for all σ.

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SLIDE 99

A Light-First Scheduler For Our Example

For the upper bound we used: Whenever S ≥ k then W ≥ k · wmin . X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε We have v =

  • 1.4

2.2

  • , so wmin = wX.

We say, X is the lightest type. Light-First Scheduler: Process the lightest type (here: X) whenever it is in the pool. In our example, the light-first scheduler guarantees: at any time at most one X-task in the pool. Hence, with the light-first scheduler: Whenever S ≥ k then W ≥ 1 · wX + (k−1) · wY .

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-100
SLIDE 100

A Light-First Scheduler For Our Example

For the upper bound we used: Whenever S ≥ k then W ≥ k · wX . X

0.2

֒ − → XY X

0.8

֒ − → ε Y

0.3

֒ − → YY Y

0.1

֒ − → X Y

0.6

֒ − → ε We have v =

  • 1.4

2.2

  • , so wmin = wX.

We say, X is the lightest type. Light-First Scheduler: Process the lightest type (here: X) whenever it is in the pool. In our example, the light-first scheduler guarantees: at any time at most one X-task in the pool. Hence, with the light-first scheduler: Whenever S ≥ k then W ≥ 1 · wX + (k−1) · wY .

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-101
SLIDE 101

A Light-First Scheduler For Our Example

Previously, for all online schedulers σ: c 2.2k ≤ Pr

  • Sσ ≥ k
  • ≤ 1.4

1.4k Now, for the light-first scheduler lf: Pr

  • Slf ≥ k
  • ≤

1.4 1.4 · 2.2k−1 So, lf achieves the optimal tail bound: Pr

  • Slf ≥ k
  • ∈ Θ

1 2.2k

  • Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger

Space-Efficient Scheduling of Stochastically Generated Tasks

slide-102
SLIDE 102

A Light-First Scheduler For Our Example

Previously, for all online schedulers σ: W ≥ k · wX c 2.2k ≤ Pr

  • Sσ ≥ k
  • ≤ 1.4

1.4k Now, for the light-first scheduler lf: Pr

  • Slf ≥ k
  • ≤

1.4 1.4 · 2.2k−1 So, lf achieves the optimal tail bound: Pr

  • Slf ≥ k
  • ∈ Θ

1 2.2k

  • Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger

Space-Efficient Scheduling of Stochastically Generated Tasks

slide-103
SLIDE 103

A Light-First Scheduler For Our Example

Previously, for all online schedulers σ: W ≥ k · wX c 2.2k ≤ Pr

  • Sσ ≥ k
  • ≤ 1.4

1.4k Now, for the light-first scheduler lf: W ≥ 1 · wX + (k−1) · wY Pr

  • Slf ≥ k
  • ≤

1.4 1.4 · 2.2k−1 So, lf achieves the optimal tail bound: Pr

  • Slf ≥ k
  • ∈ Θ

1 2.2k

  • Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger

Space-Efficient Scheduling of Stochastically Generated Tasks

slide-104
SLIDE 104

Light-First Schedulers in General

More general: Light-First Scheduler: Process the lightest type available in the pool. This strategy improves the general bound on Pr

  • S ≥ k
  • .

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

slide-105
SLIDE 105

Light-First Schedulers in General

More general: Light-First Scheduler: Process the lightest type available in the pool. This strategy improves the general bound on Pr

  • S ≥ k
  • .

Q: What is a light type? A: A type whose component in the vector v is small. with v ≥ f(v)

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 106

Light-First Schedulers in General

More general: Light-First Scheduler: Process the lightest type available in the pool. This strategy improves the general bound on Pr

  • S ≥ k
  • .

Q: What is a light type? A: A type whose component in the vector v is small. with v ≥ f(v) Q: Well, what is a light type, intuitively? A: Depends on v, but if we compute v in a simple way, then the lightest type is the one with the smallest expected time (!)

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 107

Comparison

Notation:

  • p = the optimal offline scheduler

σ = any online scheduler 0 < d < 1 In the subcritical case: tail bound Expectation Sop Pr

  • Sop ≥ k
  • ∈ O
  • d2k

finite Sσ Pr

  • Sσ ≥ k
  • ∈ Θ
  • dk

finite

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 108

Comparison

Notation:

  • p = the optimal offline scheduler

σ = any online scheduler 0 < d < 1 In the subcritical case: tail bound Expectation Sop Pr

  • Sop ≥ k
  • ∈ O
  • d2k

finite Sσ Pr

  • Sσ ≥ k
  • ∈ Θ
  • dk

finite In the critical case: tail bound Expectation Sop Pr

  • Sop ≥ k
  • ∈ O
  • dk

finite Sσ Pr

  • Sσ ≥ k
  • ∈ Ω

1

k

  • infinite

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 109

Conclusions

We have studied the space consumption of stochastically generated tasks. Such task systems require scheduling. The performance of the optimal offline scheduler is closely linked with the convergence speed of Newton’s method. The performance of the online schedulers is closely linked with fixed points of the function f. Light-First schedulers are good online schedulers. For critical systems, finite expectation is

  • nly achieved by offline schedulers.

One can efficiently approximate expectations.

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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SLIDE 110

End of Talk

Thank you!

Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks