Competitive Freshness Algorithms for Wait free Objects Wait-free - - PowerPoint PPT Presentation

Competitive Freshness Algorithms for Wait free Objects Wait-free Objects

Peter Damaschke, Phuong Ha & Philippas Tsigas

Presentation at the Euro-Par 2006

29th Aug. – 1st Sept. 2006, Dresden, Germany.

Wait-free data objects

Introduction Modeling the problem Deterministic algorithm Randomized algorithm Concl sions

• Concurrent data objects

– Consistency!

Conclusions

Consistency!

• Solutions:
• Solutions:

– Mutual exclusion?

⇒ risks of lock-convoy deadlock & priority inversion ⇒ risks of lock convoy, deadlock & priority inversion

– Non-blocking synchronization Non blocking synchronization

• Wait-free:

– every operation is guaranteed to finish in a limited number of steps

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steps. ⇒ Suitable for real-time systems

Freshness

Introduction Modeling the problem Deterministic algorithm Randomized algorithm Concl sions

• Reactive systems need read-operations that both respond

fast and return fresh values

Conclusions

W(0) A W(1) B R(0 or 1) C ( ) w3 sensor3 e3 sensor2 sensor1 w1 w2 e1 e2

3

r e0+d CPU0 s0 e0+D e0 r0

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Earlier work

Introduction Modeling the problem Deterministic algorithm Randomized algorithm Concl sions

• Freshness in databases

Conclusions

• Freshness in caching systems
• Freshness for concurrent data objects

Freshness for concurrent data objects

– single-writer-to-single-reader asynch. comm.

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Contributions

Introduction Modeling the problem Deterministic algorithm Randomized algorithm Concl sions

• The first paper that attacks the freshness for

lti it lti d h d bj t

Conclusions

multi-writer multi-reader shared objects

• Competitive freshness

– An optimal deterministic algorithm An optimal deterministic algorithm – A nearly-optimal randomized algorithm

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Road-map

Introduction Modeling the problem Deterministic algorithm Randomized algorithm Concl sions

• Introduction

Conclusions

Introduction

• Modeling the problem

Optimal deterministic algorithm

• Optimal deterministic algorithm
• Nearly-optimal randomized algorithm
• Conclusions

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Model

Introduction Modeling the problem Deterministic algorithm Randomized algorithm Concl sions

• Assumptions:

– An upper bound D on operation execution time

Conclusions

An upper bound D on operation execution time

• Freshness

e f

d d

| | = |) (| e k f

d d =

Freshness

• Constraints

d fd M f ed ≤ ≤

− |

|

1

M f M ≤ ≤

and

) (d h fd

• Constraints

sensor3 w3 e

d f d

d ≤

≤ M f D

d ≤

≤

and

sensor2 sensor1 w1 w2 e1 e2 e3 d: delay, 1 ≤ d ≤ D+1 |e |: # fresh values

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CPU0 s0 e1 e0-1+d e0+D e0 |ed|: # fresh values M: # concurrent writes at e0

Freshness as an online game

Introduction Modeling the problem Deterministic algorithm Randomized algorithm Concl sions

freshness (log) lnD

M f D M

d

ln ln ln ln ≤ ≤ − D fd ln ln ≤ ≤

Conclusions

d D f d e

d d

ln ln ln ln ln

1

− ≤ ≤ −

−

Online game:

X

g

• player (read operation)

vs.

• malicious adversary

time (log) time (log) lnD

Freshness

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A deterministic algorithm

Introduction Modeling the problem Deterministic algorithm Randomized algorithm Concl sions

D M fd ≥

Algorithm: The read accepts the first

f f

Analysis:

Conclusions

f lnD

D M f c

T

/

1 ≈

f lnD

T

f M c ≈

2

f2 x fT fT - ε f1 x fT p x p2 lnD t lnD t p1 x

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D c c = = ⇒

2 1

Lower bound

D

Introduction Modeling the problem Deterministic algorithm Randomized algorithm Concl sions

• Adversary’s strategy:

– Start with c=ln(D)/2 &

Conclusions

Start with c ln(D)/2 & decrease c at unit speed until the player stops. At this time

f lnD

this time,

• if c > 0, c jumps to the max.
• if c ≤ 0, c keeps decreasing

ln D

f2 f1

• Case 1

f1 - p1 = ln D / 2

2

p1

• Case 2:

f2 - p2 ≥ ln D / 2

lnD t p2X

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⇒ Comp. ratio = e(f-p) ≥

p2

D

A randomized algorithm

Introduction Modeling the problem Deterministic algorithm Randomized algorithm Concl sions

• Ideas:

P t b bilit th

Conclusions

– Put a probability on the freshness c when it starts to go down.

f lnD

• Algorithm

h g

g

– When c is decreasing, put on it a probability

c a

D p ln / 2 =

– If the game is over (i.e. h=g), put the rest r on the current c

lnD t1 t

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A randomized algorithm

Introduction Modeling the problem Deterministic algorithm Randomized algorithm Concl sions

• Competitive ratio

Conclusions

D c 2 2 ln 1 ln − + = 2 ln 1 ln + ⎯ ⎯ → ⎯

∞ →

D

D

O ti l d i d ti (l D)/2

D 2 ln 1 − +

• Optimal randomized comp. ratio (ln D)/2,

asymptotically

(cf. TR-CS-2005:17)

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Conclusions

Introduction Modeling the problem Deterministic algorithm Randomized algorithm Concl sions

• The first paper that defines the freshness

problem for wait-free data objects

Conclusions

problem for wait-free data objects.

• Competitive freshness

– An optimal deterministic algorithm – A nearly-optimal randomized algorithm

• Contributions to the online search problem

– New general models

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New general models

Thank you for your attention! Thank you for your attention!

A randomized algorithm

Introduction Modeling the problem Deterministic algorithm Randomized algorithm Concl sions

• Comp. ratio

D c 2 2 ln 1 ln − + =

Conclusions

• Randomized online search →

lf lnD

D 2 ln 1+ Randomized online search → deterministic one-way trading:

– exchanging some fraction of money ≈ stopping the search with

f g

money ≈ stopping the search with that probability

• Conventions:

c

– distributed money on axis lf – T(x): density of exchanged money ⇒ player’s profit = ∫ (x.T(x))

lnD t

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p y p ( ( ))

Randomized algorithm

Introduction Modeling the problem Deterministic algorithm Randomized algorithm Concl sions

• Ideas:

– Exchange /////// when c starts

Conclusions

Exchange /////// when c starts to go down.

lf lnD

• comp. ratio

f g

p

c a

• Optimal comp. ratio O(ln D)

(cf TR-CS-2005:17)

lnD t

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(cf. TR CS 2005:17)

L t f (fi l l )

Analysis

Introduction Modeling the problem Deterministic algorithm Randomized algorithm Concl sions

• Let x = f – c (final value)
• Observations:

– T=2 on [c,f] or c=f – ∑ (gaps with T=0) ≤ r f f

Conclusions

∑ (gaps with T=0) ≤ r

• Player’s profit

x x T=2

⎟ ⎟ ⎞ ⎜ ⎜ ⎛ + + ≥

∫ ∫

+ − − − 2 / ) ln (

2 2 min .

D r t t x t

dt e re dt e f

worst T=0 x+r r

⎟ ⎠ ⎜ ⎝

∫ ∫

+ , r x x r

f ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + > D f 2 2 ln 1

• Adversary profit: f.ln D

ti

T=2 x r

⎠ ⎝ D D ln

⇒ comp. ratio

• Optimal comp ratio O(ln D)

2 ln D r +

D D c / 2 2 ln 1 ln − + =

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• Optimal comp. ratio O(ln D)

(cf. TR-CS-2005:17)