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

Introduction Wait-free data objects Modeling the problem Deterministic algorithm Randomized algorithm Concl sions Conclusions • Concurrent data objects – Consistency! 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 steps. ⇒ Suitable for real-time systems EuroPar'06 2

Introduction Modeling the problem Freshness Deterministic algorithm Randomized algorithm Concl sions Conclusions • Reactive systems need read-operations that both respond fast and return fresh values W(0) A W(1) B R(0 or 1) C ( ) w 3 sensor 3 e 3 3 w 2 sensor 2 e 2 w 1 sensor 1 e 1 r r 0 CPU 0 e 0 e 0 +d s 0 e 0 +D EuroPar'06 3

Introduction Modeling the problem Earlier work Deterministic algorithm Randomized algorithm Concl sions Conclusions • Freshness in databases • Freshness in caching systems • Freshness for concurrent data objects Freshness for concurrent data objects – single-writer-to-single-reader asynch. comm. EuroPar'06 4

Introduction Modeling the problem Contributions Deterministic algorithm Randomized algorithm Conclusions Concl sions • The first paper that attacks the freshness for multi-writer multi-reader shared objects lti it lti d h d bj t • Competitive freshness – An optimal deterministic algorithm An optimal deterministic algorithm – A nearly-optimal randomized algorithm EuroPar'06 5

Introduction Modeling the problem Road-map Deterministic algorithm Randomized algorithm Concl sions Conclusions • Introduction Introduction • Modeling the problem • Optimal deterministic algorithm Optimal deterministic algorithm • Nearly-optimal randomized algorithm • Conclusions EuroPar'06 6

Introduction Modeling the problem Model Deterministic algorithm Randomized algorithm Concl sions Conclusions • Assumptions: – An upper bound D on operation execution time An upper bound D on operation execution time | (| e | |) k e d = = d d f d f f d f • Freshness Freshness d ( d ) d h | − | e d M M ≤ ≤ d ≤ ≤ ≤ ≤ d ≤ ≤ and and • Constraints • Constraints 1 f f f f M M d d D w 3 sensor 3 e e 3 w 2 sensor 2 e 2 w 1 d : delay, 1 ≤ d ≤ D+1 sensor 1 e 1 e 1 |e | : # fresh values |e d | : # fresh values M : # concurrent writes at e 0 CPU 0 s 0 e 0 e 0 -1+d e 0 +D EuroPar'06 7

Introduction Modeling the problem Freshness as an online game Deterministic algorithm Randomized algorithm Conclusions Concl sions freshness (log) ≤ − ≤ ≤ ≤ ln 0 ln ln ln ln ln M f d D D f M ln D d − ≤ ≤ − ln ln ln ln ln e d f D d − 1 d d Online game: g • player (read operation) vs. X • malicious adversary time (log) time (log) ln D 0 Freshness EuroPar'06 8

Introduction Modeling the problem A deterministic algorithm Deterministic algorithm Randomized algorithm Conclusions Concl sions M f d ≥ Algorithm: The read accepts the first D Analysis: f f f f M f c ≈ 1 ≈ f 2 T c x ln D ln D 2 f / M D T f T f 1 x f T - ε p 2 x f T p p 1 x ln D ln D 0 0 t t ⇒ = = c c D 1 2 EuroPar'06 9

Introduction Modeling the problem Lower bound D Deterministic algorithm Randomized algorithm Concl sions Conclusions • Adversary’s strategy: – Start with c=ln(D)/2 & Start with c ln(D)/2 & f decrease c at unit speed ln D until the player stops. At this time this time, • if c > 0 , c jumps to the max. f 1 • if c ≤ 0 , c keeps decreasing ln D f 2 2 • Case 1 f 1 - p 1 = ln D / 2 p 1 ln D • Case 2: 0 t f 2 - p 2 ≥ ln D / 2 p 2 X p 2 ⇒ Comp. ratio = e (f-p) ≥ D EuroPar'06 10

Introduction Modeling the problem A randomized algorithm Deterministic algorithm Randomized algorithm Concl sions Conclusions • Ideas: – Put a probability on the P t b bilit th f freshness c when it starts to ln D go down. g h • Algorithm g – When c is decreasing, put on a c = 2 / ln p D it a probability – If the game is over (i.e. h=g ), put the rest r on the current c ln D t t 1 0 EuroPar'06 11

Introduction Modeling the problem A randomized algorithm Deterministic algorithm Randomized algorithm Concl sions Conclusions • Competitive ratio ln ln D D → ∞ = ⎯ ⎯ → ⎯ D c + 2 1 ln 2 + + − − 1 1 ln ln 2 2 D • Optimal randomized comp. ratio ( ln D)/2, ( l D)/2 O ti l d i d ti asymptotically (cf. TR-CS-2005:17) EuroPar'06 12

Introduction Modeling the problem Conclusions Deterministic algorithm Randomized algorithm Conclusions Concl sions • The first paper that defines the freshness problem for wait-free data objects. 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 New general models EuroPar'06 13

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

Introduction Modeling the problem A randomized algorithm Deterministic algorithm Randomized algorithm Concl sions Conclusions ln D = c • Comp. ratio 2 1 + + − 1 ln ln 2 2 lf D ln D • Randomized online search → Randomized online search → g deterministic one-way trading: f – exchanging some fraction of money ≈ stopping the search with money ≈ stopping the search with that probability c • Conventions: – distributed money on axis lf – T(x) : density of exchanged money ln D 0 t ⇒ player’s profit = ∫ (x.T(x)) ( ( )) p y p EuroPar'06 15

Introduction Modeling the problem Randomized algorithm Deterministic algorithm Randomized algorithm Concl sions Conclusions • Ideas: – Exchange /////// when c starts Exchange /////// when c starts to go down. lf ln D g comp. ratio p f a c • Optimal comp. ratio O(ln D) ln D 0 t (cf TR-CS-2005:17) (cf. TR CS 2005:17) EuroPar'06 16

Introduction Modeling the problem Analysis Deterministic algorithm Randomized algorithm Conclusions Concl sions • Let x = f – c (final value) L t f (fi l l ) • Observations: f f – T=2 on [c,f] or c=f – ∑ (gaps with T=0) ≤ r ∑ (gaps with T=0) ≤ r T=2 • Player’s profit x x ⎛ ⎞ + ( ln ) / 2 r D x ⎜ ⎟ ∫ ∫ ∫ ∫ − − − ≥ + + t t t . min 2 2 f f e dt re e dt ⎜ ⎜ ⎟ ⎟ ⎝ ⎠ , r x + r T=0 0 x r worst ⎛ ⎞ 2 > + − ⎜ ⎟ 1 ln 2 f x+r x r ⎝ ⎝ ⎠ ⎠ D D Adversary profit: f.ln D • T=2 ln ln D D ⇒ comp. ratio = ti r + c ln D + − 1 ln 2 2 / D 2 Optimal comp ratio O(ln D) Optimal comp. ratio O(ln D) • • (cf. TR-CS-2005:17) EuroPar'06 17