O(log n / log log n) RMRs Randomized Mutual Exclusion

1. O(log n / log log n) RMRs Randomized Mutual Exclusion Danny Hendler Philipp Woelfel PODC 2009 Ben-Gurion University University of Calgary

2. Talk outline • Prior art and our results • Basic Algorithm (CC) • Enhanced Algorithm (CC) • Pseudo-code • Open questions

3. Most Relevant Prior Art • Best upper bound for mutual exclusion: O(log n) RMRs (Yang and Anderson, Distributed Computing '96). • A tight Θ(n log n) RMRs lower bound for deterministic mutex(Attiya, Hendler and Woelfel, STOC '08) • Compare-and-swap (CAS) is equivalent to read/write for RMR complexity(Golab, Hadzilacos, Hendler and Woelfel, PODC '07)

4. Our Results Randomized mutual exclusion algorithms (for both CC/DSM) that have: • O(log N / log log N) expected RMR complexity against a strong adversary, and • O(log N)deterministic worst-case RMR complexity Separation in terms of RMR complexity between deterministic/randomized mutual exclusion algorithms

5. Shared-memory scheduling adversary types • Oblivious adversary: Makes all scheduling decisions in advance • Weak adversary: Sees a process' coin-flip only after the process takes the following step, can change future scheduling based on history • Strong adversary: Can change future scheduling after each coin-flip / step based on history

6. Talk outline • Prior art and our results • Basic algorithm (CC model) • Enhanced Algorithm (CC model) • Pseudo-code • Open questions

7. Basic Algorithm – Data Structures Δ Δ=Θ(log n / log log n) Δ-1 Key idea: Processes apply randomized promotion Δ 1 2 1 0 1 2 n

8. Basic Algorithm – Data Structures (cont'd) Δ Promotion Queue notified[1…n] pi1 pi2 pik Δ-1 lock{P,} Per-node structure apply: <v1,v2, …,vΔ> Δ 1 2 1 0 1 2 n

9. Basic Algorithm – Entry Section Δ Δ-1 i CAS(, i) Lock=  i apply: <v1, , …,vΔ> 1 0 i

10. Basic Algorithm – Entry Section: scenario #2 Δ Δ-1 Failure CAS(,i) Lock=q i apply: <v1, , …,vΔ> 1 0 i

11. Basic Algorithm – Entry Section: scenario #2 Δ Δ-1 Lock=q i apply: <v1, , …,vΔ> await (n.lock=) || apply[ch]=) 1 0 i

12. Basic Algorithm – Entry Section: scenario #2 Δ Δ-1 apply: Lock=q i <v1, , …,vΔ> await (n.lock=) ||apply[ch]=) 1 0 i

13. Basic Algorithm – Entry Section: scenario #2 Δ Δ-1 await (notified[i) =true) 1 0 CS i

14. Basic Algorithm – Exit Section Δ Δ-1 Climb up from leaf until last node capturedin entry section  Lock=p apply: <v1, q, …,vΔ> 1 0 Lottery p

15. Basic Algorithm – Exit Section Δ Perform a lottery on the root Δ-1  Lock=p apply: <v1, , …,vΔ> Promotion Queue q 1 s 0 t p

16. Basic Algorithm – Exit Section Δ t CS Δ-1 await (notified[i) =true) Promotion Queue q 1 s 0 t i

17. Basic Algorithm – Exit Section (scenario #2) Δ Free Root Lock Δ-1 Promotion Queue EMPTY 1 0 i

18. Basic Algorithm – Properties Lemma: mutual exclusion is satisfied Proof intuition: when a process exits, it either • signals a single process without releasing the root's lock, or • if the promoted-processes queue is empty, releases the lock • When lock is free, it is captured atomically by CAS

19. Basic Algorithm – Properties (cont'd) Lemma: Expected RMR complexity is Θ(log N / log log N) await (n.lock=) || apply[ch]=) A waiting process participates in a lotteryevery constant number of RMRs incurred here Probability of winning a lottery is 1/Δ Expected #RMRs incurred before promotion is Θ(log N / log log N)

20. Basic Algorithm – Properties (cont'd) • Mutual Exclusion • Expected RMR complexity:Θ(log N / log log N) • Non-optimal worst-case complexity and (even worse) starvation possible.

21. Talk outline • Prior art and our results • Basic algorithm (CC) • Enhanced Algorithm (CC) • Pseudo-code • Open questions

22. The enhanced algorithm. Key idea Quit randomized algorithm after incurring ‘'too many’’ RMRS and then execute a deterministic algorithm. Problems • How do we count the number of RMRs incurred? • How do we “quit” the randomized algorithm?

23. Enhanced algorithm: counting RMRs problem await (n.lock=) || apply[ch]=) The problem: A process may incur here an unbounded number of RMRs without being aware of it.

24. Counting RMRs: solution Key idea Perform both randomized and deterministic promotion Lock=p apply: <v1, q, …,vΔ> token: • Increment promotion token whenever releasing a node • Perform deterministic promotion according to promotion index in addition to randomized promotion

25. The enhanced algorithm: quitting problem Upon exceeding allowed number of RMRs, why can't a process simply release captured locks and revert to a deterministic algorithm?? Δ 1 2 Waiting processes may incur RMRs without participating in lotteries! 1 2 N

26. Quitting problem: solution Δ Add a deterministicΔ-process mutex object to each node Δ-1 Per-node structure lock{P,} Δ apply: 1 2 <v1,v2, …,vΔ> token: 1 MX: Δ-process mutex 0 1 2 n

27. Quitting problem: solution (cont'd) Per-node structure lock{P,} apply: <v1,v2, …,vΔ> token: MX: Δ-process mutex • After incurring O(log Δ) RMRs on a node, compete for the MX lock. Then spin trying to capture node lock. • In addition to randomized and deterministic promotion, an exiting process promotes also the process that holds the MX lock, if any.

28. Quitting problem: solution (cont'd) • After incurring O(log Δ) RMRs on a node, compete for the MX lock. Then spin trying to capture node lock. Worst-case number of RMRs = O(Δ log Δ)=O(log n)

29. Talk outline • Prior art and our results • Basic algorithm (CC) • Enhanced Algorithm (CC) • Pseudo-code • Open questions

30. Data-structures the i'th leaf i'th

31. The entry section i'th

32. The exit section i'th

33. Open Problems • Is this best possible? • For strong adversary? • For weak adversary? • For oblivious adversary? • Is there an abortable randomized algorithm? • Is there an adaptive one?