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CPSC 668 Set 8: More Mutex with Read/Write Variables 1
CPSC 668Distributed Algorithms and Systems
Fall 2009
Prof. Jennifer Welch
CPSC 668 Set 8: More Mutex with Read/Write Variables 2
Number of R/W Variables
• Bakery algorithm used 2n shared read/write variables.
• Tournament tree algorithm used 3n shared read/write variables.
• Can we do (asymptotically) better, in terms of fewer variables?
• No!
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Lower Bound on Number of VariablesTheorem (4.19): Any no-deadlock
mutual exclusion algorithm using read/write variables must use at least n shared variables.
Proof Strategy: Show by induction on n there must be at least n variables.For each n, there is a configuration in which n variables are covered: means some processor is about to write to it.
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Appearing Quiescent• Two configurations C and D are P-similar if
each processor in P has same state in C as in D and each shared variable has same value in C as in D.
• A configuration is quiescent if all processors are in remainder section.
• To make the induction go through, the configuration whose existence we prove must appear quiescent to a set of processors:– C is P-quiescent if there is a quiescent
configuration D such that C and D are P-similar
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Warm-Up Lemma
Before a processor can enter its CS, it must write to an uncovered variable.
Lemma (4.17): If C is pi-quiescent, then there is a pi-only schedule such that
• pi is in CS in (C) and
• during exec(C,), pi writes to a variable that is not covered in C.
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Proof of Warm-Up Lemma (a)
• Since C is pi-quiescent, it looks the same to pi as some quiescent D.
• By ND, some pi-only schedule exists starting at D in which pi enters CS.
• When starts at C, pi also enters CS.
pi in CS by NDD
quiescent
Cpi-quiescent
pi in CS
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Proof of Warm-up Lemma (b)
• Suppose in contradiction when is executed starting at C, pi writes to the set of variables W but all the variables in W are covered in C.
• Let P be the set of processors covering the variables in W.
1C Eone step by eachproc in P; over-writes W
Q2
successivelyinvoke ND tocause all procsto be in remainder;pi takes no step
some pj (not pi)takes steps alone;by ND eventuallypj enters CS
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Proof of Warm-up Lemma (b)
1C Eoverwrites W
Q2
successivelyinvoke ND
pj-only pj in CS
1 E'overwrites W
Q'2
successivelyinvoke ND
pj-only pj in CS,
pi in CS
Only difference between C and C' are the writes by pi, but
those values are overwritten in 1 so the info is lost.
C'
pi-only,writes to W
pi inCS
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Main Result
For all k between 1 and n, for all quiescent C, there exists D s.t.
• D is reachable from C by steps of p0,…,pk-1 only
• p0,…,pk-1 cover k distinct variables in D
• D is {pk,…,pn-1}-quiescent.implies desired result when k = n
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Proof of Main Result - Basis
By induction on k.
Basis: k = 1. Must show for all quiescent C, there exists D s.t.
– D is reachable from C by steps of p0 only
– p0 covers a variable in D
– D looks quiescent to the other procs.
• By warm-up lemma (a), if p0 takes steps alone, it eventually writes to some var.
• Desired D is just before p0 's first write.
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Proof of Main Result - Induction
Assume for k, show for k+1.
C C1any qui.config.
only p0 topk-1 take steps
p0 to pk-1
cover W;pk to pn-1 qui.
0
pk-only
pk coversx not in W
p0 to pk-1 overwrite W,become quiescent
D1'pk in entrylooks qui.to rest
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Proof of Main Result - Induction
C C1any qui.config.
only p0 topk-1 take steps
p0 to pk-1
cover W;pk to pn-1 qui.
0
pk-only
pk coversx not in W
p0 to pk-1 o'write W,become quiescent
D1'pk in entrylooks qui.to rest
D1qui.
C2
p0 to pk-1
onlyp0 to pk-1
cover W;pk to pn-1 qui.
C2'
p0 to pk
cover Wand x;pk+1 to pn-1 qui.
but why is the same setof k vars covered again?
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Proof of Main Result - Fix• The result of applying to D1 might result in a
different set of k variables, W', being covered instead of W.
• If W' includes x, we have not succeeded in covering an additional variable.
• To fix this problem, repeatedly apply inductive hypothesis to get
C1,D1,C2,D2,C3,D3,…• Since number of variables is finite, there exist i and j
such that in Ci and Cj the same set of k variables is covered.
• Then apply same argument as before, replacing C1 and C2 with Ci and Cj.
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Fast Mutual Exclusion
• The read/write mutex algorithms we've seen so far require a processor to access f(n) variables in the entry section even if no contention.
• It would be nice to have a fast algorithm: if no competition, a processor enters CS in O(1) steps.
• Even better would be an adaptive algorithm: performance depends on number of currently competing processors, not total number.
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Fast Mutual Exclusion
• Note that multi-writer shared variables are required to be fast.
• Combine two mechanisms:– provide fast entry when no contention– provide no deadlock with there is
contention
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Contention Detector Overview
• A doorway mechanism captures a set of processors that are concurrently accessing the detector
• Use a race to choose a unique one of the captured processors to "win"
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Contention Detector
Uses two shared variables, door and race.
Initially door = "open", race = -1.
1 race := id2 if door = "closed" then return "lose"3 else4 door := "closed"5 if race = id then return "win"6 else return "lose"
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Analysis of Contention Detector
Claim: At most one processor wins the contention detector.
Why?• Let K be set of procs. that read "open" from door in Line 2.
• Let pj be proc. that writes to race most recently before door is first set to "closed".
• No node pi other than pj can win:– If pi is not in K, it loses in Line 2.– If pi is in K, it writes race before pj does but
checks again (Line 5) after pj 's write and loses.
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Analysis of Contention Detector
Claim: If pi executes the contention detector alone, then pi wins.
Why?
• Trace through the code when there is no concurrency.
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Ensuring No Deadlock
• If there is concurrency, it is possible that no processor wins the contention detector.
• To ensure progress:– nodes that lose the contention detector participate
in an n-processor ME alg.– The winner of the n-processor alg. competes with
the (potential) winner of the contention detector using a 2-processor ME alg.
– Winner of 2-processor alg. can enter CS
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Ensuring No Deadlock
contentiondetector
n-proc. mutex
2-proc. mutex
critical section
lose
win
play role of p0 play role of p1
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Discussion of Fast Mutex
• Be careful about the exit section: contention detector needs to be reset properly
• This is a modular presentation: doesn't specify particular n-proc and 2-proc subroutine mutex algorithms
• Not adaptive: even if only 2 procs are contending, execute the potentially expensive n-proc algorithm