Contention-Related Crash Failures Anaïs Durand LIP6, Sorbonne Université, Paris April 1st, 2019 1 / 25 Anaïs Durand Contention-Related Crash Failures
Set Agreement and Renaming in the Presence of Contention-Related Crash Failures SSS 2018 Joint work with: Michel Raynal Gadi Taubenfeld 2 / 25 Anaïs Durand Contention-Related Crash Failures
Computational Model � Asynchronous deterministic system � n processes p 1 , . . . , p n � Atomic read/write registers � 0 ≤ t < n process crashes � Participation required 3 / 25 Anaïs Durand Contention-Related Crash Failures
Process crashes 2 kinds of process crashes usually considered: � Initially dead processes � “Classical” ( any-time ) crashs: no constraints 4 / 25 Anaïs Durand Contention-Related Crash Failures
Contention-Related Crash Failures [ Taubenfeld,18 ] � Contention = # processes that accessed a shared register ≈ # processes that started to compute � λ = predefined contention threshold � 2 possible definitions: No crashes contention λ No crashes contention λ 5 / 25 Anaïs Durand Contention-Related Crash Failures
Contention-Related Crash Failures [ Taubenfeld,18 ] � Contention = # processes that accessed a shared register ≈ # processes that started to compute � λ = predefined contention threshold � 2 possible definitions: λ -constrained crashes No crashes contention λ No crashes contention λ 5 / 25 Anaïs Durand Contention-Related Crash Failures
Contention-Related vs. Any-Time Crash Failures � Consensus: ◮ [ Fischer et al. , 85 ] : Impossible with one any-time crash failure. ◮ [ Taubenfeld, 18 ] : Algorithm that tolerates one ( n − 1 ) -constrained crash failure for n > 1. � k -Set Agreement, 1 ≤ k < n : ◮ [ Borowsky, Gafni, 93 ] : Impossible with k any-time crash failures. ◮ [ Taubenfeld, 18 ] : Algorithm that tolerates ℓ + k − 2 ( n − ℓ ) -constrained crash failures for ℓ ≥ 1 and n ≥ 2 ℓ + k − 2. 6 / 25 Anaïs Durand Contention-Related Crash Failures
Motivation Consider a problem P that can be solved with t any-time crash failures, but impossible with t + 1 any-time crash failures. Given λ , can P be solved with both t 1 λ -constrained and t 2 ≤ t any-time crash failures, with t 1 + t 2 > t ? We consider here: k -set agreement (for k ≥ 2) and renaming 7 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement 8 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement [ Chaudhuri,90 ] Definition � One-shot object � Operation propose ( v ) : propose value v and return a decided value � Properties: ◮ Validity: decided value ∈ proposed values ◮ Agreement: ≤ k decided values ◮ Termination: every correct process decides 9 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement Algorithm: Properties � λ = n − k � k ≥ 2 � k = m + f , m ≥ 0, f ≥ 1 total # of faults t = 2 m + f − 1 = k + m − 1 λ -constrained crashes 2 m any-time crashes f − 1 [ Borowsky, Gafni, 93 ] : Impossible with k any-time crash failures. 10 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement: Parameters Parameters f and m allow the user to tune the proportion of each type of crash failures. max #any-time (= k -1 ) 0 k -1 m t = k − 1 1 k f � k � 0 k -1 2 m � k � k � k � � � t = 2 + − 1 1 k 2 2 2 f 0 k -1 m t = 2 k − 2 1 k f max # λ -constrained (= 2 k -2 ) 11 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement: Shared Registers (1 / 2) � DEC : atomic register, initially ⊥ � PART [ 1 . . . n ] : snapshot object, initially [ down , . . . , down ] ◮ Atomic (linearizable) operations write () and snapshot () ◮ ≈ array of single-writer multi-reader atomic registers PART [ 1 . . . n ] such that: • p i invokes write ( v ) = writes v into PART [ i ] • p i invokes snapshot () = obtains the value of the array PART [ 1 . . . n ] as if it read simultaneously and instantaneously all its entries 12 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement: Shared Registers (2 / 2) � MUTEX [ 1 ] : one-shot deadlock-free f -mutex � MUTEX [ 2 ] : one-shot deadlock-free m -mutex ◮ Operations acquire () and release () (invoked at most once) ◮ Properties: • Mutual exclusion: ≤ m processes simultaneously in critical section • Deadlock-freedom: if < m processes crashes, then ≥ 1 process invoking acquire () terminates its invocation 13 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement Algorithm (1/2) operation propose ( in i ) is PART . write ( up ) ; % signal participation (1) 14 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement Algorithm (1/2) operation propose ( in i ) is PART . write ( up ) ; % signal participation (1) repeat (2) part i := PART . snapshot () ; % wait for n − t (3) count i := |{ x such that part i [ x ] = up }| ; % participants (4) until count i ≥ n − t end repeat ; (5) 14 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement Algorithm (1/2) operation propose ( in i ) is PART . write ( up ) ; % signal participation (1) repeat (2) part i := PART . snapshot () ; % wait for n − t (3) count i := |{ x such that part i [ x ] = up }| ; % participants (4) until count i ≥ n − t end repeat ; (5) if count i ≤ λ then % split processes into groups (6) group i := 2; % � MUTEX [ 2 ] (m-mutex) (7) else (8) group i := 1; (9) % � MUTEX [ 1 ] (f-mutex) end if (10) 14 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement Algorithm (1/2) operation propose ( in i ) is PART . write ( up ) ; % signal participation (1) repeat (2) part i := PART . snapshot () ; % wait for n − t (3) count i := |{ x such that part i [ x ] = up }| ; % participants (4) until count i ≥ n − t end repeat ; (5) if count i ≤ λ then % split processes into groups (6) group i := 2; % � MUTEX [ 2 ] (m-mutex) (7) else (8) group i := 1; (9) % � MUTEX [ 1 ] (f-mutex) end if (10) launch in // the threads T 1 and T 2 ; (11) 14 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement Algorithm (2/2) thread T 1 is % wait for a decided value loop forever (12) if DEC � = ⊥ then (13) return ( DEC ); (14) end if ; (15) end loop ; (16) 15 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement Algorithm (2/2) thread T 1 is % wait for a decided value loop forever (12) if DEC � = ⊥ then (13) return ( DEC ); (14) end if ; (15) end loop ; (16) thread T 2 is % decide a value if enters its CS if group i = 1 ∨ m > 0 then (17) MUTEX [ group i ] . acquire () ; (18) if DEC = ⊥ then (19) DEC := in i ; (20) end if (21) MUTEX [ group i ] . release () ; (22) return ( DEC ) ; (23) end if ; (24) 15 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement Algorithm: Validity & Agreement thread T 1 is loop forever (12) if DEC � = ⊥ then (13) return ( DEC ); a Decided value = DEC (14) end if ; (15) end loop ; (16) thread T 2 is if group i = 1 ∨ m > 0 then (17) MUTEX [ group i ] . acquire () ; (18) if DEC = ⊥ then (19) DEC := in i ; (20) end if (21) MUTEX [ group i ] . release () ; (22) return ( DEC ) ; (23) end if ; (24) 16 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement Algorithm: Validity & Agreement thread T 1 is loop forever (12) if DEC � = ⊥ then (13) return ( DEC ); a Decided value = DEC (14) end if ; (15) end loop ; (16) b DEC assigned to proposed thread T 2 is values in i in CS if group i = 1 ∨ m > 0 then (17) MUTEX [ group i ] . acquire () ; (18) if DEC = ⊥ then (19) DEC := in i ; (20) end if (21) MUTEX [ group i ] . release () ; (22) return ( DEC ) ; (23) end if ; (24) 16 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement Algorithm: Validity & Agreement thread T 1 is loop forever (12) if DEC � = ⊥ then (13) return ( DEC ); a Decided value = DEC (14) end if ; (15) end loop ; (16) b DEC assigned to proposed thread T 2 is values in i in CS if group i = 1 ∨ m > 0 then (17) MUTEX [ group i ] . acquire () ; (18) c MUTEX [ 1 ] � ≤ f � = values if DEC = ⊥ then (19) MUTEX [ 2 ] � ≤ m � = values DEC := in i ; (20) end if (21) ⇒ ≤ f + m = k decided values MUTEX [ group i ] . release () ; (22) return ( DEC ) ; (23) end if ; (24) 16 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement Algorithm: Termination (1/5) PART . write ( up ) ; (1) repeat (2) part i := PART . snapshot () ; (3) count i := |{ x such that part i [ x ] = up }| ; (4) until count i ≥ n − t end repeat ; (5) a ≤ t crashes + participation required � eventually count i ≥ n − t at every correct process p i 17 / 25 Anaïs Durand Contention-Related Crash Failures
k -Set Agreement Algorithm: Termination (1/5) PART . write ( up ) ; (1) repeat (2) part i := PART . snapshot () ; (3) count i := |{ x such that part i [ x ] = up }| ; (4) until count i ≥ n − t end repeat ; (5) if count i ≤ λ then (6) group i := 2; (7) else (8) group i := 1; (9) end if (10) a ≤ t crashes + participation required � eventually count i ≥ n − t at every correct process p i b ≤ n − k processes with count i ≤ n − k = λ when leaving loop (2)-(5) � ≤ n − k processes in group 2 17 / 25 Anaïs Durand Contention-Related Crash Failures
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