Adaptive redundant residue system for cloud computing: a processor-oblivious and fault-oblivious technology

Transcription

1 Adaptive redundant residue system for cloud computing: a processor-oblivious and fault-oblivious technology Jean-Louis ROCH INRIA MOAIS team, LIG lab. Grenoble Université, France Joint work with C. Pernet [1], T. Roche [1, 4], S. Varrette [3, 2], S. Jafar [2], A. Krings [2], B. Cunche [4], M. Khonji[1], T. Stalinski[1] Computer Science Dept, University of Western Ontario Monday January 24, 2011

2 Plan Introduction High performance computing and cloud Related works on trusting the clouds Cloud computing and performance ABFT Redundant residue codes Homomorphic residue system Over Z: Mandelbaum algorithm Over K[X]: Reed Solomon point of view Generalization Adaptive approach First approach and the gap Experiments Early termination

3 Context: High Performance Interactive Computation INRIA - LIG Moais team Performance: multi-criteria trade-off computation latency and computation bandwidth Application domains: interactive simulations (latency), computer algebra (bandwidth),...

4 HPC platforms From low computation latency to high computation bandwidth Parallel chips & multi/many-core architectures: multicore cpu, GP-GPU, FPGA, MPSoCs Servers: Multi-processor with sharedi memory (CPUs + GPUs+...) Clusters: 72% of top 500 machines, Heterogeneous (CPUs + GPUs +...) E.g. Tianhe-1A, ranked 1 in TOP500 nov 2010: 4.7 PFlops Global computing platforms: grids, P2P, clouds E.g. BOINC : in April 2010= 5.1 PFlops

5 Cloud / Global computing platforms Applications Data sharing (1980 s), Data storage, Computation (1990 s) Unbound computation bandwidth Volunteer Computing: steal idle cycles through the Internet Folding@Home 5 PFLOPS, as of March 17, 2009 MilkyWay@Home PFLOPS, as of April 2010 Thanks to GPUs!

6 Grid or cloud? Beyond computation bandwidth, two important criteria: granularity: how small is the smallest bit of computational resource that you can buy; speed of scaling: how long it takes to increase the size of available resources.

7 Global computing architecture and trust Open platforms are subject to attacks: machine badly configured, over clocking, malicious programs, client patched and redistributed: possibly large scale Parallel Application f1 f4 f1 f2 f3 f4 f5 s1 s1 s2 f3 f2 user INTERNET s2 f5

8 Global computing architecture and trust Open platforms are subject to attacks: machine badly configured, over clocking, malicious programs, client patched and redistributed: possibly large scale Parallel Application f1 f4 f1 f2 f3 f4 f5 s1 s1 s2 f3 f2 user modified result INTERNET hacker s2 f5

9 Global computing platforms: drawbacks Peers volatility Peers can join/leave at any time. Faults, crashes As the number of nodes increases, the number of faults, process crashes increases as well. Trust in Peers Malicious Peers (intentionally or infected by malwares). Random Crashes Random Forgeries Byzantine behaviour (e.g., Peers collusion)

10 Related work: trusting the Cloud Volatility Replication, (partial) Re-execution, Checkpoint/restart Trust Challenges / blacklisting [Sramenta&ak 01] Replication of tasks: BOINC: credit evaluation Byzantine agreement: n processes for n/3 faulty (one-third faulty replicas [Lamport82s] ) Check / Verification using postconditions (on the output) [Blum97]

11 Related work: trusting the Cloud Volatility Replication, (partial) Re-execution, Checkpoint/restart Trust Challenges / blacklisting [Sramenta&ak 01] expensive Replication of tasks: BOINC: credit evaluation Attacks can be adjusted Byzantine agreement: n processes for n/3 faulty (one-third faulty replicas [Lamport82s] ) often expensive Check / Verification using postconditions (on the output) [Blum97] not always possible

12 Related work: trusting the Cloud Without trust assumptions, basic properties (eg integrity, atomicity, weak consistency,...) cannot be guaranteed But, by maintaining a small amount of trusted memory and trusted computation units, well-known cryptographic methods reduce the need for trust in the storage [Cachin&al. ACM SIGACT 2009] integrity by storing small hash (hash tree for huge data) authentication of data proofs of retrievability (POR) and of data possession (PDP)

13 Considered cloud computing platform = Two disjoint set of resources: U: The cloud, unreliable R: Resources blindly trusted by the user (reliable), but with limited computation bandwidth interconnected through a stable memory Safe Resources Controler Verifier Checkpoint Server Unsafe Resources Grid INTERNET user

14 Plan Introduction High performance computing and cloud Related works on trusting the clouds Cloud computing and performance ABFT Redundant residue codes Homomorphic residue system Over Z: Mandelbaum algorithm Over K[X]: Reed Solomon point of view Generalization Adaptive approach First approach and the gap Experiments Early termination

15 How to take benefit of the Cloud for HPiC? Performance and correction Performance: massive parallelism (work depth) workstealing, adaptive scheduling [Leiserson&al. 2007] Correction of the results: proof verifications on R (e.g. randomized checking) Fault tolerance: to support resilience and/or errors ABFT: Algorithm Based Fault Tolerance

16 Execution time model on the cloud Resource average computation Total computation Usage type bandwidth per proc bandwidth Cloud U Π U Π tot U Client R Π R Π tot R bandwidth: number of unit operations per second computation certification Bound on the time required for computation+certification Based on work-stealing, with high probability [Bender-Rabin02] : Execution time W ( ) c Dc Π tot + O + W ( ) r Dr Π U U Π tot + O. Π R R Notations and target context: W c (resp. D c ) = total work (resp. depth) executed on the cloud U; W r (resp. D r ) = total work (resp. depth) executed on the client R; Target context: W c = O(W 1+ɛ 1 ); D c W c ; W r, D r D c. Correction: may be computed either on R, or on U, or on both.

17 Plan Introduction High performance computing and cloud Related works on trusting the clouds Cloud computing and performance ABFT Redundant residue codes Homomorphic residue system Over Z: Mandelbaum algorithm Over K[X]: Reed Solomon point of view Generalization Adaptive approach First approach and the gap Experiments Early termination

18 ABFT: Algorithmic Based Fault Tolerance Idea: incorporate redundancy in the algorithm [Huang&Abraham 98] [Saha 2006] [Dongarra & al. 2006] use properties specific to the problem

19 ABFT: Algorithmic Based Fault Tolerance Idea: incorporate redundancy in the algorithm [Huang&Abraham 98] [Saha 2006] [Dongarra & al. 2006] use properties specific to the problem Example: Matrix-vector product A u ua

20 ABFT: Algorithmic Based Fault Tolerance Idea: incorporate redundancy in the algorithm [Huang&Abraham 98] [Saha 2006] [Dongarra & al. 2006] use properties specific to the problem Example: Matrix-vector product A Id R u ua x

21 ABFT: Algorithmic Based Fault Tolerance Idea: incorporate redundancy in the algorithm [Huang&Abraham 98] [Saha 2006] [Dongarra & al. 2006] use properties specific to the problem Example: Matrix-vector product A Id R u ua x pre-compute the product B = A [ I R ] compute x = ub in parallel decode/correct x

22 Summary Typical considered global computation Interactive computation between R (reliable) and the cloud U (unreliable); On U: Computations loosely coupled and fault tolerant On R: submission of the computation correction (if not too much errors) verification (if not too much errors) homorphic residue system

23 Plan Introduction High performance computing and cloud Related works on trusting the clouds Cloud computing and performance ABFT Redundant residue codes Homomorphic residue system Over Z: Mandelbaum algorithm Over K[X]: Reed Solomon point of view Generalization Adaptive approach First approach and the gap Experiments Early termination

24 Homomorphic residue system One-to-one mapping Φ from E to E 1... E n that preserves the algebraic structure Φ : x x 1,... x n : projection Φ 1 (x 1,..., x n ): lifting For a fixed computation P (straight-line program): For input x: Φ(P x ) = Φ(P) x1,..., Φ(P) xn. Classical examples: residue number/polynomial systems: Polynomials: Φ(Q) = (Q(a 1 ),..., Q ( a n )) [evaluation/interpolation] Integers: Φ(x) = (x mod a 1,..., x mod a n ) a 1,..., a n : residue number system (relatively prime)

25 Integers: Chinese remainder algorithm Z/(n 1... n k )Z Z/n 1 Z Z/n k Z Computation of y = f (x) over Z begin Compute a bound β on max( f ); Pick n 1,... n k, pairwise prime, s.t. n 1... n k > β; for i = 1... k do Compute y i = f (x mod n i ) mod n i Compute y = CRT(y 1,..., y k ) end i CRT : Z/n 1 Z Z/n k Z Z/(n 1... n k )Z (x 1,..., x k ) k i=1 x iπ i Y i mod Π Π = k i=1 n i where Π i = Π/n i Y i = Π 1 i mod n i

26 Integers: Chinese remainder algorithm Z/(n 1... n k )Z Z/n 1 Z Z/n k Z Computation of y = f (x) over Z begin Compute a bound β on max( f ); Pick n 1,... n k, pairwise prime, s.t. n 1... n k > β; for i = 1... k do Compute y i = f (x mod n i ) mod n i ; /* Evaluation */ Compute y = CRT(y 1,..., y k ); /* Interpolation */ end i CRT : Z/n 1 Z Z/n k Z Z/(n 1... n k )Z (x 1,..., x k ) k i=1 x iπ i Y i mod Π Π = k i=1 n i where Π i = Π/n i Y i = Π 1 i mod n i

27 Chinese remaindering and evaluation/interpolation Evaluate P in a Reduce P modulo X a

28 Chinese remaindering and evaluation/interpolation Evaluate P in a Reduce P modulo X a Polynomials Evaluation: P mod X a Evaluate P in a Interpolation: P = k i=1 Q Q j i (X a j ) j i (a i a j )

29 Chinese remaindering and evaluation/interpolation Evaluate P in a Reduce P modulo X a Polynomials Evaluation: P mod X a Evaluate P in a Interpolation: P = k i=1 Q Q j i (X a j ) j i (a i a j ) Integers N mod m Evaluate N in m N = k i=1 a i j i m j( j i m j) 1[m i ]

30 Chinese remaindering and evaluation/interpolation Evaluate P in a Reduce P modulo X a Polynomials Evaluation: P mod X a Evaluate P in a Interpolation: P = k i=1 Q Q j i (X a j ) j i (a i a j ) Integers N mod m Evaluate N in m N = k i=1 a i j i m j( j i m j) 1[m i ] Analogy: complexities over Z over K [X] For a program with T algebr. algebraic operations: size of coefficients O ( log result T algebr. ) degree of polynomials O ( deg(result) T algebr. )

31 Chinese remaindering and evaluation/interpolation Evaluate P in a Reduce P modulo X a Polynomials Evaluation: P mod X a Evaluate P in a Interpolation: P = k i=1 Q Q j i (X a j ) j i (a i a j ) Integers N mod m Evaluate N in m N = k i=1 a i j i m j( j i m j) 1[m i ] Analogy: complexities over Z over K [X] For a program with T algebr. algebraic operations: size of coefficients O ( log result T algebr. ) det(n, A ) = O (n log A n ω ) degree of polynomials O ( deg(result) T algebr. ) det(n, d) = O (nd n ω )

32 Early termination Classic Chinese remaindering bound β on the result Choice of the n i : such that n 1... n k > β deterministic algorithm

33 Early termination Classic Chinese remaindering bound β on the result Choice of the n i : such that n 1... n k > β deterministic algorithm Early termination For each new modulo n i : reconstruct y i = f (x) mod n 1 n i If y i = y i 1 terminated if n i chosen at random: randomized algorithm (Monte Carlo)

34 Early termination Classic Chinese remaindering bound β on the result Choice of the n i : such that n 1... n k > β deterministic algorithm Early termination For each new modulo n i : reconstruct y i = f (x) mod n 1 n i If y i = y i 1 terminated if n i chosen at random: randomized algorithm (Monte Carlo) Advantage: Adaptive number of moduli depending on the output value Interesting when pessimistic bound: sparse/structured matrices,... no bound available

35 Redundant residues codes Principle: Chinese remaindering based parallelization Byzantines faults affecting some modular computations Fault tolerant reconstruction Algorithm Based Fault Tolerance (ABFT)

36 Plan Introduction High performance computing and cloud Related works on trusting the clouds Cloud computing and performance ABFT Redundant residue codes Homomorphic residue system Over Z: Mandelbaum algorithm Over K[X]: Reed Solomon point of view Generalization Adaptive approach First approach and the gap Experiments Early termination

37 Mandelbaum algorithm over Z Chinese Remainder Theorem x Z x 1 x 2... x k where p 1 p k > x and x i = x mod p i i

38 Mandelbaum algorithm over Z Chinese Remainder Theorem x Z x 1 x 2... x k x k+1... x n where p 1 p n > x and x i = x mod p i i

39 Mandelbaum algorithm over Z Chinese Remainder Theorem x Z x 1 x 2... x k x k+1... x n where p 1 p n > x and x i = x mod p i i Definition { (n, k)-code: C = { x < p1... p (x 1,..., x n ) Z p1 Z pn s.t.!x, k x i = x mod p i i }

40 Principle Property X C iff X < Π k. Π n = p 1 p n p 1 p 2... p k p k+1... p n Π k = p 1 p k Redundancy : r = n k

41 Principle Transmission Channel Computation

42 Principle Noisy Transmission Channel Unsecured Computation

43 Principle Noisy Transmission Channel Unsecured Computation Input A Encoding A = (A 1,... A n) Computation Solution x < M n x < M k x = (r 1,..., r n) Correction x = (x 1,..., x n) Decoding

44 Properties of the code Error model: Error: E = X X Error support: I = {i 1... n, E 0 mod p i } Impact of the error: Π F = i I p i

45 Properties of the code Error model: Error: E = X X Error support: I = {i 1... n, E 0 mod p i } Impact of the error: Π F = i I p i Detects up to r errors: If X = X + E with X C, #I r, X > Π k. Redundancy r = n k, distance: r + 1 can correct up to r 2 errors in theory More complicated in practice...

46 Correction i / I : E mod p i = 0 E is a multiple of Π V : E = Z Π V = Z i / I gcd(e, Π) = Π V Mandelbaum 78: rational reconstruction X = X E = X Z Π v X Π = X Π Z Π F X Π Z Π F 1 2Π 2 F Z Π F = E X Π is a convergent of Π rational reconstruction of X mod Π

47 Correction capacity Mandelbaum 78: 1 symbol = 1 residue Polynomial algorithm if e (n k) log p min log 2 log p max+log p min worst case: exponential (random perturbation) Goldreich Ron Sudan 99 weighted residues equivalent Guruswami Sahai Sudan 00 invariably polynomial time

48 Correction capacity Mandelbaum 78: 1 symbol = 1 residue Polynomial algorithm if e (n k) log p min log 2 log p max+log p min worst case: exponential (random perturbation) Goldreich Ron Sudan 99 weighted residues equivalent Guruswami Sahai Sudan 00 invariably polynomial time Interpretation: Errors have variable weights depending on their impact Π F = i I p i Example: X = 20, p 1 = 2, p 2 = 3, p 3 = error on X mod 2, or X mod 3, can be corrected but not on X mod 101

49 Plan Introduction High performance computing and cloud Related works on trusting the clouds Cloud computing and performance ABFT Redundant residue codes Homomorphic residue system Over Z: Mandelbaum algorithm Over K[X]: Reed Solomon point of view Generalization Adaptive approach First approach and the gap Experiments Early termination

50 Analogy with Reed Solomon Gao02 Reed-Solomon decoding by extended Euclidean Alg: Chinese Remaindering over K [X] p i = X a i Encoding = evaluation in a i Decoding = interpolation Correction = Extended Euclidean algorithm étendu

51 Analogy with Reed Solomon Gao02 Reed-Solomon decoding by extended Euclidean Alg: Chinese Remaindering over K [X] p i = X a i Encoding = evaluation in a i Decoding = interpolation Correction = Extended Euclidean algorithm étendu Generalization for p i of degrees > 1 Variable impact, depending on the degree of p i Necessary unification [Sudan 01,...]

52 Plan Introduction High performance computing and cloud Related works on trusting the clouds Cloud computing and performance ABFT Redundant residue codes Homomorphic residue system Over Z: Mandelbaum algorithm Over K[X]: Reed Solomon point of view Generalization Adaptive approach First approach and the gap Experiments Early termination

53 Generalized point of view: amplitude code Over a Euclidean ring A with a Euclidean function ν Distance : A A R + (x, y) i x y[p i ] log 2 ν (P i ) Definition (n,k) amplitude code C = {x A : ν(x) < κ}, n = log 2 Π, k = log 2 κ.

54 Generalized point of view: amplitude code Over a Euclidean ring A with a Euclidean function ν Distance : A A R + (x, y) i x y[p i ] log 2 ν (P i ) Definition (n,k) amplitude code C = {x A : ν(x) < κ}, n = log 2 Π, k = log 2 κ. Property (Quasi MDS) d > n k in general, and d n k + 1 over K [X]. correction rate = maximal amplitude of an error that can be corrected

55 Advantages Generalization over any Euclidean ring Natural representation of the amount of information No need to sort moduli Finer correction capacities

56 Advantages Generalization over any Euclidean ring Natural representation of the amount of information No need to sort moduli Finer correction capacities Adaptive decoding: taking advantage of all the available redundancy Early termination: with no a priori knowledge of a bound on the result

57 Interpretation of Mandelbaum s algorithm Remark Rational reconstruction Algorithm Partial Extended Euclidean Property The Extended Euclidean Algorithm, applied to (E, Π) and to (X = X + E, Π), performs the same first iterations until r i < Π V. u i 1 E + v i 1 Π = Π v u i E + v i Π = 0 u i 1 X + v i 1 Π = r i 1 u i X + v i Π = r i u i X = r i

58 Amplitude decoding, with static correction capacity Amplitude based decoder over R Data: Π, X Data: τ R + τ < ν(π) 2 : bound on the maximal error amplitude Result: X R: corrected message s.t. ν(x)4τ 2 ν(π) begin α 0 = 1, β 0 = 0, r 0 = Π; α 1 = 0, β 1 = 1, r 1 = X ; i = 1; while (ν(r i ) > ν(π)/2τ) do Let r i 1 = q i r i + r i+1 be the Euclidean division of r i 1 by r i ; α i+1 = α i 1 q i α i ; β i+1 = β i 1 q i β i ; i = i + 1; return X = r i β i end reaches the quasi-maximal correction capacity

59 reaches the quasi-maximal correction capacity requires a a priori knowledge of τ How to make the correction capacity adaptive? Amplitude decoding, with static correction capacity Amplitude based decoder over R Data: Π, X Data: τ R + τ < ν(π) 2 : bound on the maximal error amplitude Result: X R: corrected message s.t. ν(x)4τ 2 ν(π) begin α 0 = 1, β 0 = 0, r 0 = Π; α 1 = 0, β 1 = 1, r 1 = X ; i = 1; while (ν(r i ) > ν(π)/2τ) do Let r i 1 = q i r i + r i+1 be the Euclidean division of r i 1 by r i ; α i+1 = α i 1 q i α i ; β i+1 = β i 1 q i β i ; i = i + 1; return X = r i β i end

60 Adaptive approach Multiple goals: With a fixed n, the correction capacity depends on a bound on X pessimistic estimate how to take advantage of all the available redundancy? redondance effective utilisable X borne sur X redondance utilisee

61 Adaptive approach Multiple goals: With a fixed n, the correction capacity depends on a bound on X pessimistic estimate how to take advantage of all the available redundancy? redondance effective utilisable X borne sur X redondance utilisee Allow early termination: variable n and unknown bound

62 Plan Introduction High performance computing and cloud Related works on trusting the clouds Cloud computing and performance ABFT Redundant residue codes Homomorphic residue system Over Z: Mandelbaum algorithm Over K[X]: Reed Solomon point of view Generalization Adaptive approach First approach and the gap Experiments Early termination

63 A first adaptive approach Termination criterion in the Extended Euclidean alg.: α i+1 Π β i+1 E = 0 E = α i+1 Π/β i+1 test if β j divides Π check if X satisfies:ν(x) ν(π) 4ν(β j ) 2 But several candidates are possible discrimination by a post-condition on the result

64 A first adaptive approach Termination criterion in the Extended Euclidean alg.: α i+1 Π β i+1 E = 0 E = α i+1 Π/β i+1 test if β j divides Π check if X satisfies:ν(x) ν(π) 4ν(β j ) 2 But several candidates are possible discrimination by a post-condition on the result Example x = 23 with 0 error x = 2 with 1 error p i x i 2 3 2

65 Detecting a gap α i Π β i (X + E) = r i α i Π β i E = r i + β i X r i β i X

66 Detecting a gap α i Π β i (X + E) = r i α i Π β i E = r i + β i X r i β i X

67 Detecting a gap α i Π β i (X + E) = r i α i Π β i E = r i + β i X r i β i X

68 Detecting a gap α i Π β i (X + E) = r i α i Π β i E = r i + β i X r i β i X X = r i /β i At the final iteration: ν(r i ) ν(β i X) If necessary, a gap appears between r i 1 and r i.

69 Detecting a gap α i Π β i (X + E) = r i α i Π β i E = r i + β i X r i β i X 2 g X = r i /β i At the final iteration: ν(r i ) ν(β i X) If necessary, a gap appears between r i 1 and r i. Introduce a blank 2 g in order to detect a gap > 2 g

70 Detecting a gap α i Π β i (X + E) = r i α i Π β i E = r i + β i X r i β i X 2 g X = r i /β i At the final iteration: ν(r i ) ν(β i X) If necessary, a gap appears between r i 1 and r i. Introduce a blank 2 g in order to detect a gap > 2 g

71 Detecting a gap α i Π β i (X + E) = r i α i Π β i E = r i + β i X r i β i X 2 g X = r i /β i At the final iteration: ν(r i ) ν(β i X) If necessary, a gap appears between r i 1 and r i. Introduce a blank 2 g in order to detect a gap > 2 g

72 Detecting a gap α i Π β i (X + E) = r i α i Π β i E = r i + β i X r i β i X 2 g X = r i /β i At the final iteration: ν(r i ) ν(β i X) If necessary, a gap appears between r i 1 and r i. Introduce a blank 2 g in order to detect a gap > 2 g

73 Detecting a gap α i Π β i (X + E) = r i α i Π β i E = r i + β i X r i β i X 2 g X = r i /β i At the final iteration: ν(r i ) ν(β i X) If necessary, a gap appears between r i 1 and r i. Introduce a blank 2 g in order to detect a gap > 2 g Property Loss of correction capacity: very small in practice Test of the divisibility for the remaining candidates Strongly reduces the number of divisibility tests

74 Plan Introduction High performance computing and cloud Related works on trusting the clouds Cloud computing and performance ABFT Redundant residue codes Homomorphic residue system Over Z: Mandelbaum algorithm Over K[X]: Reed Solomon point of view Generalization Adaptive approach First approach and the gap Experiments Early termination

75 Experiments Size of the error g = 2 1/446 1/765 1/1118 2/1183 2/4165 1/7907 g = 3 1/244 1/414 1/576 2/1002 2/2164 1/4117 g = 5 1/53 1/97 1/153 2/262 1/575 1/1106 g = 10 1/1 1/3 1/9 1/14 1/26 1/35 g = 20 1/1 1/1 1/1 1/1 1/1 1/1 Table: Number of remaining candidates after the gap detection: c /d means d candidates with a gap > 2 g, and c of them passed the divisibility test. n 6001 (3000 moduli), κ 201 (100 moduli).

76 Experiments Divisibility Gap g=2 Gap g=5 Gap g=10 Threshold T= Time (s) Size of the errors Figure: Comparison for n (m = 1300 moduli of 20 bits), κ 6001 (300 moduli) and τ (about 500 moduli).

77 Experiments 5 Time (s) Gap g=5 Gap g=7 Gap g=10 Gap g=20 Threshold T= Size of the errors Figure: Comparison for n (m = moduli of 20 bits), κ (8500 moduli) and τ (500 moduli). Gap: Euclidean Algorithm down to the end overhead

78 Plan Introduction High performance computing and cloud Related works on trusting the clouds Cloud computing and performance ABFT Redundant residue codes Homomorphic residue system Over Z: Mandelbaum algorithm Over K[X]: Reed Solomon point of view Generalization Adaptive approach First approach and the gap Experiments Early termination

79 Early termination Input Output X Master Secured workers Fork tasks P2P workers Empty or X residues Certifier List of candidates Decoder Y CRT Lifter Figure: Fault tolerant distributed computation with early termination

80 Conclusion Residue systems : technology suited to cloud computing: parallelism, fault-tolerant, verification New metric for redundant residue codes: Unification with finer bounds on the correction capacities Enables adaptive decoding Adaptative decoding and early termination Gap method: limited overhead, better performances Framework for interactive computing with a cloud Perspective ABFT in exact linear algebra, determinant [LinBox] ABFT with (exact) floating point arithmetic Theoretical efficiency of the gap method, Generalization to adaptive list decoding [Sudan, Guruswami]

81 Jean-Louis Roch s References for this talk Majid Khonji, Clément Pernet, Jean-Louis Roch, Thomas Roche, and Thomas Stalinski. Output-sensitive decoding for redundant residue systems. In ISSAC 10: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, pages , New York, NY, USA, Jul ACM. Axel W. Krings, Jean-Louis Roch, Samir Jafar, and Séebastien Varrette. A Probabilistic Approach for Task and Result Certification of Large-scale Distributed Applications in Hostile Environments. In European Grid Conference EGC 2005 ( Amsterdam, The Netherlands, February Springer-Verlag, LNCS. Jean-Louis Roch and Sébastien Varrette. Probabilistic Certification of Divide & Conquer Algorithms on Global Computing Platforms. Application to Fault-Tolerant Exact Matrix-Vector Product. In Parallel Symbolic Computation 07 (PASCO 07), London, Ontario, Canada, July ACM. Thomas Roche, Jean-Louis Roch, and Matthieu Cunche. Algorithm-based fault tolerance applied to P2P computing networks. In The First International Conference on Advances in P2P Systems, pages , Sliema, Malta, Oct IEEE.