Resilient Dynamic Programming

Transcription

1 Resilient Dynamic Programming Irene Finocchi, Saverio Caminiti, and Emanuele Fusco Dipartimento di Informatica, Sapienza Università di Roma via Salaria, Rome, Italy. {finocchi, caminiti, Kickoff AlgoDEEP Bertinoro, Italia. April (task C.1.1)

2 Outline 1 Introduction 2 A resilient framework for dynamic programming 3 Testing and experimental validation

3 Memories and faults Why should we care about memory faults in algorithm design? Memory faults happen: a large cluster of computers with a few gigabytes per node can experience one bit error every few minutes [Sah06]. Memory faults are harmful: undetected memory faults cause data corruption to spread; (potentially safety critical, e.g., avionics). Hardware solutions may be inadequate: fault-tolerant memory chips does not guarantee complete fault coverage; (expensive system halt upon detection of uncorrectable errors interruptions of service) [JNW08].

4 From liars to data corruption Algorithmic research related to memory errors has focused mainly on sorting and searching problems: late 70 s: Rényi [Rén94] and Ulam [Ula77]: twenty questions game against a liar, handling noise in binary search. Yao and Yao [YY85], and then [AU91, LM99, LMP97]: destructive faults in fault-tolerant sorting networks, comparison gates can destroy one of the input values.... [FI04] sorting in the faulty RAM model.

5 Faulty memories: an adversarial model Memory in a faulty-ram of word-size w is divided in three classes: a large unreliable memory: an adaptive adversary of unlimited computational power can modify up to δ memory words; O(1) safe memory words: the adversary can read but not modify this memory; O(1) private memory words: the adversary cannot even read this memory.

6 Local dependency dynamic programming edit distance Let e i,j be the edit distance between the prefix up to the i-th symbol of the input string X and the prefix up to the j-th symbol of the input string Y. e i,j := { ei 1,j 1 if i, j > 0 and x j = y i 1 + min {e i 1,j, e i,j 1, e i 1,j 1 } if i, j > 0 and x j y i (e 0,j = j, e i,0 = i.)

7 Correctness requirements Correctness of sorting and searching required only on uncorrupted values. In our setting, such a relaxed definition of correctness does not seem to be natural.

8 Correctness requirements Correctness of sorting and searching required only on uncorrupted values. In our setting, such a relaxed definition of correctness does not seem to be natural. We seek algorithms that correctly compute the edit distance between the two input strings, in spite of memory faults.

9 Tools Majority. Table decomposition. Fingerprinting.

10 Majority A variable can be made resilient by making 2δ + 1 copies. As at most δ of them can be altered by the adversary, the majority value is the correct value. The majority value can be read in time O(δ) and space O(1) [BM91].

11 Table decomposition The DP table is split in blocks of size δ δ. The boundaries of each block are written reliably in the faulty memory. δ 2 values result in roughly 5δ 2 memory words.

12 Fingerprinting A fingerprint for a column is computed as: ϕ k = v 1 v 2... v δ mod p where p is a prime number uniformly chosen at randomly in interval [n c 1, n c ] (where c is an appropriate constant).

13 Fingerprinting A fingerprint for a column is computed as: ϕ k = v 1 v 2... v δ mod p where p is a prime number uniformly chosen at randomly in interval [n c 1, n c ] (where c is an appropriate constant). Using logical shifts and Horner s rule, each fingerprint can be incrementally computed while generating the values v h : for h = 1 to δ do ϕ = ((ϕ 2 w ) + v h ) mod p end for

14 Block computation B i 1,j 1 B i 1,j B i,j 1 B i,j The first column of a block is computed reading reliably all values it depends from. ϕ 1

15 Block computation B i 1,j 1 B i 1,j B i,j 1 B i,j While computing the first column, fingerprint ϕ 1 is also computed. ϕ 1

16 Block computation B i 1,j 1 B i 1,j B i,j 1 B i,j While computing the first column, fingerprint ϕ 1 is also computed. ϕ 1

17 Block computation B i 1,j 1 B i 1,j B i,j 1 Bi,j While computing column k + 1, we produce two fingerprints, ϕ k+1 and ϕ k. ϕ k ϕ k ϕk+1

18 Block computation B i 1,j 1 B i 1,j B i,j 1 Bi,j Fingerprint ϕ k is then compared with ϕ k (i.e., the fingerprint produced while computing column k). ϕ k ϕ k ϕk+1

19 Block computation B i 1,j 1 B i 1,j B i,j 1 Bi,j If ϕ k ϕ k, the block is recomputed from scratch. ϕ k ϕ k ϕk+1

20 As a result we have: Theorem The edit distance between two strings of length n and m, with n m, can be correctly computed, with high probability, in: O(nm + αδ 2 ) time; O(nm) space, when δ is polynomial in n.

21 Generalizing Theorem A d-dimensional local dependency dynamic programming table M of size n d can be correctly computed, with high probability, in: O(n d + αδ d ) time; O(n d + nδ) space, when the actual number α δ of memory faults occurring during the computation is polynomial in n. (Edit distance, longest common subsequence, sequence alignment,...)

22 faultylib We are developing a library to test program behavior in presence of memory faults. Plugging in the library should be very easy: existing C/C++ code should require minimal changes to be tested with our library. Implementation of different (and meaningful) adversaries should be easy....

23 faultylib: usage FaultyUInt M[n+1u][m+1u]; // An n+1 X m+1 matrix of // faulty unsigned int... for (unsigned int i = 1; i <= n; i++) { for (unsigned int j = 1; j <= m; j++) { M[i][j] = min(1 + min(m[i-1][j], M[i][j-1]), M[i-1][j-1] + ((x[i-1]==y[j-1])? 0 : 1)); } }...

24 faultylib: faulty types implementation template <typename T> class Faulty : public FaultyBase {... private: T _val; T read() const { FaultyMM::getInstance()->faultBeforeRead(&_val, sizeof(t), context); return _val; } void write(t v) { _val = v; FaultyMM::getInstance()->faultAfterWrite(&_val, sizeof(t), context); } }... typedef Faulty<unsigned int> FaultyUInt;

25 faultylib: overriding operators... //Assignment operator template <typename Targ> Faulty & operator=(const Targ & v) { write((t)v); return *this; }... //OR template <typename Targ> bool operator (const Targ & v) const { return (read() (T)v); } }...

26 faultylib: adversaries implementation class REDAdversary : public Adversary {... virtual void faultafterwrite(void * location, size_t s, Context * cnt) { if ((cnt!= NULL) && (cnt->tag == EDMATRIX_TAG)) { MatrixContext * m = (MatrixContext *)cnt; unsigned int * i = (unsigned int *)location; if (m->getindex(0) == 3) if (m->getindex(1) == 7) *i = *i +3; } }...

27 Thanks! Thank you for your attention!

28 References [AU91] [BM91] [FI04] S. Assaf and E. Upfal. Fault tolerant sorting networks. SIAM J. Discrete Math., 4(4): , R. S. Boyer and J. S. Moore. Mjrty: A fast majority vote algorithm. In Automated Reasoning: Essays in Honor of Woody Bledsoe, pages , Irene Finocchi and Giuseppe F. Italiano. Sorting and searching in the presence of memory faults (without redundancy). In László Babai, editor, STOC, pages ACM, [JNW08] B. L. Jacob, S. W. Ng, and D. T. Wang. Memory Systems: Cache, DRAM, Disk. [LM99] Morgan Kaufmann, F. T. Leighton and Y. Ma. Tight bounds on the size of fault-tolerant merging and sorting networks with destructive faults. SIAM J. Comput., 29(1): , [LMP97] F. T. Leighton, Y. Ma, and C. G. Plaxton. Breaking the θ(n log 2 n) barrier for sorting with faults. J. Comput. Syst. Sci., 54(2): , [Rén94] [Sah06] [Ula77] [YY85] A. Rény. A diary on information theory. J. Wiley and Sons, Original publication: Napló az információelméletröl, Gondolat, Budapest, G. K. Saha. Software based fault tolerance: a survey. Ubiquity, 7(25), S. M. Ulam. Adventures of a mathematician. Charles Scribner s Sons, New York, A. C. Yao and F. F. Yao. On fault-tolerant networks for sorting. SIAM J. Comput., 14(1): , 1985.