Date post: | 20-Jan-2017 |
Category: |
Engineering |
Upload: | stephen-larroque |
View: | 218 times |
Download: | 1 times |
Some advertising first
School on Graph Theory, Algorithms and Applications Erice (Sicily), September 25 - October 3, 2011
Please send your best students /postdocs /junior ….
Outline of the Talk
1. Motivation and Model 2. Resilient Algorithms: • Sorting and Searching
3. Resilient Data Structures • Priority Queues • Dictionaries
4. Conclusions and Open Problems
4
Memory Errors
Memory error: one or multiple bits read differently from how they were last written.
Many possible causes: • electrical or magnetic interference (cosmic rays) • hardware problems (bit permanently damaged) • corruption in data path between memories and processing units
Errors in DRAM devices concern for a long time [May & Woods 79, Ziegler et al 79, Chen & Hsiao 84, Normand 96, O’Gorman et al 96, Mukherjee et al 05, … ]
5
Memory Errors
Soft Errors: Randomly corrupt bits, but do not leave any physical damage --- cosmic rays
Hard Errors: Corrupt bits in a repeatable manner because of a physical defect (e.g., stuck bits) --- hardware problems
6
Error Correcting Codes (ECC)
Error correcting codes (ECC) allow detection and correction of one or multiple bit errors
Typical ECC is SECDED (i.e., single error correct, double error detect)
Chip-Kill can correct up to 4 adjacent bits at once
ECC has several overheads in terms of performance (33%), size (20%) and money (10%).
ECC memory chips are mostly used in memory systems for server machines rather than for client computers
7
Impact of Memory Errors
Consequence of a memory error is system dependent
1. Correctable errors : fixed by ECC
2. Uncorrectable errors :
2.1. Detected : Explicit failure (e.g., a machine reboot)
2.2. Undetected : 2.2.1. Induced failure (e.g., a kernel panic) 2.2.2. Unnoticed (but application corrupted, e.g., segmentation fault, file not found, file not readable, … )
8
How Common are Memory Errors? [Schroeder et al 2009] experiments 2.5 years (Jan 06 – Jun 08) on Google fleet (104 machines, ECC memory)
Memory errors are NOT rare events! 12
Memory Errors
Not all machines (clients) have ECC memory chips.
Increased demand for larger capacities at low cost just makes the problem more serious – large clusters of inexpensive memories
Need of reliable computation in the presence of memory faults
13
Memory Errors
• Memory errors can cause security vulnerabilities: Fault-based cryptanalysis [Boneh et al 97, Xu et al 01, Bloemer & Seifert 03] Attacking Java Virtual Machines [Govindavajhala & Appel 03] Breaking smart cards [Skorobogatov & Anderson 02, Bar-El et al 06]
• Avionics and space electronic systems: Amount of cosmic rays increase with altitude (soft errors)
Other scenarios in which memory errors have impact (and seem to be modeled in an adversarial setting):
14
Recap on Memory Errors
1. Memory errors are NOT rare: even a small cluster of computers with few GB per node can experience one bit error every few minutes.
18
I know my PIN number: it’s my name I can’t remember…
Memory Errors
Mem. size Mean Time Between Failures
512 MB 2.92 hours 1 GB 1.46 hours 16 GB 5.48 minutes 64 GB 1.37 minutes 1 TB 5.13 seconds
In the field study, Google researchers observed mean error rates of 2,000 – 6,000 per GB per year (25,000 – 75,000 FIT/Mbit)
19
Recap on Memory Errors
2. Memory errors can be harmful: uncorrectable memory errors cause some catastrophic event (reboot, kernel panic, data corruption, …)
20
I’m thinking of getting back into crime, Luigi. Legitimate business is too corrupt…
A small example
Classical algorithms may not be correct in the presence of (even very few) memory errors
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
A
B
Out
An example: merging two ordered lists
Θ(n) Θ(n)
Θ(n2) inversions
... 11 12 20 13
80
... 2 3 4 9 10 80
21
Recap on Memory Errors
3. ECC may not be available (or may not be enough): No ECC in inexpensive memories. ECC does not guarantee complete fault coverage; expensive; system halt upon detection of uncorrectable errors; service disruption; etc… etc…
22
Resilient Algorithms and Data Structures
Resilient Algorithms and Data Structures: Capable of tolerating memory errors on data (even
throughout their execution) without sacrificing correctness, performance and storage space
Make sure that the algorithms and data structures we design are capable of dealing with memory errors
23
Faulty- Memory Model [Finocchi, I. 04]
• Memory fault = the correct data stored in a memory location gets altered (destructive faults)
• Faults can appear at any time in any memory location simultaneously
• Assumptions: – Only O(1) words of reliable memory (safe memory) – Corrupted values indistinguishable from correct ones
Wish to produce correct output on uncorrupted data (in an adversarial model)
• Even recursion may be problematic in this model.
24
Terminology
δ = upper bound known on the number of memory errors (may be function of n)
α = actual number of memory errors (happen during specific execution)
Note: typically α ≤ δ
All the algorithms / data structure described here need to know δ in advance
25
Other Faulty Models
Design of fault-tolerant alg’s received attention for 50+ years
Liar Model [Ulam 77, Renyi 76,…]
Comparison questions answered by a possibly lying adversary. Can exploit query replication strategies.
Fault-tolerant sorting networks [Assaf Upfal 91, Yao Yao 85,…]
Comparators can be faulty. Exploit substantial data replication using fault-free data replicators.
Parallel Computations [Huang et al 84, Chlebus et al 94, …]
Faults on parallel/distributed architectures: PRAM or DMM simulations (rely on fault-detection mechanisms)
26
Other Faulty Models
Robustness in Computational Geometry [Schirra 00, …]
Faults from unreliable computation (geometric precision) rather than from memory errors
Noisy / Unreliable Computation [Bravermann Mossel 08]
Faults (with given probability) from unreliable primitives (e.g., comparisons) rather than from memory errors
Memory Checkers [Blum et al 93, Blum et al 95, …]
Programs not reliable objects: self-testing and self-correction. Essential error detection and error correction mechanisms.
………………………………………
27
Outline of the Talk
1. Motivation and Model 2. Resilient Algorithms: • Sorting and Searching
3. Resilient Data Structures • Priority Queues • Dictionaries
4. Conclusions and Open Problems
28
Resilient Sorting
We are given a set of n keys that need to be sorted
Q1. Can sort efficiently correct values in presence of memory errors? Q2. How many memory errors can tolerate in the worst case if we wish to maintain optimal time and space?
Value of some keys may get arbitrarily corrupted
We cannot tell which is faithful and which is corrupted
29
Terminology
• Faithfully ordered sequence = ordered except for corrupted keys
• Resilient sorting algorithm = produces a faithfully ordered sequence (i.e., wish to sort correctly all the uncorrupted keys)
• Faithful key = never corrupted
1 2 3 4 5 6 7 8 9 10 ordered Faithfully
80
• Faulty key = corrupted
30
Trivially Resilient Resilient variable: consists of (2δ+1) copies
x1, x2, …, x2δ+1 of a standard variable x
Value of resilient variable given by majority of its copies: • cannot be corrupted by faults • can be computed in linear time and constant space
[Boyer Moore 91]
Trivially-resilient algorithms and data structures have Θ(δ) multiplicative overheads in terms of time and space
Note: Trivially-resilient does more than ECC (SECDED, Chip-Kill, ….)
31
Trivially Resilient Sorting
Can trivially sort in O(δ n log n) time during δ memory errors
Trivially Resilient Sorting
O(n log n) sorting algorithm able to tolerate only O (1) memory errors
32
Resilient Sorting
Comparison-based sorting algorithm that takes O(n log n + δ2) time to run during δ memory errors
O(n log n) sorting algorithm able to tolerate up to O ((n log n)1/2) memory errors
Any comparison-based resilient O(n log n) sorting algorithm can tolerate the corruption of at most O ((n log n)1/2) keys
Upper Bound [Finocchi, Grandoni, I. 05]:
Lower Bound [Finocchi, I. 04]:
33
Resilient Sorting (cont.)
Randomized integer sorting algorithm that takes O(n + δ2) time to run during δ memory errors
O(n) randomized integer sorting algorithm able to tolerate up to O(n1/2) memory errors
Integer Sorting [Finocchi, Grandoni, I. 05]:
34
search(5) = false
Resilient Binary Search
2 3 4 5 8 9 13 20 26 1 7 80 10
Wish to get correct answers at least on correct keys:
search(s) either finds a key equal to s, or determines that no correct key is equal to s
If only faulty keys are equal to s, answer uninteresting (cannot hope to get trustworthy answer)
35
Trivially Resilient Binary Search
Can search in O(δ log n) time during δ memory errors
Trivially Resilient Binary Search
36
Resilient Searching
Randomized algorithm with O(log n + δ) expected time [Finocchi, Grandoni, I. 05]
Deterministic algorithm with O(log n + δ) time [Brodal et al. 07]
Upper Bounds :
Lower Bounds : Ω(log n + δ) lower bound (deterministic) [Finocchi, I. 04]
Ω(log n + δ) lower bound on expected time [Finocchi, Grandoni, I. 05]
37
Resilient Dynamic Programming
Running time O(nd + δd+1) and space usage O(nd + nδ) Can tolerate up to δ = O(nd/(d+1)) memory errors
[Caminiti et al. 10]
d-dim. Dynamic Programming
38
Outline of the Talk
1. Motivation and Model 2. Resilient Algorithms: • Sorting and Searching
3. Resilient Data Structures • Priority Queues • Dictionaries
4. Conclusions and Open Problems
39
Resilient Data Structures
Algorithms affected by errors during execution
Data structures affected by errors in lifetime
Data structures more vulnerable to memory errors than algorithms:
40
Resilient Priority Queues
Maintain a set of elements under insert and deletemin
insert adds an element
deletemin deletes and returns either the minimum uncorrupted value or a corrupted value
Consistent with resilient sorting
41
Resilient Priority Queues
Upper Bound : Both insert and deletemin can be implemented
in O(log n + δ) time [Jorgensen et al. 07] (based on cache-oblivious priority queues)
Lower Bound : A resilient priority queue with n > δ elements must
use Ω(log n + δ) comparisons to answer an insert followed by a deletemin
[Jorgensen et al. 07]
42
Resilient Dictionaries
Maintain a set of elements under insert, delete and search
insert and delete as usual, search as in resilient searching:
Again, consistent with resilient sorting
search(s) either finds a key equal to s, or determines that no correct key is equal to s
43
Resilient Dictionaries
Randomized resilient dictionary implements each operation in O(log n + δ) time
[Brodal et al. 07]
More complicated deterministic resilient dictionary implements each operation in O(log n + δ) time
[Brodal et al. 07]
44
Resilient Dictionaries
Pointer-based data structures
Faults on pointers likely to be more problematic than faults on keys
Randomized resilient dictionaries of Brodal et al. built on top of traditional (non-resilient) dictionaries
Our implementation built on top of AVL trees
45
Outline of the Talk
1. Motivation and Model 2. Resilient Algorithms: • Sorting and Searching
3. Resilient Data Structures • Priority Queues • Dictionaries
4. Conclusions and Open Problems
46
Concluding Remarks
• Need of reliable computation in the presence of memory errors
• Investigated basic algorithms and data structures in the faulty memory model: do not wish to detect /correct errors, only produce correct output on correct data
• Tight upper and lower bounds in this model • After first tests, resilient implementations of
algorithms and data structures look promising
47
Future Work and Open Problems
• More (faster) implementations, engineering and experimental analysis?
• Resilient graph algorithms?
• Lower bounds for resilient integer sorting?
• Better faulty memory model?
• Resilient algorithms oblivious to δ?
• Full repertoire for resilient priority queues (delete, decreasekey, increasekey)?
48