An easy to digest intro to resilient algorithms data structures intro by Giuseppe F.Italiano, Universita di Roma "Tor Vergata" (mirror)

1. Resilient Algorithms and
Data Structures
Giuseppe F. Italiano
Università di Roma “Tor Vergata”

2. Some advertising first
School on Graph Theory, Algorithms and Applications
Erice (Sicily), September 25 - October 3, 2011
Please send your best students /postdocs /junior ….

3. Memory Errors
Common in Practice
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4. 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
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5. 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, … ]
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6. 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
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7. 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
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8. 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, … )
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9. Impact of Memory Errors
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10. How Common are Memory Errors?
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11. How Common are Memory Errors?
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12. 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!
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13. 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
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14. 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):
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15. Memory Errors in Space
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16. Memory Errors in Space
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17. Memory Errors in Space
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18. 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…

19. 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)
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20. 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…

21. A small example
Classical algorithms may not be correct in the
presence of (even very few) memory errors
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A
B
Out
An example: merging two ordered lists
Θ(n)
Θ(n)
Θ(n2)
inversions
...
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80
...
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80
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22. 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…
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23. 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
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24. 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.
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25. 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
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26. 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)
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27. 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.
………………………………………
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28. 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
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29. 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
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30. 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
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ordered
Faithfully
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• Faulty key = corrupted
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31. 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, ….)
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32. 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
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33. 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]:
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34. 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]:
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35. search(5) = false
Resilient Binary Search
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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)
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36. Trivially Resilient Binary Search
Can search in O(δ log n) time during δ memory errors
Trivially Resilient Binary Search
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37. 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]
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38. 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
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39. 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
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40. 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:
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41. 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
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42. 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]
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43. 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
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44. 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]
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45. 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
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46. 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
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47. 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
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48. 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)?
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49. Thank You!
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My memory’s terrible these days…