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PROBABILISTIC ALGORITHMS for fun and pseudorandom profit
Tyler Treat / 12.5.2015
ABOUT THE SPEAKER
➤ Backend engineer at Workiva
➤ Messaging platform tech lead
➤ Distributed systems
➤ bravenewgeek.com @tyler_treat
Time
Data
Batch (days, hours)
Meh, data (I can store this on
my laptop)
Streaming (minutes, seconds)
Oi, data! (We’re gonna need a bigger boat…)
Real-Time™ (I need it now, dammit!)
Big data™ (IoT, sensors)
/dev/null
Time
Data
Batch (days, hours)
Meh, data (I can store this on
my laptop)
Streaming (minutes, seconds)
Oi, data! (We’re gonna need a bigger boat…)
Real-Time™ (I need it now, dammit!)
Big data™ (IoT, sensors)
Not Interesting
Kinda Interesting
Pretty Interesting
/dev/null
http://bravenewgeek.com/stream-processing-and-probabilistic-methods/
THIS TALK IS NOT
➤ About Samza, Storm, Spark Streaming et al.
➤ Strictly about stream-processing techniques
➤ Mathy
➤ Statistics-y
THIS TALK IS
➤ About basic probability theory
➤ About practical design trade-offs
➤ About algorithms & data structures
➤ About dealing with large or unbounded datasets
➤ A marriage of CS & engineering
OUTLINE
➤ Terminology & context
➤ Why probabilistic algorithms?
➤ Bloom filters & variants
➤ Count-min sketch
➤ HyperLogLog
Randomized Algorithms
Las Vegas Algorithms Monte Carlo Algorithms
Random Input
Correct result Gamble on speed
Deterministic speed Gamble on result
Randomized Algorithms
Las Vegas Algorithms Monte Carlo Algorithms
Random Input
Correct result Gamble on speed
Deterministic speed Gamble on result
DEFINING SOME TERMINOLOGY
➤ Online - processing elements as they arrive
➤ Offline - entire dataset is known ahead of time
➤ Real-time - hard constraint on response time
➤ A priori knowledge - something known beforehand
BATCH VS STREAMING
➤ Batch
➤ Offline
➤ Heuristics/multiple passes
➤ Data structures less important
documents search index
BATCH VS STREAMING
➤ Streaming
➤ Online, one pass
➤ Usually real-time (but not necessarily)
➤ Potentially unbounded
transactions
caches
fraud
analytics
3 DATA INTEGRATION QUESTIONS
➤ How do you get the data?
➤ How do you disseminate the data?
➤ How do you process the data?
3 DATA INTEGRATION QUESTIONS
➤ How do you get the data (quickly)?
➤ How do you disseminate the data (quickly)?
➤ How do you process the data (quickly)?
Denormalization is critical to performance at scale.
How to count the number of distinct document views across Wikipedia?
10b531cb-914c-4b3e-ac1d-11678dd72f7a
3,042,568
16-byte GUID 8-byte integer
10b531cb-914c-4b3e-ac1d-11678dd72f7a
5d5d5a78-f98f-4eee-bc83-762b3c78f1ea
3558d299-45ef-4fc9-b9ec-902e4943c7f8
6febb745-c987-4c51-afd2-90a55f357d7b
6f3f199e-4cc3-4c68-9d2a-00c31eb199f3
3,042,568
1,250,763
982,531
24,703,289
7,401,050
Wikipedia has ~38 million pages.
38,000,000 pages x
(16-byte guid + 8-byte integer)
≈ 1GB
➤ Not unreasonable for modern hardware
➤ Held in memory for lifetime of process so will move to old GC generations—expensive to collect!
➤ Now we want to track views per unique IP address
➤ >4 billion IPv4 addresses
➤ Naive solutions quickly become intractable
DISTRIBUTED SYSTEMS TRADE-OFFS
Consistency
Availability
Partition Tolerance
DATA PROCESSING TRADE-OFFS
Time
Accuracy
Space
HAVE YOUR CAKE AND EAT IT TOO?
Stream Processing
Batch Processing
App
The “Lambda Architecture”
Probabilistic algorithms trade accuracy for space and performance.
“Sketching” data structures make this trade by storing a summary of the dataset when storing it entirely is prohibitively expensive.
Bloom FiltersB. H. Bloom.
Space/Time Trade-offs in Hash Coding with Allowable Errors. 1970.
Answers a simple question: is this element a member of a set?
S ⊆ 𝕌 x ∈ S
SET MEMBERSHIP
➤ Is this URL malicious?
➤ Is this IP address blacklisted?
➤ Is this word contained in the document?
➤ Is this record in the database?
➤ Has this transaction been processed?
Hash Table
entry for each member
Bit Array
bit for each element in universe
0101101000101110010100100110101…
BLOOM FILTERS
➤ Bloom filters store set memberships
➤ Answers “not in set” or “probably in set”
Bloom Filter Secondary Store
Do you have key 1?
no
no
Do you have key 2?
Here’s key 2
yes
Necessary access
Here’s key 2
Do you have key 3?
no
yes
Unnecessary access
no
yes
no
BLOOM FILTERS
➤ 2 operations: add, lookup
➤ Allocate bit array of length m
➤ k hash functions
➤ Configure m and k for desired false-positive rate
BLOOM FILTERS
➤ Add element:
➤ Hash with k functions to get k indices
➤ Set bits at each index
➤ Lookup:
➤ Hash with k functions to get k indices
➤ Check bit at each index
➤ If any bit is unset, element not in set
BLOOM FILTERS
➤ Benefits:
➤ More space-efficient than hash table or bit array
➤ Can determine trade-off between accuracy and space
➤ Drawbacks:
➤ Some elements potentially more sensitive to false positives than others (solvable by partitioning)
➤ Can’t remove elements
➤ Requires a priori knowledge of the dataset
➤ Over-provisioned filter wastes space
Bloom filters are great for efficient offline processing, but what about streaming?
BLOOM FILTERS WITH A TWIST
➤ Rotating Bloom filters
➤ e.g. remember everything in the last hour
➤ Scalable Bloom Filters
➤ Dynamically allocating chained filters
➤ Stable Bloom Filters
➤ Continuously evict stale data
Scalable Bloom FiltersP. S. Almeida, C. Baquero, N. Preguiça, D. Hutchison.
Scalable Bloom Filters. 2007.
l
P0 P0 P0 P0
P0 = error prob. of 1 filter
l = # filters
P = compound error prob.
P=1- (1-P0)i=0
l-1
P0 = 0.1
P=1- (1-P0)i=0
l-1
SCALABLE BLOOM FILTERS
➤ Questions:
➤ When to add a new filter?
➤ How to place a tight upper bound on P?
SCALABLE BLOOM FILTERS
➤ When to add a new filter?
➤ Fill ratio p = # set bits / # bits
➤ Add new filter when target p is reached
➤ Optimal target p = 0.5 (math follows from paper)
SCALABLE BLOOM FILTERS
➤ How to place a tight upper bound on P?
➤ Apply tightening ratio r to P0, where 0 < r < 1
➤ Start with 1 filter, error probability P0
➤ When full, add new filter, error probability P1=P0r
➤ Results in geometric series:
➤ Series converges on target error probability P
P0 = 0.1 r = 0.5
P
P=1- (1-P0r i)i=0
l-1
SCALABLE BLOOM FILTERS
➤ Add elements to last filter
➤ Check each filter on lookups
➤ Tightening ratio r controls m and k for new filters
SCALABLE BLOOM FILTERS
➤ Benefits:
➤ Can grow dynamically to accommodate dataset
➤ Provides tight upper bound on false-positive rate
➤ Can control growth rate
➤ Drawbacks:
➤ Size still proportional to dataset
➤ Additional computation on adds (negligible amortized)
Stable Bloom FiltersF. Deng, D. Rafiei.
Approximately Detecting Duplicates for Streaming Data using Stable Bloom Filters. 2006.
DUPLICATE DETECTION
➤ Query processing
➤ URL crawling
➤ Monitoring distinct IP addresses
➤ Advertiser click streams
➤ Graph processing
Bloom filters are remarkably useful for dealing with graph data.
GRAPH PROCESSING
➤ Detecting cycles
➤ Pruning search space
➤ E.g. often used in bioinformatics
➤ Storing chemical structures, properties, and molecular fingerprints in filters to optimize searches and determine structural similarities
➤ Rapid classification of DNA sequences as large as the human genome
GRAPH PROCESSING
➤ Store crawled nodes in memory
➤ Set of nodes may be too large to fit in memory
➤ Store crawled nodes in secondary storage
➤ Too many searches to perform in limited time
Precisely eliminating duplicates in an unbounded stream isn’t feasible with
limited space and time.
Efficacy/Efficiency Conjecture: In many situations, a quick answer with an allowable error rate is better than a precise one that is slow.
Staleness Conjecture: In many situations, more recent data has more value than stale data.
STABLE BLOOM FILTERS
➤ Discards old data to make room for new data
➤ Replace bit array with array of d-bit counters
➤ Initialize counters to zero
➤ Maximum counter value Max = 2d - 1
STABLE BLOOM FILTERS
➤ Add element:
➤ Select P random counters and decrement by one
➤ Hash with k functions to get k indices
➤ Set counters at each index to Max
➤ Lookup:
➤ Hash with k functions to get k indices
➤ Check counter at each index
➤ If any counter is zero, element not in set
STABLE BLOOM FILTERS
➤ Classic Bloom filter a special case of SBF w/ d=1, P=0
➤ Tight upper bound on false positives
➤ FP rate asymptotically approaches configurable fixed constant (stable-point property)
➤ See paper for math and parameter settings
➤ Evicting data introduces false negatives
STABLE BLOOM FILTERS
➤ Benefits:
➤ Fixed memory allocation
➤ Evicts old data to make room for new data
➤ Provides tight upper bound on false positives
➤ Drawbacks:
➤ Introduces false negatives
➤ Additional computation on adds
Count-Min SketchG. Cormode, S. Muthukrishnan.
An Improved Data Stream Summary: The Count-Min Sketch and its Applications. 2003.
Can we count element frequencies using sub-linear space?
page views
94.136.205.1
132.208.90.15
54.222.151.15
7
4
11
COUNT-MIN SKETCH
➤ Approximates frequencies in sub-linear space
➤ Matrix with w columns and d rows
➤ Each row has a hash function
➤ Each cell initialized to zero
➤ When element arrives:
➤ Hash for each row
➤ Increment each counter by 1
➤ freq(element) = min counter value
COUNT-MIN SKETCH
➤ Why the minimum?
➤ Possibility for collisions between elements
➤ Counter may be incremented by multiple elements
➤ Taking minimum counter value gives closer approximation
COUNT-MIN SKETCH
➤ Benefits:
➤ Simple!
➤ Sub-linear space
➤ Useful for detecting “heavy hitters”
➤ Easy to track top-k by adding a min-heap
➤ Drawbacks:
➤ Biased estimator: may overestimate, never underestimates
➤ Better suited to Zipfian distributions & rare events
HyperLogLogP. Flajolet, É. Fusy, O. Gandouet, F. Meunier.
HyperLogLog: the analysis of a near-optimal cardinality estimation algorithm. 2007.
How do we count distinct things in a stream?
COUNTING PROBLEMS
➤ E.g. how many different words are used in Wikipedia?
➤ Counter per element explodes memory
➤ Usually requires memory proportional to cardinality
➤ Can we approximate cardinality with constant space?
HYPERLOGLOG
➤ The name: can estimate cardinality of set w/ cardinality Nmax using loglog(Nmax) + O(1) bits
➤ Hash element to integer
➤ Count number of leading 0’s in binary form of hash
➤ Track highest number of leading 0’s, n
➤ Cardinality ≈ 2n+1
HYPERLOGLOG
➤ stream = [“foo”, “bar”, “baz”, “qux”]
➤ h(“foo”) = 10100001
➤ h(“bar”) = 01110111
➤ h(“baz”) = 01110100
➤ h(“qux”) = 10100011
➤ n = 1
➤ |stream| ≈ 2n+1 = 22 = 4
It’s actually not magic but just a few really clever observations.
With 50/50 odds, how long will it take to flip 3 heads in a row? 20? 100?
HYPERLOGLOG
➤ Replace “heads” and “tails” with 0’s and 1’s
➤ Count leading consecutive 0’s in binary form of hash
➤ E.g. imagine a 4-bit hash, 16 possible values: ➤ 0000 4 leading 0’s
➤ 0001 3 leading 0’s
➤ 0011, 0010 2 leading 0’s
➤ 0100, 0111, 0110, 0101 1 leading 0’s
➤ 1111, 1110, 1001 1010, 1101, 1100 1011, 1000 0 leading 0’s
➤ Assume good hash function → 1/16 odds for each permutation
HYPERLOGLOG
➤ Track highest number of leading 0’s, n
➤ n = 0 → 8/16=1/2 odds
➤ n = 1 → 4/16=1/4 odds
➤ n = 2 → 2/16=1/8 odds
➤ n = 3 → 1/16 odds
➤ Cardinality ≈ how many things did we have to look?
➤ E.g. highest count = 1 → 1/4 odds → cardinality 4
HYPERLOGLOG
➤ 1/2 of all binary numbers start with 1
➤ Each additional bit cuts the probability in half:
➤ 1/4 start with 01
➤ 1/8 start with 001
➤ 1/16 start with 0001
➤ etc.
➤ P(run of length n) = 1 / 2n+1
➤ Seeing 001 has 1/8 probability, meaning we had to look at approximately 8 things til we saw it (cardinality 8)
➤ Cardinality ≈ prob-1 (reciprocal of probability)
What about outliers?
HYPERLOGLOG
➤ Use multiple buckets
➤ Use first few bits of hash to determine bucket
➤ Use remaining bits to count 0’s
➤ Each bucket tracks its own count
➤ Take harmonic mean of all buckets to get cardinality ➤ min(x1…xn) ≤ H(x1…xn) ≤ n min(x1…xn)
01011010001011100101001001101010bucket counting space
http://highscalability.com/blog/2012/4/5/big-data-counting-how-to-count-a-billion-distinct-objects-us.html
Number of distinct words in all of Shakespeare's work
HYPERLOGLOG
➤ Benefits:
➤ Constant memory
➤ Super fast (calculating MSB is cheap)
➤ Can give accurate count with <1% error
➤ Drawbacks:
➤ Has a margin of error (albeit small)
What did we learn?
Data processing has trade-offs.
Probabilistic algorithms trade accuracy for speed and space.
Often we only care about answers that are mostly correct but available now.
Sometimes the “right” answer is impossible to compute or simply doesn’t exist.
But mostly…
Probabilistic algorithms are just damn cool.
What about the code?
ALGORITHM IMPLEMENTATIONS
➤ Algebird - https://github.com/twitter/algebird ➤ Bloom filter
➤ Count-min sketch
➤ HyperLogLog
➤ stream-lib - https://github.com/addthis/stream-lib ➤ Bloom filter
➤ Count-min sketch
➤ HyperLogLog
➤ Boom Filters - https://github.com/tylertreat/BoomFilters ➤ Bloom filter
➤ Scalable Bloom filter
➤ Stable Bloom filter
➤ Count-min sketch
➤ HyperLogLog
OTHER COOL PROBABILISTIC ALGORITHMS
➤ Counting Bloom filter (and many other Bloom variations)
➤ Bloomier filter (encode functions instead of sets)
➤ Cuckoo filter (Bloom filter w/ cuckoo hashing)
➤ q-digest (quantile approximation)
➤ t-digest (online accumulation of rank-based statistics)
➤ Locality-sensitive hashing (hash similar items to same buckets)
➤ MinHash (set similarity)
➤ Miller–Rabin (primality testing)
➤ Karger’s algorithm (min cut of connected graph)
@tyler_treat
github.com/tylertreat
bravenewgeek.com
Thanks
We’re hiring!
BIBLIOGRAPHYAlmeida, P., Baquero, C., Preguica, N., Hutchison, D. 2007. Scalable Bloom Filters; http://gsd.di.uminho.pt/members/cbm/ps/dbloom.pdf
Bloom, B. 1970. Space/Time Trade-offs in Hash Coding with Allowable Errors; https://www.cs.upc.edu/~diaz/p422-bloom.pdf
Cormode, G., & Muthukrishnan, S. 2003. An Improved Data Stream Summary: The Count-Min Sketch and its Applications; http://dimacs.rutgers.edu/~graham/pubs/papers/cm-full.pdf
Deng, F., & Rafiei, D. 2006. Approximately Detecting Duplicates for Streaming Data using Stable Bloom Filters; https://webdocs.cs.ualberta.ca/~drafiei/papers/DupDet06Sigmod.pdf
Flajolet, P., Fusy, É, Gandouet, O., Meunier, F. 2007. HyperLogLog: The analysis of a near-optimal cardinality estimation algorithm; http://algo.inria.fr/flajolet/Publications/FlFuGaMe07.pdf
Stranneheim, H., Käller, M., Allander, T., Andersson, B., Arvestad, L., Lundeberg, J. 2010. Classification of DNA sequences using Bloom filters. Bioinformatics, 26(13); http://bioinformatics.oxfordjournals.org/content/26/13/1595.full.pdf
Tarkoma, S., Rothenberg, C., & Lagerspetz, E. 2011. Theory and Practice of Bloom Filters for Distributed Systems. IEEE Communications Surveys & Tutorials, 14(1); https://gnunet.org/sites/default/files/TheoryandPracticeBloomFilter2011Tarkoma.pdf
Treat, T. 2015. Stream Processing and Probabilistic Methods: Data at Scale; http://bravenewgeek.com/stream-processing-and-probabilistic-methods
