Cuckoo Filter: Practically Better than Bloom

Introduction
Cuckoo Filter
Analysis
Conclusion
Cuckoo Filter: practically better than Bloom
AED 2013/2014 Presentation
Alessandro Lenzi
Master in Computer Science and Networking
Università di Pisa and Scuola Superiore Sant’Anna
February 23, 2015
From Cuckoo Filter: Practically Better than Bloom
by Fan, Andersen, Kaminsky and Mitzenmacher [6]
Alessandro Lenzi Cuckoo Filter: practically better than Bloom

Introduction
Cuckoo Filter
Analysis
Conclusion
Table of contents
1 Introduction
Bloom Filters and extensions
2 Cuckoo Filter
3 Analysis
Insertion failure probability
Space analysis
Comparison with Bloom Filters
Benchmark

Introduction
Cuckoo Filter
Analysis
Conclusion
Section 1
Introduction

Introduction
Cuckoo Filter
Analysis
Conclusion
Lookup, Classiﬁcation and approximate set membership
Approximate set membership tests are often used for high
speed packet processing
Allow to rule out elements not in the set in a fast and space
eﬃcient manner
Both Lookup and Monitor require fast membership tests

Introduction
Cuckoo Filter
Analysis
Conclusion
Application of approximate set membership
Search on Preﬁx Lengths using Bloom Filters[1]
Associate a Bloom Filter to each possible preﬁx length
Perform parallel membership queries on all possible lengths.

Introduction
Cuckoo Filter
Analysis
Conclusion
BUFFALO [2]

Introduction
Cuckoo Filter
Analysis
Conclusion
BUFFALO [2]
Entire lookup in fast memory, at the expense of slightly longer
paths in worst case

Introduction
Cuckoo Filter
Analysis
Conclusion
BUFFALO [2]
Entire lookup in fast memory, at the expense of slightly longer
paths in worst case
Build a Bloom Filter BF(h) for each next hop h

Introduction
Cuckoo Filter
Analysis
Conclusion
Subsection 1

Introduction
Cuckoo Filter
Analysis
Conclusion
Bloom Filters
The most famous approximate set membership data structure.
Allows to represent an item in the set with 1.44log2(1/ε) bit, very
close to the information-theoretic minimum, log2(1/ε).

Introduction
Cuckoo Filter
Analysis
Conclusion
Bloom Filters
The probability of false positive ε can be made arbitrarily small at
the expense of space eﬃciency.
How Bloom Filters Work
The data structure is a bit array BF of size n

Introduction
Cuckoo Filter
Analysis
Conclusion
Bloom Filters
When x is inserted, BF[hi (x)] is set, for 1 ≤ i ≤ k

Introduction
Cuckoo Filter
Analysis
Conclusion
Bloom Filters
When x is inserted, BF[hi (x)] is set, for 1 ≤ i ≤ k
Lookup(x) returns true when
BF[h1(x)] = 1∧BF[h2(x)] = 1∧...∧BF[hk(x)] = 1.

Introduction
Cuckoo Filter
Analysis
Conclusion
Bloom Filter limitations
Classical Bloom Filter don’t allow deletion of elements from
the set without inserting false negatives.
Counting Bloom Filters allow deletions, using counters of c
bits.

Introduction
Cuckoo Filter
Analysis
Conclusion
bits.
Every lookup query causes k cache misses, impairing lookup
time

Introduction
Cuckoo Filter
Analysis
Conclusion
bits.
time
Blocked Bloom Filters provide spatial locality by forcing all k
hashes belonging to element x to fall on the same cache-line.

Introduction
Cuckoo Filter
Analysis
Conclusion
bits.
time
The number of cache misses falls to 1 for lookup query,
increasing false positive rates.

Introduction
Cuckoo Filter
Analysis
Conclusion
bits.
time
It is not possible to list elements in the set using only the
Bloom Filter data structure

Introduction
Cuckoo Filter
Analysis
Conclusion
bits.
time
Need of another way to store the elements to check in case of
false positive or to retrieve the set.

Introduction
Cuckoo Filter
Analysis
Conclusion
bits.
time
Need of another way to store the elements to check in case of
false positive or to retrieve the set.
Invertible Bloom Lookup Table [7] allow for deletion and
listing (successful w.h.p) of elements.

Introduction
Cuckoo Filter
Analysis
Conclusion
Section 2
Cuckoo Filter

Introduction
Cuckoo Filter
Analysis
Conclusion
Cuckoo Hashing [3] [4]
Exploits the power of 2-choices to provide lookup of an
element in a hash table in O(1) worst case, rather than in
expectation.
An element x can be stored either in T[h1(x)] or T[h2(x)].

Introduction
Cuckoo Filter
Analysis
Conclusion
expectation.
In case both positions are occupied:

Introduction
Cuckoo Filter
Analysis
Conclusion
expectation.
Select a victim v between T[h1(x)] and T[h2(x)]

Introduction
Cuckoo Filter
Analysis
Conclusion
expectation.
Try to store v in T[h1(v)] or T[h2(v)], otherwise repeat the
procedure as done for x.

Introduction
Cuckoo Filter
Analysis
Conclusion
expectation.
In case a threshold n on the maximum number of movements
is reached, rebuild the hash table from scratch.

Introduction
Cuckoo Filter
Analysis
Conclusion
expectation.
In case a threshold n on the maximum number of movements
is reached, rebuild the hash table from scratch.
Further improvements from buckets in place of single element
storage.

Introduction
Cuckoo Filter
Analysis
Conclusion
Cuckoo Filter - The idea
Intuition
Cuckoo Hashing can be used to check for set membership in
O(1), but is not as spatially eﬃcient as approximate set
membership data structures.
For set membership queries, store a ﬁngerprint, rather than
(key,value) pairs!

Introduction
Cuckoo Filter
Analysis
Conclusion
Intuition
(key,value) pairs!
Partial-key Cuckoo Hashing
Storing ﬁngerprint makes it impossible to ﬁnd alternative
locations and perform rehashing.

Introduction
Cuckoo Filter
Analysis
Conclusion
Intuition
(key,value) pairs!
Partial-key Cuckoo Hashing
Storing fingerprint makes it impossible to find alternative
locations and perform rehashing.
Partial key cuckoo hashing derivers alternative location using
x’s fingerprint f and the current position i = h1(x) or h2(x):
h2(x) = h1(x)⊕hash(f )

Introduction
Cuckoo Filter
Analysis
Conclusion
Cuckoo Filter - the data structure
m is the number of buckets
b is the number of entries (ﬁngerprints) for each bucket
f is the number of bits used to represent a ﬁngerprint

Introduction
Cuckoo Filter
Analysis
Conclusion
Cuckoo Filter - Insert
The ﬁngerprint of x, fx = ﬁngerprint(x) can be stored either in
bucket i = hash(x) or j = i ⊕hash(fx ) if one has less than b
elements. Hash fx used to spread colliding elements

Introduction
Cuckoo Filter
Analysis
Conclusion
If both are full, fv is picked at random between the 2b
ﬁngerprints elements stored in T[i] and T[j]

Introduction
Cuckoo Filter
Analysis
Conclusion
Replace fv in bucket h ∈ {i,j} with fx

Introduction
Cuckoo Filter
Analysis
Conclusion
Try to insert fv in j = h⊕hash(fv ), otherwise kick out another
element and repeat for at most maxKicks elements.

Introduction
Cuckoo Filter
Analysis
Conclusion
Try to insert fv in j = h⊕hash(fv ), otherwise kick out another
element and repeat for at most maxKicks elements.
Copies or elements with same ﬁngerprint can be inserted up to
2b times.
To increase the limit we can add counters.

Introduction
Cuckoo Filter
Analysis
Conclusion
Lookup and Delete Operations
Lookup
Given fx = ﬁngerprint(x), check for it in buckets i = hash(x)
and j = i ⊕hash(fx )
The cost of the operation is O(2b), with b usually small.
Delete

Introduction
Cuckoo Filter
Analysis
Conclusion
Lookup
Delete
Deleting x requires removing from T an instance of fx

Introduction
Cuckoo Filter
Analysis
Conclusion
Lookup
Delete
The delete procedure avoids the false deletion problem.
However requires that x must have been inserted previously.

Introduction
Cuckoo Filter
Analysis
Conclusion
Lookup
Delete
The delete procedure avoids the false deletion problem.
However requires that x must have been inserted previously.
The cost of the operation is O(2b).

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Section 3
Analysis

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Subsection 1

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Insertion failure
Insertion failure in cuckoo hashing [5]
In CH, insert(x) can fail only if, in the sequence
x1 = x,x2 = v,...,xp of elements removed and then reinserted in
the hash ∃i,j ∈ [0,p] : xi = xj .
Insertion failure in cuckoo ﬁlter

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Insertion failure
Obiously, a similar condition holds for CF.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Insertion failure
However insert(x) always fails if there are 2b +1 elements
with the same candidate buckets i and j.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Insertion failure
However insert(x) always fails if there are 2b +1 elements
with the same candidate buckets i and j.
This event probability dominates the insertion failure
probability

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Insertion Failure Probability (1)
CH’s insertion failure analysis cannot be used for CF, because:
Candidates positions for an element are not independent as,
given a certain i = hash(x), j may assume at most 2f diﬀerent
values. Thus it is often chosen only among a subset of
positions.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
positions.
For large f and m not too big, partial key hashing is similar to
cuckoo.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
positions.
cuckoo.
Otherwise insertion failure probability increases.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
positions.
cuckoo.
Otherwise insertion failure probability increases.
It is possible to provide a lower bound on the insertion failure
probability by calculating P{|Si,j | = 2b +1}, where Si,j = {x : x’s
candidate buckets are i and j}.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Insertion failure probability (2)
Since fx length will be much smaller than its hash one, we will
assume no collisions on ﬁngerprint’s hashing.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
For two elements x and y to collide they must:

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
1 Have the same ﬁngerprint ( fx = fy ), which occurs with
probability 1
2f and

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
probability 1
2f and
2 Have i or j in common, happens with probability 2
m

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
probability 1
2f and
m
Thus P{|Si,j | = q} = n
q ( 2
2f m
)q−1

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
probability 1
2f and
m
Thus P{|Si,j | = q} = n
q ( 2
2f m
)q−1
Insertion failure probability is at least n
2b+1 ( 2
2f m
)2b = Ω n
4bf
for m = cn

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Bits per element
Minimum Fingerprint Size
For a non-trivial insertion failure probability we need 4bf = Ω(n),
or, in other words, f = Ω(log(n/b))
Comparison with Bloom Filter
In CF Ω(log(n/b)) bits are used to represent an element,
against the ln(1/ε) = O(1) required by Bloom Filter.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Bits per element
CF is asymptotically worst, but in practice the denominator b
saves the day!

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Bits per element
saves the day!
With ε < 3%, the number of bits required to represent an
element with CF is less than the one required by BF!

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Bits per element
saves the day!
With ε < 3%, the number of bits required to represent an
element with CF is less than the one required by BF!
In practice f , for b large enough can be treated as a
reasonable sized constant.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Subsection 2
Space analysis

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Amortized space cost
False Positive Probability
A false positive occurs when searching for x if either bucket i or j
contains fy = fx .

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
contains fy = fx .
P{false positive for x} = 1−P{all diﬀerent from fx } = 1−P{an
element is diﬀerent from fx }2b = 1−(1− 1
2f )2b ≈ 2b
2f

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
contains fy = fx .
2f )2b ≈ 2b
2f
If we want a certain error probability ε, given b:

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
contains fy = fx .
2f )2b ≈ 2b
2f
f ≥ log2(2b/ε) = log2(2b)+log2(1/ε)

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
contains fy = fx .
2f )2b ≈ 2b
2f
Finally, the cost for representing a single element, or amortized
space cost C is deﬁned as:

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
contains fy = fx .
2f )2b ≈ 2b
2f
C =
table size
items stored
=
f ×b ×m
n
=
f
α
≤
log2(2b)+log2(1/ε)
α

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
contains fy = fx .
2f )2b ≈ 2b
2f
C =
table size
items stored
=
f ×b ×m
n
=
f
α
≤
log2(2b)+log2(1/ε)
α
With α = n
b×m load factor.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Space Optimization: Buckets semi sorting
All operations are insensible to the relative ordering of
elements in the buckets
Since the number of sorted sequences is 2f +b−1
b , a table
containing all them can be pre-computed.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
b , a table
Buckets can be replaced with an oﬀset in the table, achieving
some compression.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
b , a table
some compression.
If such table is retained in fast memory, the further indirection
penalty is negligible.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
b , a table
some compression.
If such table is retained in fast memory, the further indirection
penalty is negligible.
Example with b = 4 and f = 4
In this case an uncompressed bucket occupies 16 bits. However,
since the sorted buckets are at most 3876, they can be represented
with only 12 bits, sparing a bit for element.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Subsection 3

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Space Occupancy
Load Factor
Crucial for the space
eﬃciency of the cuckoo
ﬁlter
α increases with b, and
b = 4 is already enough
to reach close to optimal
load factors.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Space Occupancy
Load Factor
ﬁlter
load factors.
Since C ≤ log2(1/ε)+log2(2b) /α, we can see that for typical
scenarios C < 1.44log2(1/ε), making CF more space eﬃcient than
BF.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Space Occupancy
Load Factor
ﬁlter
load factors.
Since C ≤ log2(1/ε)+log2(2b) /α, we can see that for typical
scenarios C < 1.44log2(1/ε), making CF more space eﬃcient than
BF. Bit × item cost decreases further with semi-sorting
optimization.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
ε, space efficiency and lookup cost
With empirical (close
to optimal) load
factors, cuckoo filter
is more efficient than
bloom filters for
target error rates
smaller than 3%.
In terms of cache-misses, CF outperforms BF for meaningful
values of ε

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
to optimal) load
bloom ﬁlters for
target error rates
smaller than 3%.
values of ε
In BF positive queries incour in k > 2 misses for ε < 25%

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
to optimal) load
bloom ﬁlters for
target error rates
smaller than 3%.
values of ε
negative queries have on average the same cost.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
to optimal) load
bloom ﬁlters for
target error rates
smaller than 3%.
values of ε
negative queries have on average the same cost.
In both cases maintaining a ﬁxed ε might require expansion as
the set grows.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Subsubsection 1
Benchmark

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Experiment
Random 64-bit elements, with 64 bits hash
Filters conﬁgured to have the same space occupancy of 192MB
Metrics

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Experiment
Metrics
Space eﬃciency, achieved false positive rate, construction rate

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Experiment
Metrics
Space eﬃciency, achieved false positive rate, construction rate
Insert, Delete and Lookup throughput

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Cuckoo Filter
bits × item: 12.60
ε: 0.19%
Bloom Filter
bits × item: 13
ε: 0.19%

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Cuckoo Filter
bits × item: 12.60
ε: 0.19%
ss-CF optimization allows
for lower ε (0.09%) as one
more bit can be encoded.
Bloom Filter
bits × item: 13
ε: 0.19%

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Cuckoo Filter
bits × item: 12.60
ε: 0.19%
CF construction rate:
5×106keys/sec; ss-CF
construction rate:
3.13×106keys/sec
Bloom Filter
bits × item: 13
ε: 0.19%
BF construction rate:
3.91×106keys/sec

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Cuckoo Filter
bits × item: 12.60
ε: 0.19%
construction rate:
3.13×106keys/sec
Bloom Filter
bits × item: 13
ε: 0.19%
3.91×106keys/sec
Blocked Bloom Filter
optimization has a
construction rate f
7.64×106keys/sec

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Cuckoo Filter
bits × item: 12.60
ε: 0.19%
construction rate:
3.13×106keys/sec
Bloom Filter
bits × item: 13
ε: 0.19%
3.91×106keys/sec
Blocked Bloom Filter
optimization has a
construction rate f
7.64×106keys/sec
Semi Sorted implementation of Cuckoo Filter outperforms all
competitors in all metrics, except construction rate.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Lookup performance
Experiment characterized by a fraction p of items belonging to the
set.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Lookup performance
set.
Drop in performance for p in the range 50% - 75% of CF due to
branch misprediction.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Lookup performance
set.
ss-CF is slower due to decoding, but still outperforms BF.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Lookup performance
set.
ss-CF is slower due to decoding, but still outperforms BF.
Both are stable independently of the load factor.

Introduction
Cuckoo Filter
Analysis
Conclusion
Space analysis
Insert and Delete Performance
Insert performance is widely
aﬀected by load factor.
Throughput for CF and variant
ss-CF is inversely proportional to
α.
CF delete performance
signiﬁcantly outperform all Bloom
Filter based alternatives. ss-CF is
slower. Both maintain constant
performance across changing α.

Introduction
Cuckoo Filter
Analysis
Conclusion
Section 4
Conclusion

Introduction
Cuckoo Filter
Analysis
Conclusion
Conclusion
Cuckoo Filter and its semi sorted optimization are viable
alternatives to replace Bloom Filer for small false positive
rates.
Space eﬃciency and presence of delete operation makes it
viable for dynamic and highly changing scenarios (e.g. sensor
networks)

Introduction
Cuckoo Filter
Analysis
Conclusion
Conclusion
Cuckoo Filter and its semi sorted optimization are viable
alternatives to replace Bloom Filer for small false positive
rates.
Space eﬃciency and presence of delete operation makes it
viable for dynamic and highly changing scenarios (e.g. sensor
networks)
The elevate lookup performance makes them viable for network
application requiring high speed approximate set membership.

Introduction
Cuckoo Filter
Analysis
Conclusion
Dharmapurikar, Krishnamurthy and Taylor Longest Preﬁx
Match using Bloom Filters, SIGCOMM 2003.
Yu, Fabrikant and Rexford BUFFALO: Bloom Filter Forwarding
Architecture for Large Organizations, CoNEXT’09, December
1-4, 2009, Rome, Italy.
Pagh: Cuckoo Hashing for Undergraduates.
Pagh and Rodle Cuckoo Hashing Proceedings of European
Symposium on Algorithms, 2001.
Chen An Overview of Cuckoo Hashing.
Fan, Andersen, Kaminsky, Mitzenmacher Cuckoo Filter:
Practically Better Than Bloom, CoNEXT ’14, 02-05 December
2014, Sidney, Australia.

Introduction
Cuckoo Filter
Analysis
Conclusion
Godritch and Mitzenmacher Invertible Bloom Lookup Table,
arXiv, 3 May 2011.
Broder and Mitzenmacher Network Applications of Bloom
Filters: A Survey.
Deri High-Speed Dynamic Packet Filtering

Cuckoo Filter: Practically Better than Bloom

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