The Bloom Paradox

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The Bloom Paradox
Ori Rottenstreich Joint work with Yossi Kanizo
and Isaac Keslassy Technion, Israel

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Problem Definition
x
y
user

• Requirement A data structure in user with fast
  answer to
• Solutions
• O(n) Searching in a list
• O(log(n)) Searching in a sorted list
• O(1) But with false positives / negatives

x
M central memory with all elements
y
cost 10
cost 1
S local cache
cost 10
v
u
z
y
x
z
x
y
user
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Two Possible Errors

• False Positive but the data
  structure answers
• Results in a redundant access to the local cache.
• Additional cost of 1.
• False Negative but the data structure
  answers
• Results in an expensive access to the central
  memory instead of the local cache.
• Additional cost of 10-19.

y
x
4
Bloom Filters (Bloom, 1970)

• Initialization Array of zero bits.
• Insertion Each of the elements is hashed
  times, the corresponding bits are set.
• Query Hashing the element, checking that all
  bits are set.
• False positive rate (probability) of
  
• No false negatives

0
0
0
0
0
0
0
0
0
0
0
0
y
x
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
z
x
w
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Bloom Filters are Widely Used

• Cache/Memory Framework
• Packet Classification
• Intrusion Detection
• Routing
• Accounting
• Beyond networking Spell Checking, DNA
  Classification
• Can be found in
• Google's web browser Chrome
• Google's database system BigTable
• Facebook's distributed storage system Cassandra
• Mellanox's IB Switch System

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Outline

• Introduction to Bloom Filters
• The Bloom Paradox
• The Variable-Increment Counting Bloom Filter

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The Bloom Paradox
Sometimes, it is better to disregard the Bloom
filter results, and in fact not to even query it,
thus making the Bloom filter useless.
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Example
Bloom filter

• Parameters
• Extreme case without locality All elements with
  equal probability of
• belonging to the cache.
• Toy example

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The Bloom Paradox

• Parameters
• Let be the set of elements that the Bloom
  filter indicates are in
• In particular, no false negatives ?
• Intuition

Bloom filter
B Bloom filter
user

M central memory with all elements
cost 10
cost 1
S local cache
cost 10
v
u
z
y
x
z
x
. .
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The Bloom Paradox

• Parameters
• Let be the set of elements that the Bloom
  filter indicates are in
• In particular, no false negatives ?
• Surprise

B Bloom filter

M central memory with all elements
cost 10
cost 1
S local cache
cost 10
v
u
z
y
x
z
x
. .
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The Bloom Paradox

• Parameters
• Let be the set of elements that the Bloom
  filter indicates are in
• In particular, no false negatives ?
• Surprise
• The Bloom filter indicates the membership of
• elements. Only
  of them are indeed in .
  

B Bloom filter

. .
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The Bloom Paradox

• When the Bloom filter states that ,
  it is wrong with probability
• Average cost if we listen to the Bloom filter
• Average cost if we dont
• The Bloom filter is useless!

?
Dont listen to the Bloom filter


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Outline

• Introduction to Bloom Filters
• The Bloom Paradox
• The Variable-Increment Counting Bloom Filter

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Counting Bloom Filters (CBFs)

• Bloom filters do not support deletions of
  elements. Simply resetting bits might cause false
  negatives.
• The solution Counting Bloom filters - Storing
  array of counters instead of bits.
• Insertion Incrementing counters by one.
• Deletion Decrementing counters by one.
• Query Checking that counters are positive.
• The same false positive probability.
• Require too much memory, e.g. 57 bits per element
  for .

y
x
1
1
1
1
1
1
0
0
0
0
0
0
1
0
1
0
0
0
y
x
1
1
1
1
1
1
0
1
0
2
0
0
1
0
1
0
0
1
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Intuition for Variable Increments

• Upon query, we should consider the exact values
  of the counters and not just their positiveness
• Can we design a deterministic scheme that
  exploits the exact values of the counters?
• Idea Use variable increments to encode the
  element identity

0
3
8
1
0
5
2
0
1
0
1
2
z
y
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Architecture

• Each hash entry contains a pair of counters
• , fixed increments ? number of elements in
  entry (as in CBF)
• , variable increments ? weighted sum of
  elements
• weights from a pre-determined set

• We use two sets of hash functions
• The first set uses
  hash functions with range
• , i.e. it points to the set of
  entries.
• The second set uses
  hash functions with range , i.e.
  it points to the set
  .

2
7
8
9
4
5
6
1
3
5
3
3
4
2
3
0
3
c1
2
34
9
6
26
26
17
21
0
25
c2
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Insertion

• Insertion
• At each entry , the two counters are
  updated as follows.
• from the
  set
• Example 1

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7
8
9
4
5
6
1
3
5
3
3
4
2
3
0
3
c1
2
3 4
0 1
3 4
4 5
34
9
13
26
17
17
21
0
25
c2
25 29
30 43
30 34
0 8
4
8
4
13
x
z
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Query

• Query ( with
  )
• We ask whether
• 17 can be a sum of 2 elements from the set
  including 4
• 30 can be a sum of 3 elements from the set
  including 8
• No
• How should we pick the set of variable
  increments?

y
8?
4?
y?

• We should use Sequences!

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Bh Sequences

• Definition 1
• Let be a
  sequence of positive integers.
• Then, is a sequence iff all the sums
• with are
  distinct.
• Example 2
• 
  
• All the sums of elements of are
  distinct
• Therefore, is a sequence.
• sequences are widely used in
  error-correcting codes.

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The Bh-CBF Scheme Query

• Example 3
  is a sequence
• Since , then the Bh-CBF can
  determine that

4?
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The Bh-CBF Scheme Operations
The Bh-CBF Scheme Query

• Example 3
  is a sequence

• Here, and then necessarily
• Since , the Bh-CBF can
  determine that

4?
8?
4?
y?
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The Bh-CBF Scheme Operations
The Bh-CBF Scheme Query

• Example 3
  is a sequence

• Since , the
  Bh-CBF cannot exclude that

4?
13?
4?
8?
4?
z?
y?
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Experimental Results

• Internet trace (equinix-chicago) with real hash
  functions.
• For the Bh-CBF,
  (with ).

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Experimental Results

• Internet trace (equinix-chicago) with real hash
  functions.
• For the Bh-CBF,
  (with ).
• For the VI-CBF,
  and . .
  

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Concluding Remarks

• The Bloom Paradox
• Discovery of the Bloom paradox
• Importance of the a priori membership probability
• The Variable-Increment Counting Bloom Filter
• Can extend many variants of the counting Bloom
  filter
• First time sequences are presented in
  networking applications

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Thank You