Perfect Hashing for Network Applications

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Perfect Hashing for Network Applications
Yi Lu, Balaji Prabhakar, Flavio Bonomi
Stanford University Cisco Systems
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Organization of Talk

• Motivation
• Limitations of existing algorithms
• The Algorithm
• Encoding size
• Construction time guarantees
• Simulation results

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Organization of Talk

• Motivation
• Limitations of existing algorithms
• The Algorithm
• Encoding size
• Construction time guarantees
• Simulation results

4
Hash Tables

• Review hash
• Universe U, subset S µ U, h U ? finite range
  integer set
• good h, spread U evenly, eg. a uniform random
  number

h

• Hash tables
• Fundamental data structure in many network
  applications
• Route lookup
• Packet Classification
• Per-flow state maintenance
• Network monitoring

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Fast AND Big

• Requirement
• Line speed is fast
• eg. 10 Gbs line, 50ns per packet
• Hash tables are big
• eg. 100,000 entries, each 32 bytes, for a
  lookup table
• Available Technology
• On-chip memory fast but small
• eg. 7 ns, 5 Mbits
• Off-chip memory big but slow
• eg. 40 ns, 500 Mbits

required
On-chip
Off-chip
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Minimize off-chip access and space

• Sol Hash table resides off-chip. Use on-chip
  processing to reduce off-chip access.
• Song et. al. (05)
• Kirsch and Mitzenmacher. (05)

• Why not build a minimal perfect hash table from
  the beginning?

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Definition

• Perfect Hash Function
• A function h mapping U into the integers is said
  to be perfect for S if, when restricted to S, it
  is injective.
• Minimal Perfect Hash Function
• Let Sn. A perfect hash function h is minimal
  if h(S) equals 0,...,n-1.

• Minimize off-chip access (1 access always) and
  space.

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Organization of Talk

• Motivation
• Limitations of existing algorithms
• The Algorithm
• Encoding size
• Construction time guarantees
• Simulation results

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Slow Construction

• Mehlhorns construction (84)
• Space
• Construction Time
• Fox et. al. (92) A faster algorithm
• Space 2.5 bits per entry
• Construction Time 6 hours on a NeXT station for
  3.8 million entries
• Our construction.
• Space 8.6 bits per entry
• Construction Time 7.7 seconds on a Pentium4 for
  3.8 million entries

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Needle in Haystack

• Knuth Among all projections of S to the range
  0,..., n-1, how many are injective? (S n)
• total number of such projections nn
• number of injective projections n!
  (2pn)1/2nn/en
• ratio (2pn)1/2 / en
• Ratio gets exponentially small as n gets large.
• Cleverly search for a needle in a haystack

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Our Approach

• Avoids searching for the function.
• Start with a random set of hash functions.
• Algorithm randomly divides the set S into
  subgroups, each becoming the domain of one of the
  functions
• Trade space for speed

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Organization of Talk

• Motivation
• Limitations of existing algorithms
• The Algorithm
• Encoding size
• Construction time guarantees
• Simulation results

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Counting Bloom Filter (CBF)

• Main component the counting Bloom filter
• Keep track of how many entries hashed to each
  address
• The concept of unique address

CBF
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Counting Bloom Filter (CBF)

• Main component the counting Bloom filter
• Keep track of how many entries hashed to each
  address
• The concept of unique address

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Counting Bloom Filter (CBF)

• Main component the counting Bloom filter
• Keep track of how many entries hashed to each
  address
• The concept of unique address

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Counting Bloom Filter (CBF)

• Main component the counting Bloom filter
• Keep track of how many entries hashed to each
  address
• The concept of unique address

CBF
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Construction
CBF
indicator
s2, s4,s5, s1,s3,s6
s1,s3, s6

• Entries without a unique address in one CBF is
  carried over to the next
• A different set of hash functions for each CBF

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Construction
indicator
off-chip list

• The off-chip structure is a simple list. There
  are no empty slots as in a hash table.
• On-chip structure -gt encoding

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Organization of Talk

• Motivation
• Limitations of existing algorithms
• The Algorithm
• Encoding size
• Construction time guarantees
• Simulation results

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Minimum Encoding Size

• Theorem 1
• The minimum number of bits needed to provide n
  entries with one unique address each, with random
  hashing, goes to en as n becomes large.
• It is achievable with an infinite number of CBFs
  with geometrically decreasing size, each with a
  single hash function.

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A practical tradeoff point

• Minimum encoding size requires an infinite number
  of CBFs in the worst case, hence an infinite
  number of hash evaluations
• Tradeoff between space and number of CBFs for 95
  of entries. Over-provide in last CBF to
  accommodate the remaining 5.

• A little increase of space cuts the number of
  CBFs in half (space 4n)

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Organization of Talk

• Motivation
• Limitations of existing algorithms
• The Algorithm
• Encoding size
• Construction time guarantees
• Simulation results

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Construction Time

• A failure occurs when not all entries find a
  unique address.
• How big should the last CBF be so that the
  failure probability is small?

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Construction Time

• Theorem 2
• Let n be the number of entries remaining for the
  last CBF, and m be the space assigned for the
  section. Then the probability of failure can be
  made double-exponentially small in m, and the
  optimal number of hash functions in this section
  is k ln2 m/n.
• T time for one pass
• average construction time T / (1-Pfail)

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Organization of Talk

• Motivation
• Limitations of existing algorithms
• The Algorithm
• Encoding size
• Construction time guarantees
• Simulation results

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Simulation results

• Design Parameters
• 5 CBFs. (4 optimal, plus 1 for the last 5)
• Space ratio 1.560.740.350.171.5 total size
  8.6n
• Number of hash functions 1, 1, 1, 1, 12
• Results
• Probability of failure 0.0012
• Construction time 7.73 seconds for 3.8 million
  entries
• A typical Ethernet table with 100K entries
  requires 125 milliseconds of construction time

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Extension dynamic perfect hashing

• Non-minimal perfect hash function
• Accommodates incremental insertion and deletion
• Space remains O(n). Design 20 bits per entry.
• Constant-time insertion / deletion / lookup

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Summary

• A new approach to minimal perfect hashing via
  counting Bloom filters.
• Avoid searching by generating random subgroups
  for pre-determined hash functions, and as a
  result, speed up construction
• The resulting construction is hardware-friendly
  and fits the need of high-speed network
  applications well

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Backup 1 Lookup
indicator
counter
off-chip list

• The off-chip structure is a simple list. There
  are no empty slots as in a hash table.
• On-chip structure -gt encoding