Cuckoo Hashing : Hardware Implementations

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Cuckoo Hashing Hardware Implementations

• Adam Kirsch
• Michael Mitzenmacher

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Motivation

• Hash tables are ubiquitous.
• Highly useful in router hardware.
• Measurement and monitoring tasks.
• Desiderata
• Few (parallel) memory accesses.
• High space utilization.
• Low failure probability.
• Hardware-level simplicity.
• What are good hash table designs for hardware?

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State of the Art Multiple Choice Hashing

• Each element placed in least loaded of d
  locations. (If 1 element/cell, look for 1 empty
  cell out of d.)

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Cuckoo Hashing and Moves

• Cuckoo hashing paradigm give each element d
  choices, and move elements among choices as
  needed.

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Original Cuckoo Hashing

• 2 subtables, left and right. Each element gets
  one location per subtable.
• Place new element in left subtable.
• If element already there, kick it out, move to
  right subtable.
• If element already there, kick it out, move to
  left subtable
• Until everything placed.
• Works with high probability as long as load is
  less than ½.

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Better Cuckoo Hashing

• More choices
• More elements per bucket
• Generally kick out a random item.
• Such schemes are not fully analyzed.

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Whats Wrong with Cuckoo Hashing?

• Lots of moves per insert in worst case.
• Average is constant.
• But maximum is Omega(log n) with non-trivial
  (inverse-poly) probability.
• Router hardware settings may need bounded
  number of memory accesses per insert.

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Moves Needed per Insertion
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The Power of One Move

• Previous work (submitted) How much gain from
  allowing just one move?
• Framework allow small content-addressable
  memory (CAM) to handle unsolvable collisions max
  0.2.
• Multiple schemes analyzed.
• With 4 choices, insertions only (no deletions),
  factor of 2 or larger improvement in space.

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Pros/Cons of One Move Systems

• Pros
• Simple to implement
• Efficient
• High space utilization for insertion-only
• Analyzable and optimizable
• Cons
• Performance suffers in settings with churn
• Better space utilization possible with more moves

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The New Idea

• Use the CAM as a queue for move operations.
• Lookup check the hash table and the CAM-queue.
• Try move operations from queue as available.
• Move attempt 1 parallel memory lookup.
• De-amortization
• Use queue to make worst-case performance same as
  average-case performance.

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Queue Policy

• Key point better to give priority to new
  insertions over moves.
• New moves have d choices moves effectively have
  d 1.
• Intuition suggests older items may be less likely
  to be successfully placed.
• True in practice.
• Full priority queue may be too complex.
• Simple strategy new items placed at front,
  failed moves places at back.

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Probability of Success vs. Age
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Experimental Evaluation

• Table of size 32768, 4 subtables.
• Target utilization u.
• Insert 32678u elements, then alternate
  insertions/deletions to get to steady state.
• Allow ops queue operations (parallel memory
  operations) per insertion.

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Analysis

• Currently we do not know how to analyze such
  systems.
• For d gt 2 choices, lots of open questions in
  cuckoo hashing analysis.
• Analyzing d 2 may be possible, but very low
  space utilization.
• See Kutzelnigg, asymptotic analysis of cuckoo
  hashing.
• Need to understand distribution of move
  operations/element to analyze queue.

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Conclusions and Open Questions

• Moving elements leads to much better space
  utilization in hash tables, at a price.
• Cuckoo hashing appears implementable, with
  per-insert move guarantees based on
  de-amortization via a CAM queue.
• Analysis in an idealized model?
• Even analysis for basic cuckoo hashing open.
• Performance on real traffic?
• Bursty insertions/deletions?
• Distribution of element lifetimes?
• Proper sizing of CAM queue?
• How does overflow probability scale?