Multiple Choice Hash Tables with Moves on Deletes and Inserts

1
Multiple Choice Hash Tables with Moves on Deletes
and Inserts

• Adam Kirsch
• Michael Mitzenmacher

2
Hashing Modern Perspective

• For many situations (e.g., hardware for routers)
  multiple choice hash tables are state-of-the-art.
• Each item gets d possible hash locations, placed
  in one.
• Moving items among choices (e.g., cuckoo hashing)
  greatly improves space utilization.
• Only cost may take many moves per insert.

3
Previously

• Schemes that move at most 1 item per insertion.
• Limit cost of cuckoo hashing.
• Schemes that batch move operations in a queue.
• Amortize cost of cuckoo hashing.
• Using content addressable memories (CAMs) to
  reduce chance of overflow.
• Small CAMs yield big gains.

4
Contributions

• Consider potential of moving items on deletions.
• Focus on one move per deletion/insertion.
• Examine alternative approach using weaker hashing
  from KTC, Peacock Hashing.
• Analyze limits of performance.

5
Multilevel Hash Table BK90

• Use a multilevel hash table (MHT)
• Can store n elements with d log log n O(1)
  levels in O(n) space with high probability
• Example with d 4 hash functions

Level
1
2
x
3
Skew more elements placed by early hash
functions (double exponential decay)
4
6
Second Chance (SC) Scheme

• Standard MHT fills from top down
• elements cascade from table to table.
• We try to slow cascade at every step.

x
Standard MHT Insertion
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Second Chance (SC) Scheme

• Standard MHT fills from top down
• elements cascade from table to table.
• We try to slow cascade at every step.

x
8
Second Chance (SC) Scheme

• Standard MHT fills from top down
• elements cascade from table to table.
• We try to slow cascade at every step.

x
9
CAMs

• Last few collisions hard to stop.
• Can waste lots of space on few items.
• Solution content addressable memory.
• CAMs fully asociative.
• Hold small numbers of items.

10
Moves on Deletions

• Harder to manage.
• What item to move up?

Level
1
2
x
3
4
11
Hint-Based Approach

• Each cell stores hint for where an item to move
  on delete is held.
• Hints can be kept fairly small.
• About log n bits.
• Various hint approaches possible.
• We found replace hint on any collision works
  well.
• May depend on item lifetime distribution, etc.
• One move, recursive move variations.

12
Simulation Data

• No current method of analysis for hints.
• Use simulations. 10,000 trials per data point.
• MHT levels decreasing in size by factor of 2.
  Plus small CAM.
• With n items, top level has size n.
• Space usage just above 50.
• Load table to n elements, alternate
  inserts/deletes for 218 steps.
• Exponentially distributed lifetimes.
• Goal how many hash functions needed?

13
Simulation Results
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Lessons from Simulations

• No moves very weak.
• Second Chance (move on insert) more powerful than
  hint-based move on delete.
• But the two combine well.
• Four hash functions better than 50 load, small
  CAM.

15
Alternative Weak Hashes

• To avoid hints, overflow at each bucket splits to
  two buckets at next level.
• Each bucket receives from four buckets.
• Less spreading of items, but know where to look
  on deletes.
• Conjecture loss of randomness implies weak
  performance.

16
Picturing Weak Hashes
17
Two Idealized Schemes

• Each bucket holds random item, splits rest.
• Each bucket counts items passed to bucket A and
  bucket B at next level, greedily holds item from
  bucket with larger count.
• Assume invariants kept over insertions/deletions
  at all times.
• Can be analyzed recursively level by level.
• Get distribution of bucket loads at each level.
• Obtain average case peformance.

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Results
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Conclusions

• Weak hashes, based on buckets, much less
  effective than hints.
• Even under optimistic assumptions.
• One move approaches effective.
• Move on insert/delete complement each other.
• Need methods for analysis.
• Challenging dependencies hard to get exact
  numbers.