Transcript Multiple Choice Hash Tables with Moves on Deletes and Inserts

Multiple Choice Hash Tables with
Moves on Deletes and Inserts
Adam Kirsch
Michael Mitzenmacher
Hashing : Modern Perspective
• For many situations (e.g., hardware for
routers) multiple choice hash tables are stateof-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.
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.
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.
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
Level
– Example with d = 4 hash functions
1
2
x
Skew: more elements placed
by early hash functions
(double exponential decay)
3
4
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
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
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
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.
Moves on Deletions
• Harder to manage.
• What item to move up?
Level
1
2
x
3
4
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.
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?
Simulation Results
Scheme
No moves
Items
Hash
Average
Functions
Stash
32768
13
4.225
Second Chance 32768
Max
Stash
31
6
0.001
2
Hint, 1 Move
32768
7
0.013
3
Hint, Moves
32768
6
0.246
7
Hint,1Move+SC 32768
4
4.678
18
Hint,Moves+SC 32768
4
0.911
9
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.
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.
Picturing Weak Hashes
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.
Results
Scheme
Items
Hash
Functions
Average
Stash
Random
32768
6
2.470
Greedy
32768
6
0.182
Second
Chance
32768
6
0.001
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.