The Limit of Buffering Dynamic External Hashingyike/hashinglbslides.pdf · 2009-09-01 · 4-1...
Transcript of The Limit of Buffering Dynamic External Hashingyike/hashinglbslides.pdf · 2009-09-01 · 4-1...
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Dynamic External Hashing:The Limit of Buffering
August 13, 2009
Qin Zhang
Hong Kong University of Science & Technology
Joint work with Zhewei Wei and Ke Yi
SPAA 2009
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(internal) Hashing!
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universalhashing
dynamichashing
distributedhashing
externalhashing
perfecthashing
One of the most important datastructures in computer science!
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External hashing!
A cell has b (= 2) words
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externalhashing
universalhashing
dynamichashing
distributedhashing
perfecthashing Extremely useful in
Database System!
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4-1
Model and problem
Model: External memory model.Block size b words, cache size mwords. Cost is “number of blocksread/write (I/Os)”
Problem: Maintain a hash table tosupport update and query.
Try to understand “the inherenttradeoff between queries and up-dates”
Hashing
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5-1
Results
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Hashing in the internal memory is well under-stood (under random inputs).
Knuth, 1970s: tq = 12(1+1/(1−α)), tu = 1+
12(1 + 1/(1− α)2). α: load factor: minimum
storage should be use/storage actually used
Previous results
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6-2
Hashing in the internal memory is well under-stood (under random inputs).
Knuth, 1970s: tq = 12(1+1/(1−α)), tu = 1+
12(1 + 1/(1− α)2). α: load factor: minimum
storage should be use/storage actually used
Previous results
In external memory (random inputs)
Knuth, 1970s: Expected average cost of aquery is 1 + 1/2Ω(b) I/Os, provided the loadfactor α is less than a constant smaller than1. Update has a similar bound.
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7-1
Exact Numbers Calculated by D. E. Knuth
The Art of Computer Programming, volume 3, 1998, page 542
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8-1
Can we batch updates?
However, in external memory model, disk reads/writesare expensive and powerful. Can we hope for lower than1 I/O per update?
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8-2
Can we batch updates?
However, in external memory model, disk reads/writesare expensive and powerful. Can we hope for lower than1 I/O per update?
No, if there is no space in main memory for buffering.But, not the case in reality!
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8-3
Can we batch updates?
However, in external memory model, disk reads/writesare expensive and powerful. Can we hope for lower than1 I/O per update?
No, if there is no space in main memory for buffering.But, not the case in reality!
Maybe yes, if we have an Ω(b) main memory for buffer-ing! Like numerous problems in external memory, e.g.stack. More: priority queue, buffer tree ...
Can the amortized update cost be something likeO(1/bc) (for some 0 < c ≤ 1) for hashing?
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8-4
Can we batch updates?
However, in external memory model, disk reads/writesare expensive and powerful. Can we hope for lower than1 I/O per update?
No, if there is no space in main memory for buffering.But, not the case in reality!
Maybe yes, if we have an Ω(b) main memory for buffer-ing! Like numerous problems in external memory, e.g.stack. More: priority queue, buffer tree ...
Can the amortized update cost be something likeO(1/bc) (for some 0 < c ≤ 1) for hashing?
Conjectured by Jensen and Pagh (2007):
The insertion cost must be Ω(1) I/Os if the query cost is
required to be O(1) I/Os.
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9-1
Our results
1 + 1/2Ω(b)
1−O(1/b(c−1)/6)
Ω(bc−1)
O(bc−1)
Ω(1)
O(1)
Insertion
Query
1 + Θ(1/b)
1 + Θ(1/bc), c < 11
upper bounds
lower bounds
1+Θ(1/bc)c > 1
expectedamortized
expected average
Due to Knuth (on random inputs)
successful queries
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9-2
Our results
1 + 1/2Ω(b)
1−O(1/b(c−1)/6)
Ω(bc−1)
O(bc−1)
Ω(1)
O(1)
Insertion
Query
1 + Θ(1/b)
1 + Θ(1/bc), c < 11
upper bounds
lower bounds
1+Θ(1/bc)c > 1
expectedamortized
expected average
Due to Knuth (on random inputs)
successful queries
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9-3
Our results
1 + 1/2Ω(b)
1−O(1/b(c−1)/6)
Ω(bc−1)
O(bc−1)
Ω(1)
O(1)
Insertion
Query
1 + Θ(1/b)
1 + Θ(1/bc), c < 11
upper bounds
lower bounds
1+Θ(1/bc)c > 1
Almost a completeunderstanding forsuccessful queries!
expectedamortized
expected average
Due to Knuth (on random inputs)
successful queries
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10-1
Other related results
Upper bounds
Remove ideal hash function assumption [Carter and Wegman1979], making query worst-case [i.e. Fredman, Komlos andSzemeredi, 1984] ... (internal)
Queries and updates in 1 + O(1/b12 ) I/Os with α = 1 −
O(1/b12 ) [Jensen and Pagh, 2007]. (external, no memory)
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10-2
Other related results
Upper bounds
Remove ideal hash function assumption [Carter and Wegman1979], making query worst-case [i.e. Fredman, Komlos andSzemeredi, 1984] ... (internal)
Lower bounds (internal)
Very sparse, only with some strong requirements, e.g., the al-gorithm is deterministic and query is worst-case [Dietzfelbingeret. al. 1994].
Queries and updates in 1 + O(1/b12 ) I/Os with α = 1 −
O(1/b12 ) [Jensen and Pagh, 2007]. (external, no memory)
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10-3
Other related results
Upper bounds
Remove ideal hash function assumption [Carter and Wegman1979], making query worst-case [i.e. Fredman, Komlos andSzemeredi, 1984] ... (internal)
Lower bounds (internal)
Very sparse, only with some strong requirements, e.g., the al-gorithm is deterministic and query is worst-case [Dietzfelbingeret. al. 1994].
Lower bounds in other dynamic external memory problems
Only known are query-update tradeoffs for the predecessor[Fagerberg and Brodal 2003], range reporting [Yi 2009].
Queries and updates in 1 + O(1/b12 ) I/Os with α = 1 −
O(1/b12 ) [Jensen and Pagh, 2007]. (external, no memory)
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11-1
Technical details:
Lowerbounds
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12-1
Preliminaries
U = 0, 1, . . . , u− 1: universe. |U | = u.
m: size of main memory. n: total number of items.b: size of one block. All in words.
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12-2
Preliminaries
U = 0, 1, . . . , u− 1: universe. |U | = u.
Some mild assumptions
Atomic elements
n ≥ Ω(m log u · b2c
)for some constant c > 0
m: size of main memory. n: total number of items.b: size of one block. All in words.
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12-3
Preliminaries
U = 0, 1, . . . , u− 1: universe. |U | = u.
Deterministic data structure + a random distrib. of inputs
(Via a method similar to Yao’s Minimax Principle) =⇒Randomized data structure
Some mild assumptions
Atomic elements
n ≥ Ω(m log u · b2c
)for some constant c > 0
m: size of main memory. n: total number of items.b: size of one block. All in words.
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13-1
Observations
Two extreme cases
One exterme: only use afixed mapping for all items.
1, 11
21, 31
32 13, 73
43, 23
63, 33
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Update is expensive!b = 2
15, 55
45
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13-2
Observations
Two extreme cases
One exterme: only use afixed mapping for all items.
1, 11
21, 31
32 13, 73
43, 23
63, 33
null
Update is expensive!
Another exterme: for every bitems come, write to a newblock.
Too manypossiblemappings.b = 2
15, 55
45
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13-3
Observations
Two extreme cases
One exterme: only use afixed mapping for all items.
1, 11
21, 31
32 13, 73
43, 23
63, 33
null
Update is expensive!
Also easy to see
If with only the information in memory, the hash table cannotlocate the item, then querying it takes at least 2 I/Os.
Another exterme: for every bitems come, write to a newblock.
Too manypossiblemappings.b = 2
15, 55
45
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14-1
The abstraction
Consider the layout of a hash table at any snapshot. Denoteall the blocks on disk by B0, B1, B2, . . . , Bd (B0 = M). Letf : U → 0, 1, . . . , d be any function computable withinmemory.
When querying x ∈ U , f(x): index of the first block the DSwill probe. If f(x) = 0, the DS will still probe the memory.
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14-2
The abstraction
Consider the layout of a hash table at any snapshot. Denoteall the blocks on disk by B0, B1, B2, . . . , Bd (B0 = M). Letf : U → 0, 1, . . . , d be any function computable withinmemory.
When querying x ∈ U , f(x): index of the first block the DSwill probe. If f(x) = 0, the DS will still probe the memory.
Memory zone M : set of itemsstored in memory. tq = 0.
Fast zone F : set ofitems x such thatx ∈ Bf(x). tq = 1.
Slow zone S:The rest of items.tq ≥ 2.
Memory
Disk
We divide items inserted into 3 zones with respect to f .
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15-1
The key idea
The hash table can employ a family F of at most2m log u distinct f ’s.
Note that the current f adopted by the hash ta-ble is dependent upon the already inserted items,but the family F has to be fixed beforehand.
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16-1
Size of the slow zone is small.
Suppose the hash table answers a successful query withan expected average cost of tq = 1 + δ I/Os. Considerthe snapshot when k random items have been inserted.
E[|S|] ≤ m + δk. Memory zone M : tq = 0
Fast zone F : tq = 1 Slow zone S: tq ≥ 2
small!
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16-2
Size of the slow zone is small.
Suppose the hash table answers a successful query withan expected average cost of tq = 1 + δ I/Os. Considerthe snapshot when k random items have been inserted.
E[|S|] ≤ m + δk.
The high-probability version.
LEMMA 1. Let φ ≥ 1/b(c−1)/4 and let k ≥ φn. Atthe snapshot when k items have been inserted, withprobability at least 1− 2φ, |S| ≤ m + δ
φk.
Memory zone M : tq = 0
Fast zone F : tq = 1 Slow zone S: tq ≥ 2
small!
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17-1
Basic idea of the lower bound proof
Consider any f : U → 0, 1, . . . , d. For i = 0, . . . , d,let αi = |f−1(i)|/u, and we call (α0, α1, . . . , αd) thecharacteristic vector of f .
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17-2
Basic idea of the lower bound proof
Consider any f : U → 0, 1, . . . , d. For i = 0, . . . , d,let αi = |f−1(i)|/u, and we call (α0, α1, . . . , αd) thecharacteristic vector of f .
After φn random insertions. Pick a fixed threshold ρ.Assume α0 ≥ α1 ≥ . . . ≥ αd
α0 α1 α2 αk αd
ρ
f
If ∃ too many large αi’s,S too large, violatingthe query requirement(LEMMA 1).
(A)
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17-3
Basic idea of the lower bound proof
Consider any f : U → 0, 1, . . . , d. For i = 0, . . . , d,let αi = |f−1(i)|/u, and we call (α0, α1, . . . , αd) thecharacteristic vector of f .
After φn random insertions. Pick a fixed threshold ρ.Assume α0 ≥ α1 ≥ . . . ≥ αd
α0 α1 α2 αk αdf αj
ρElse f is likely to distributerecent randomly inserteditems evenly, leading to ahigh insertion cost.
(B)
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17-4
Basic idea of the lower bound proof
Consider any f : U → 0, 1, . . . , d. For i = 0, . . . , d,let αi = |f−1(i)|/u, and we call (α0, α1, . . . , αd) thecharacteristic vector of f .
After φn random insertions. Pick a fixed threshold ρ.Assume α0 ≥ α1 ≥ . . . ≥ αd
α0 α1 α2 αk αd
ρ
f
If ∃ too many large αi’s,S too large, violatingthe query requirement(LEMMA 1).
αj
ρElse f is likely to distributerecent randomly inserteditems evenly, leading to ahigh insertion cost.
Both hold with very high probability, even after taking unionof all O(2m log u) different f .
(A) (B)
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18-1
Upper bounds
Easy!
Logarithmic method
+ Query start from the last (biggest) layer
+ Tricks to keep the last layer large
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19-1
Beyond hashing:Subsequent and future work
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20-1
Beyond hashing
Hashing (successful)
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20-2
Beyond hashing
Hashing (successful)
Membership
Problem: Maintain a set S ⊆ U . Given anx ∈ U , answer x ∈ S?
Again, tradeoffs between update and query
Membership: if tq ≤ 1 + δ(0 ≤ δ < 1/2), then tu ≥ Ω(1)[Yi and Zhang 2009]
(1) Without atomic assumption(2) Consider both successful andunseccessful query
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20-3
Beyond hashing
Hashing (successful)
Membership
Problem: Maintain a set S ⊆ U . Given anx ∈ U , answer x ∈ S?
Again, tradeoffs between update and query
Membership: if tq ≤ 1 + δ(0 ≤ δ < 1/2), then tu ≥ Ω(1)[Yi and Zhang 2009]
(1) Without atomic assumption(2) Consider both successful andunseccessful query
General Membership
General Hashing
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Lower bounds of other dynamic problems in thecell probe with cache setting.
1. predecessor, range-sum
2. union-find
3. . . .
More problems
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The End
T HANK YOUQ and A
The Banff National Park