Transcript (A Survey on) Priority Queues Gerth Stølting Brodal Aarhus Universty IanFest, University of Waterloo, Waterloo, Ontario, Canada, August 14-15, 2013

(A Survey on)
Priority Queues
Gerth Stølting Brodal
Aarhus Universty
IanFest, University of Waterloo, Waterloo, Ontario, Canada, August 14-15, 2013
Program
Ian Munro : A Constant Time Priority Queue
Gerth Brodal : A Constant Time Priority Queue
2
Priority Queue
Insert
ExtractMin
FindMin
Delete
DecreaseKey
Meld
ExtractMax (double-ended priority queue)
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7 ,513
1
7,542
13 ,7
3
2
7,5
42 ,4
4
66 ,6
5
13 ,7
6
7
7,5 7,5 13,7 42,4 7,5 66,6 13,7
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Communications of the ACM (1962)
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3
4
5
6
7
4
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Communications of the ACM (1964)
Heap
2
5
3
•
•
•
•
•
•
7
66
2
5
7
3
1
2
3
4
66 42
5
6
42
Simple
Implicit – single array of size n
Insert and ExtractMin O(log n)
Construction O(n)
Insert (loglog n) comparisons
Select k smallest in O(k) time
Willliams (1964)
Floyd(1964)
Gonnet, Munro (1986)
Frederickson (1993)
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Comparison Lower Bound
n x Insert + n x ExtractMin  Sorting
 Insert or ExtractMin (log n) comparisons
Comparisons
Insert ExtractMin
Heap
Binary search
Carlsson, Munro, Poblette (1988)
O(log n)
O(log n)
O(1)
O(log n)
0
O(log n)
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Priority Queues  Directions of Research
Worst-case
vs
Amortized
Comparisons
vs
Bit-tricks
Theory
vs
Implementation
O
vs
Constants
InsertExtractMin
vs
DecreaseKey,
Meld,…
Single processor
vs
Parallel
RAM
vs
Hierarchical
memory
Aware
vs
Oblivious
Simplicity
vs
Let’s-do-somethingcomplicated
Weak
vs
Strong
Implicit
vs
Space wasting
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Open Problem
Can only
store n + array
between operations
Strongly implicit
priority queue
worst-case
O(1) Insert and O(log n) ExtractMin
supporting identical elements
O(1) swaps per operation
and cache oblivious optimal ?
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ExtractMin
Implicit
Swaps
Identical
elements
log n
log n
Strong
log n
Yes
Carlsson, Munro, Poblette (1988)
1
log n
Weak
log n
Yes
Arge, Bender, Demaine, HollandMinkley, Munro (2002)
log n
log n
log n
Yes
Munro, Franceschini (2006)
log n
log n Strong
1
1
log n Strong
log n
 amortized bounds
Heaps
Harvey, Zatloukal (2004)
Brodal, Nielsen, Truelsen (2013)
Open
1
log n
Strong
log n
1
x
x
log n
x
x
Strong
1
Cache
oblivious
Insert
(Some Random) Results
Yes
Yes
Yes
Yes
x
x
Implicit dictionary
O(n) Munro, Suwanda (1980)
O(log2 n) Munro (1986)
O(log2 n/loglog n) Franceschini, Grossi, Munro, Pagli (2004)
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Some Observations
Pair encoding: Munro (1986)
42 66
66 42
66 66
Bit = 1
Bit = 0
???
 2log n elements can encode a pointer
 O(log n) time allows pointer manipulations
Strict implicit + O(1) insertions
 Insertions ”close to oblivious”
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Strict Implicit Priority Queue
O(1) Insert and O(log n) ExtractMin
…work in progress Brodal, Nielsen, Truelsen (2013)
•
•
•
•
Forest, trees powers-of-2
Looser trees
Pair-encode bits at nodes
Insertions balance using LSB(n)
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ExtractMin
Implicit
Swaps
Identical
elements
log n
log n
Strong
log n
Yes
Carlsson, Munro, Poblette (1988)
1
log n
Weak
log n
Yes
Arge, Bender, Demaine, HollandMinkley, Munro (2002)
log n
log n
log n
Yes
Munro, Franceschini (2006)
log n
log n Strong
1
1
log n Strong
log n
 amortized bounds
Heaps
Harvey, Zatloukal (2004)
Brodal, Nielsen, Truelsen (2013)
Open
1
log n
Strong
log n
1
x
x
log n
x
x
Strong
1
Cache
oblivious
Insert
(Some Random) Results
Yes
Yes
Yes
Yes
x
x
Implicit dictionary
O(n) Munro, Suwanda (1980)
O(log2 n) Munro (1986)
O(log2 n/loglog n) Franceschini, Grossi, Munro, Pagli(2004)
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