(A Survey on) Priority Queues

IanFest, University of Waterloo, Waterloo, Ontario, Canada, August 14-15, 2013

Gerth Stølting BrodalAarhus Universty

2

ProgramIan Munro : A Constant Time Priority QueueGerth Brodal : A Constant Time Priority Queue

3

Priority Queue

InsertExtractMi

n

FindMinDeleteDecreaseKeyMeldExtractMax (double-ended priority queue)

4Communications of the ACM (1962)

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• Simple• Implicit – single array of size n• Insert and ExtractMin O(log n) Willliams (1964)• Construction O(n) Floyd(1964)• Insert (loglog n) comparisons Gonnet, Munro (1986)• Select k smallest in O(k) time Frederickson (1993)

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Comparison Lower Bound

n x Insert + n x ExtractMin Sorting

Insert or ExtractMin (log n) comparisons

ComparisonsInsert ExtractMin

Heap O(log n) O(log n)Binary search O(log n) 0

Carlsson, Munro, Poblette (1988) O(1) O(log n)

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Priority Queues Directions of Research

Comparisonsvs

Bit-tricks

Single processorvs

Parallel

RAMvs

Hierarchical memory

Implicitvs

Space wasting

Insert-ExtractMin

vsDecreaseKey,

Meld,…

Weakvs

Strong

Ovs

Constants

Simplicityvs

Let’s-do-something-complicated

Worst-casevs

Amortized

Theoryvs

Implementation

Awarevs

Oblivious

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Open Problem

Strongly implicit priority queue

worst-caseO(1) Insert and O(log n) ExtractMin

supporting identical elementsO(1) swaps per operation

and cache oblivious optimal ?

Can only store n + array

between operations

Insert ExtractMin

Implicit Swaps

Identical elements

Cache oblivious

Heaps log n log n Strong log n Yes

Carlsson, Munro, Poblette (1988) 1 log n Weak log n YesArge, Bender, Demaine, Holland-Minkley, Munro (2002) log n log n log n Yes Yes

Munro, Franceschini (2006) log n log n Strong 1Harvey, Zatloukal (2004) 1 log n Strong log n YesBrodal, Nielsen, Truelsen (2013) 1 log n Strong log n

Open 1xx

log nxx

Strong 1

x

Yes Yesx

(Some Random) Results

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amortized bounds

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

42 66 66 42

Bit = 1 Bit = 0

66 66

???

Pair encoding: Munro (1986)

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 QueueO(1) Insert and O(log n) ExtractMin

• Forest, trees powers-of-2• Looser trees• Pair-encode bits at nodes• Insertions balance using LSB(n)

…work in progress Brodal, Nielsen, Truelsen (2013)

Insert ExtractMin

Implicit Swaps

Identical elements

Cache oblivious

Heaps log n log n Strong log n Yes

Carlsson, Munro, Poblette (1988) 1 log n Weak log n YesArge, Bender, Demaine, Holland-Minkley, Munro (2002) log n log n log n Yes Yes

Munro, Franceschini (2006) log n log n Strong 1Harvey, Zatloukal (2004) 1 log n Strong log n YesBrodal, Nielsen, Truelsen (2013) 1 log n Strong log n

Open 1xx

log nxx

Strong 1

x

Yes Yesx

(Some Random) Results

13

amortized bounds

Implicit dictionary O(n) Munro, Suwanda (1980)

O(log2 n) Munro (1986)O(log2 n/loglog n) Franceschini, Grossi, Munro, Pagli(2004)

Thank You !