[PPT] - Worst-Case Ef fi cient External-Memory Priority Queues Gerth PowerPoint Presentation

SLIDE 1

Worst-Case Efficient External-Memory Priority Queues

Gerth Stølting Brodal

Max–Planck–Institut f¨ ur Informatik Saarbr¨ ucken Germany

Jyrki Katajainen

Datalogisk Institut Københavns Universitet Denmark July 1998

SLIDE 2

Worst-Case Efficient External-Memory Priority Queues

Priority Queues

Maintain a set of

N elements from a totally ordered universe under

INSERT

x – Insert element x into the set

DELETEMIN

– Delete and return the minimum of the set

INSERT and DELETEMIN can be implemented with

log

N
comparisons

– Williams, 1964

G. S. Brodal, J. Katajainen

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SLIDE 3

Worst-Case Efficient External-Memory Priority Queues

External Memory Model

– Aggarwal, Vitter, 1988

A A A

CPU

registers

Internal memory

External memory External memory

Y I/O

M = Size of the internal memory B = I/O block size N = # elements in the priority queue

Complexity measures Internal : # comparisons External : # I/Os Assumptions

M

B ,

B

log

M B N M

G. S. Brodal, J. Katajainen

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SLIDE 4

Worst-Case Efficient External-Memory Priority Queues

External-Memory Merge Sorting

Theorem

N elements stored in O

N

B

blocks can be sorted with

N log

N

comparisons and

N

B log M B N B

I/Os.

Algorithm: (1) Sort in internal memory all runs of

M elements

(2) In

log M B N M iterations apply a

M

B

ary merging algorithm

to obtain lists of length

M

M

B

i.

The sorting algorithm is optimal w.r.t. both comparisons and I/Os. – Aggarwal, Vitter, 1988

G. S. Brodal, J. Katajainen

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SLIDE 5

Worst-Case Efficient External-Memory Priority Queues

Worst-Case Effeciency

Internal Each priority queue operation should require

log

N
comparisons in the worst-case

External The I/Os should be divided evenly among the operations, i.e., one I/O for every

B
log

M B N B

th operation.
G. S. Brodal, J. Katajainen

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SLIDE 6

Worst-Case Efficient External-Memory Priority Queues

External-Memory Priority Queues

Binary heap – Williams, 1964 – Random memory accesses imply one I/O every

O CPU operation.

Fishspear – Fischer, Patterson, 1994 –

O

N

B log

N
I/Os for N operations. Uses

O push down stacks.

Heap + B elements/node – Wegner, Teuhola, 1989 –

O

log
N

B

I/Os for every

Bth operation. Optimal for M

O

B .

Buffer tree * – Arge, 1995 –

O

N

B log M B N M

I/Os for N operations.

Heap with buffers * – Fadel et al., 1997 – Similar bounds as for buffer trees. * No worst-case guarantee on the individual operations

G. S. Brodal, J. Katajainen

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SLIDE 7

Worst-Case Efficient External-Memory Priority Queues

New Result

Theorem DELETEMIN and INSERT can be done with

log N

comparisons and one I/O for every

B

log

M B N B th operation.

Outline of the Data Structure

Internal part External part

MIN
NEW
Work area
Q

Q k

Sorted lists of length

M
M

B

i

MIN stores the overall

O M smallest elements fast DELETEMIN.

NEW stores the most recently inserted

O M elements fast INSERT.

[ small elements are inserted directly into MIN ] The external part stores exactly

B elements per block

[ all elemenst are larger than the elements in MIN ]

G. S. Brodal, J. Katajainen

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SLIDE 8

Operations

K

M
INSERT

x

– If

x

max MIN then swap

x and max MIN

– Insert

x into NEW

– If

jNEWj

K then perform BATCHINSERT

with

K elements from NEW.

DELETEMIN

– If MIN
then perfrom BATCHDELETE

Merge the result with NEW Move the

K smallest elements to MIN

– Delete and return

min MIN

BATCHINSERT – Create a new list of length

K and assign it rank 1

–

i

– while #list of rank

i

M

B do

Merge the lists with rank

i

Assign the resulting list rank

i

i
i
BATCHDELETE

– Fetch the

K smallest elements from external

memory to internal memory rank

L

blog

M B jLj B c

lists
M

B log M B N M

G. S. Brodal, J. Katajainen

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SLIDE 9

Worst-Case Efficient External-Memory Priority Queues

Amortized Result

Lemma The described data structure supports INSERT and DELETEMIN with amortized

log N comparisons and

B

log M B N B

I/Os.

[ Identical bounds follow by using buffer trees and external heaps ]

Worst-Case Bounds?

Incremental list merging

– Thorup, 1996

Incremental batch operations Start BATCHDELETE before MIN gets empty.

G. S. Brodal, J. Katajainen

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SLIDE 10

Incremental List Merging

Lists of rank

i L

i

L

i
L

n i i L i z

L
i

L

i
L

n i i

MERGESTEP

i

– Perform

K steps of the list merging at rank i.

– If the merging is finished, then make the resulting list

L

i a list of rank i

(list promotion)

BATCHINSERT – Create a new list with rank one –

i MERGESTEP i

G. S. Brodal, J. Katajainen

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SLIDE 11

Deletions

BATCHDELETE deletes and returns the

K least

elements of the lists.

The length of the lists decreases.
The maximum rank increases.

Possible solution: Incremental global rebuilding. Our solution: 1) If the incremental merging of rank

i finishes,

but

j

L

i j

K
M

B

i, then we do not promote
L

i

to rank

i

.

2) Let

R be the maximum rank of a list.

If the resulting list

L

R

K
M

B

R

,

then

L

R gets rank R

.

3) If

K elements are deleted from

L

i, we perform

MERGESTEP

i.

4) Always perform MERGESTEP

R after

BATCHDELETE

R
O
log

M B N M

G. S. Brodal, J. Katajainen

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SLIDE 12

Worst-Case Efficient External-Memory Priority Queues

Conclusion

The first worst case external-memory priority-queue implementation.

Open Problems

Is the definition of “worst-case” reasonable in practice ? Experimental studies. Can buffer trees be deamortized, i.e., how can buffer trees support

perations with worst-case

O

log

M B N B

I/Os for

B operations ?

G. S. Brodal, J. Katajainen

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