1 sorting. 2 motivation sells(bar,beer,price )bars(bar,addr ) joe’sbud2.50joe’smaple st....

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Sorting

2

Motivation

Sells( bar, beer, price ) Bars( bar, addr )Joe’s Bud 2.50 Joe’s Maple St.Joe’s Miller 2.75 Sue’s River Rd.Sue’s Bud 2.50Sue’s Coors3.00

Query: Find all locations that sell beer for less than 2.75.

Select Bars.addr From Sells, Bars Where (Sells.bar = Bars.bar) and (Sells.price < 2.75)

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Motivation

Sells Bars

JOIN Sells.bar = Bars.bar

PROJECTBars.addr

SELECTSells.price < 2.75

A straightforwardjoin is nested loop

Another possiblejoin is to sortthen mergethe tables

Which one is faster?Need a cost model

Illustrating the Two Different Join Algs

4

1 4 7 3 8 2

2 6 9 3 4

(4,4), (3,3), (2,2)

1 2 3 4 7 8

2 3 4 6 9

5

The I/O Model of Computation• In main memory algorithms

– we care about CPU time

• In databases time is dominated by I/O cost• Assumption: cost is given only by I/O• Consequence:

– need to redesign certain algorithms– compute cost using I/O read/writes

• Will illustrate here with sorting

6

Sorting• Illustrates the difference in algorithm design

when your data is not in main memory:– Problem: sort 1Gb of data with 1Mb of RAM.

• Arises in many places in database systems: – Data requested in sorted order (ORDER BY)– Needed for grouping operations– First step in sort-merge join algorithm– Duplicate removal– Bulk loading of B+-tree indexes.

7

2-Way Merge-sort:Requires 3 Buffers

• Pass 1: Read a page, sort it, write it.– only one buffer page is used

• Pass 2, 3, …, etc.:– three buffer pages used.

Main memory buffers

INPUT 1

INPUT 2

OUTPUT

DiskDisk

8

Two-Way External Merge Sort• Each pass we read + write

each page in file.• N pages in the file => the

number of passes

• So total cost is:

• Improvement: start with

larger runs• Sort 1GB with 1MB memory

in 10 passes

log2 1N

2 12N Nlog

Input file

1-page runs

2-page runs

4-page runs

8-page runs

PASS 0

PASS 1

PASS 2

PASS 3

9

3,4 6,2 9,4 8,7 5,6 3,1 2

3,4 5,62,6 4,9 7,8 1,3 2

2,34,6

4,7

8,91,35,6 2

2,3

4,46,7

8,9

1,23,56

1,22,3

3,4

4,56,6

7,8

9

265 317 84 962443

Two-Way External Merge Sort

2368 14 53 4 96 2 73 4 6 2 49 8 7 5 136 25

PASS 0

PASS 1

Input file

1-page runs

2-page runs

PASS 2

10

76

68 9

53

44

2

1 2

3

2 3

4 6 6

4 7

8 9

1

5

3

2

PASS 2

PASS 3

2-page runs

4-page runs

8-page runs

98

74

64

2 3

6

3

5 2

1

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Can We Do Better ?• We have more main memory• Should use it to improve performance

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Cost Model for Our Analysis

• B: Block size• M: Size of main memory• N: Number of records in the file• R: Size of one record

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External Merge-Sort• Phase one: load M bytes in memory, sort

– Result: runs of length M/R records

M bytes of main memory

DiskDisk

. .

.. . .

M/R records

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Phase Two

• Merge M/B – 1 runs into a new run• Result: runs have now M/R (M/B – 1) records


DiskDisk

. .

.. . .

Input M/B

Input 1

Input 2. . . .

Output

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Phase Three

• Merge M/B – 1 runs into a new run• Result: runs have now M/R (M/B – 1)2 records


DiskDisk

. .

.. . .

Input M/B

Input 1

Input 2. . . .

Output

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Cost of External Merge Sort• Number of passes:• Think differently

– Given B = 4KB, M = 64MB, R = 0.1KB– Pass 1: runs of length M/R = 640000

• Have now sorted runs of 640000 records– Pass 2: runs increase by a factor of M/B – 1 = 16000

• Have now sorted runs of 10,240,000,000 = 1010 records– Pass 3: runs increase by a factor of M/B – 1 = 16000

• Have now sorted runs of 1014 records• Nobody has so much data !

• Can sort everything in 2 or 3 passes !

MNRBM /log1 1/

Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 17

Extra Materials on SortingRead only after we have

covered B+ tree, and only if you want to know more

about sortingChapter 13


Why Sort?

A classic problem in computer science! Data requested in sorted order

e.g., find students in increasing gpa order Sorting is first step in bulk loading B+ tree

index. Sorting useful for eliminating duplicate

copies in a collection of records (Why?) Sort-merge join algorithm involves sorting. Problem: sort 1Gb of data with 1Mb of RAM.

why not virtual memory?


2-Way Sort: Requires 3 Buffers

Pass 1: Read a page, sort it, write it. only one buffer page is used

Pass 2, 3, …, etc.: three buffer pages used.

Main memory buffers

INPUT 1

INPUT 2

OUTPUT

DiskDisk


Two-Way External Merge Sort Each pass we read +

write each page in file. N pages in the file =>

the number of passes

So toal cost is:

Idea: Divide and

conquer: sort subfiles and merge

log2 1N

2 12N Nlog

Input file

1-page runs

2-page runs

4-page runs

8-page runs

PASS 0

PASS 1

PASS 2

PASS 3

9

3,4 6,2 9,4 8,7 5,6 3,1 2

3,4 5,62,6 4,9 7,8 1,3 2

2,34,6

4,7

8,91,35,6 2

2,3

4,46,7

8,9

1,23,56

1,22,3

3,4

4,56,6

7,8


General External Merge Sort

To sort a file with N pages using B buffer pages: Pass 0: use B buffer pages. Produce sorted

runs of B pages each. Pass 2, …, etc.: merge B-1 runs.

N B/

B Main memory buffers

INPUT 1

INPUT B-1

OUTPUT

DiskDisk

INPUT 2

. . . . . .

. . .

More than 3 buffer pages. How can we utilize them?


Cost of External Merge Sort

Number of passes: Cost = 2N * (# of passes) E.g., with 5 buffer pages, to sort 108

page file: Pass 0: = 22 sorted runs of 5

pages each (last run is only 3 pages) Pass 1: = 6 sorted runs of 20

pages each (last run is only 8 pages) Pass 2: 2 sorted runs, 80 pages and 28

pages Pass 3: Sorted file of 108 pages

1 1 log /B N B

108 5/

22 4/


Number of Passes of External Sort

N B=3 B=5 B=9 B=17 B=129 B=257100 7 4 3 2 1 11,000 10 5 4 3 2 210,000 13 7 5 4 2 2100,000 17 9 6 5 3 31,000,000 20 10 7 5 3 310,000,000 23 12 8 6 4 3100,000,000 26 14 9 7 4 41,000,000,000 30 15 10 8 5 4


Internal Sort Algorithm

Quicksort is a fast way to sort in memory.

An alternative is “tournament sort” (a.k.a. “heapsort”) Top: Read in B blocks Output: move smallest record to output

buffer Read in a new record r insert r into “heap” if r not smallest, then GOTO Output else remove r from “heap” output “heap” in order; GOTO Top


More on Heapsort

Fact: average length of a run in heapsort is 2B The “snowplow” analogy

Worst-Case: What is min length of a run? How does this arise?

Best-Case: What is max length of a run? How does this arise?

Quicksort is faster, but ...

B


I/O for External Merge Sort

… longer runs often means fewer passes!

Actually, do I/O a page at a time In fact, read a block of pages

sequentially! Suggests we should make each buffer

(input/output) be a block of pages. But this will reduce fan-out during merge

passes! In practice, most files still sorted in 2-3

passes.


Number of Passes of Optimized SortN B=1,000 B=5,000 B=10,000100 1 1 11,000 1 1 110,000 2 2 1100,000 3 2 21,000,000 3 2 210,000,000 4 3 3100,000,000 5 3 31,000,000,000 5 4 3

Block size = 32, initial pass produces runs of size 2B.


Double Buffering To reduce wait time for I/O request to

complete, can prefetch into `shadow block’. Potentially, more passes; in practice, most

files still sorted in 2-3 passes.

OUTPUT

OUTPUT'

Disk Disk

INPUT 1

INPUT k

INPUT 2

INPUT 1'

INPUT 2'

INPUT k'

block sizeb

B main memory buffers, k-way merge


Sorting Records!

Sorting has become a blood sport! Parallel sorting is the name of the game ...

Datamation: Sort 1M records of size 100 bytes Typical DBMS: 15 minutes World record: 3.5 seconds

• 12-CPU SGI machine, 96 disks, 2GB of RAM

New benchmarks proposed: Minute Sort: How many can you sort in 1

minute? Dollar Sort: How many can you sort for

$1.00?


Using B+ Trees for Sorting

Scenario: Table to be sorted has B+ tree index on sorting column(s).

Idea: Can retrieve records in order by traversing leaf pages.

Is this a good idea? Cases to consider:

B+ tree is clustered Good idea! B+ tree is not clusteredCould be a very bad

idea!


Clustered B+ Tree Used for Sorting

Cost: root to the left-most leaf, then retrieve all leaf pages (Alternative 1)

If Alternative 2 is used? Additional cost of retrieving data records: each page fetched just once.

Always better than external sorting!

(Directs search)

Data Records

Index

Data Entries("Sequence set")


Unclustered B+ Tree Used for Sorting

Alternative (2) for data entries; each data entry contains rid of a data record. In general, one I/O per data record!

(Directs search)

Data Records

Index

Data Entries("Sequence set")


External Sorting vs. Unclustered IndexN Sorting p=1 p=10 p=100

100 200 100 1,000 10,000

1,000 2,000 1,000 10,000 100,000

10,000 40,000 10,000 100,000 1,000,000

100,000 600,000 100,000 1,000,000 10,000,000

1,000,000 8,000,000 1,000,000 10,000,000 100,000,000

10,000,000 80,000,000 10,000,000 100,000,000 1,000,000,000

p: # of records per page B=1,000 and block size=32 for sorting p=100 is the more realistic value.


Summary

External sorting is important; DBMS may dedicate part of buffer pool for sorting!

External merge sort minimizes disk I/O cost: Pass 0: Produces sorted runs of size B (# buffer

pages). Later passes: merge runs. # of runs merged at a time depends on B, and

block size. Larger block size means less I/O cost per page. Larger block size means smaller # runs merged. In practice, # of runs rarely more than 2 or 3.


Summary, cont.

Choice of internal sort algorithm may matter: Quicksort: Quick! Heap/tournament sort: slower (2x), longer

runs The best sorts are wildly fast:

Despite 40+ years of research, we’re still improving!

Clustered B+ tree is good for sorting; unclustered tree is usually very bad.

1 sorting. 2 motivation sells(bar,beer,price )bars(bar,addr ) joe’sbud2.50joe’smaple st....

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