Sorting 2

External Sorting
Module 2, Lecture 6
Database Management Systems, R. Ramakrishnan
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.
INPUT 1
OUTPUT
INPUT 2
Disk
Main memory buffers
Disk
3
Two-Way External Merge Sort
Each pass we read + write

each page in file.
N pages in the file => the
number of passes
= log2 N + 1
So toal cost is:
6,2
9,4
8,7
5,6
3,1
3,4
2,6
4,9
7,8
5,6
1,3
4,7
2,3
4,6
1,3
5,6
8,9
Input file
PASS 0
1-page runs
PASS 1
2
2-page runs
PASS 2
2,3
2 N log 2 N + 1
3,4
Idea: Divide and conquer:

sort subfiles and merge
4,4
6,7
1,2
3,5
6
8,9
4-page runs
PASS 3
1,2
2,3
3,4
8-page runs
4,5
6,6
7,8
9
General External Merge Sort

More than 3 buffer pages. How can we utilize them?
To sort a file with N pages using B buffer pages:
Pass 0: use B buffer pages. Produce N / B sorted runs of B
pages each.
Pass 2, , etc.: merge B-1 runs.
INPUT 1
...
INPUT 2
...
OUTPUT
...
INPUT B-1
Disk
B Main memory buffers
Disk
5
Cost of External Merge Sort
Number of passes: 1 + log B 1 N / B

Cost = 2N * (# of passes)
E.g., with 5 buffer pages, to sort 108 page file:
Pass 0: 108 / 5 = 22 sorted runs of 5 pages each
(last run is only 3 pages)
Pass 1: 22 / 4 = 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
Number of Passes of External Sort

N
B=3 B=5 B=9

100
7
4
3
1,000
10
5
4
10,000
13
7
5
100,000
17
9
6
1,000,000
20
10
7
10,000,000
23
12
8
100,000,000
26
14
9
1,000,000,000 30
15
10
B=17 B=129 B=257

2
1
1
3
2
2
4
2
2
5
3
3
5
3
3
6
4
3
7
4
4
8
5
4
7
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 ...
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.
10
Number of Passes of Optimized Sort

N
100
1,000
10,000
100,000
1,000,000
10,000,000
100,000,000
1,000,000,000
B=1,000
1
1
2
3
3
4
5
5
B=5,000
1
1
2
2
2
3
3
4
B=10,000
1
1
1
2
2
3
3
3
Block size = 32, initial pass produces runs of size 2B.

11
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.
INPUT 1
INPUT 1'
INPUT 2
INPUT 2'
OUTPUT
OUTPUT'
Disk
INPUT k
block size
Disk
INPUT k'
B main memory buffers, k-way merge
12
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?
13
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
B+ tree is not clustered
Good idea!
Could be a very bad idea!
14
Clustered B+ Tree Used for Sorting
Cost: root to the leftmost leaf, then retrieve

all leaf pages
(Alternative 1)
If Alternative 2 is used?
Additional cost of
retrieving data records:
each page fetched just
once.
Index
(Directs search)
AAAAAAAAAAAA
AAAAAAAAAAAA
AAAA
AAAA
AAAAAAAA
AAAA
AAAAAAAA
AAAA
AAAA
AAAA
AAAA
AAAAAAAAAAAA
AAAA
AAAA
AAAA
AAAA
AAAAAAAAAAAA
AAAAAAAAAAAA
AAAAAAAAAAAA
AAAA
AAAA
AAAAAAAA
AAAA
AAAAAAAA
AAAA
AAAA
AAAA
AAAA
AAAAAAAAAAAA
AAAA
AAAA
AAAA
AAAAAAAAAAAA
AAAA
AAAAAAAAAAAA
AAAAAAAAAAAA
AAAA
AAAA
AAAAAAAA
AAAA
AAAAAAAA
AAAA
AAAA
AAAA
AAAA
AAAAAAAAAAAA
AAAA
AAAA
AAAA
AAAAAAAAAAAA
AAAA
Data Entries
("Sequence set")
Data Records
Always better than external sorting!

15
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!
Index
(Directs search)
AAAAAAAA
AAAA
AAAA
AAAA
AAAAAAAA
AAAA
AAAAAAAA
AAAAAAAA
AAAA
AAAA
AAAAAAAAAAAA
AAAA
AAAA
AAAAAAAA
AAAA
AAAA
AAAAAAAA
AAAAAAAA
AAAAAAAA
AAAA
AAAA
AAAA
AAAAAAAA
AAAA
AAAAAAAA
AAAAAAAA
AAAA
AAAAAAAA
AAAAAAAA
AAAA
AAAA
AAAAAAAA
AAAA
AAAA
AAAAAAAA
AAAAAAAA
AAAAAAAA
AAAA
AAAA
AAAA
AAAAAAAA
AAAA
AAAAAAAA
AAAAAAAA
AAAA
AAAAAAAA
AAAAAAAA
AAAA
AAAA
AAAAAAAA
AAAA
AAAA
AAAAAAAA
AAAAAAAA
Data Entries
("Sequence set")
Data Records
16
External Sorting vs. Unclustered Index

N
100
1,000
10,000
100,000
1,000,000
10,000,000
Sorting
p=1
p=10
p=100
200
2,000
40,000
600,000
8,000,000
80,000,000
100
1,000
10,000
100,000
1,000,000
10,000,000
1,000
10,000
100,000
1,000,000
10,000,000
100,000,000
10,000
100,000
1,000,000
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.
17
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.
18
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, were still
improving!
Clustered B+ tree is good for sorting;

unclustered tree is usually very bad.
19

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External Sorting

Database Management Systems, R. Ramakrishnan

A classic problem in computer science!

Sorting is first step in bulk loading B+ tree index.

Database Management Systems, R. Ramakrishnan

2-Way Sort: Requires 3 Buffers

Pass 1: Read a page, sort it, write it.

Main memory buffers

Database Management Systems, R. Ramakrishnan

Two-Way External Merge Sort

Each pass we read + write

So toal cost is:

Idea: Divide and conquer:

Database Management Systems, R. Ramakrishnan

General External Merge Sort

B Main memory buffers

Database Management Systems, R. Ramakrishnan

Cost of External Merge Sort

Number of passes: 1 + log B 1 N / B

Database Management Systems, R. Ramakrishnan

Number of Passes of External Sort

B=3 B=5 B=9

B=17 B=129 B=257

Internal Sort Algorithm

Quicksort is a fast way to sort in memory.

Top: Read in B blocks

Database Management Systems, R. Ramakrishnan

Fact: average length of a run in heapsort is 2B

What is max length of a run?

Quicksort is faster, but ...

Database Management Systems, R. Ramakrishnan

I/O for External Merge Sort

longer runs often means fewer passes!

Database Management Systems, R. Ramakrishnan

Number of Passes of Optimized Sort

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

To reduce wait time for I/O request to

B main memory buffers, k-way merge

Database Management Systems, R. Ramakrishnan

Sorting has become a blood sport!

Datamation: Sort 1M records of size 100 bytes

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

New benchmarks proposed:

Database Management Systems, R. Ramakrishnan

Using B+ Trees for Sorting

Scenario: Table to be sorted has B+ tree index on

Database Management Systems, R. Ramakrishnan

Clustered B+ Tree Used for Sorting

Cost: root to the leftmost leaf, then retrieve

Always better than external sorting!

Unclustered B+ Tree Used for Sorting

Alternative (2) for data entries; each data

External Sorting vs. Unclustered Index

p: # of records per page

External sorting is important; DBMS may dedicate

Database Management Systems, R. Ramakrishnan

Choice of internal sort algorithm may matter:

The best sorts are wildly fast: