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External Sorting: Demetris Zeinalipour

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Department of Computer Science

University of Cyprus
EPL446 Advanced Database Systems

Lecture 9
External Sorting
Chapter 13: Ramakrishnan & Gehrke

Demetris Zeinalipour
http://www.cs.ucy.ac.cy/~dzeina/courses/epl446
EPL446: Advanced Database Systems - Demetris Zeinalipour (University of Cyprus)

9-1

External Sorting Introduction


( : )
Problem: We cant sort 1TB of data with 1GB of RAM (i.e.,
more data than available memory) in main memory
Solution: Utilize an External Sorting Algorithm
External sorting refers to the sorting of a file that resides on
secondary memory (e.g., disk, flash, etc).
Internal sorting refers to the sorting of an array of data that is in
RAM (quick-, merge-, insertion-, selection-, radix-, bucket-, bubble,heap-, sort algorithms we saw in the Data Struct. & Alg. Course)

Objective: Minimize number of I/O accesses.


External Sorting is part of the Query Evaluation /
Optimization subsystem
Efficient Sorting algorithms can speed up query evaluation plans
(e.g., during joins)!
EPL446: Advanced Database Systems - Demetris Zeinalipour (University of Cyprus)

9-2

Lecture Outline
External Sorting
13.1) When does a DBMS sort Data.
13.2) A Simple Two-Way Merge-Sort
(
)
13.3) External Merge-Sort (E
)
Exclude 13.3.1: Minimizing the Number of
Runs.

13.4.2) Double Buffering (


)
13.5) Using B+Trees for Sorting

Query Optimization
and Execution

Relational Operators
Files and Access Methods
Buffer Management
Disk Space Management

DB

EPL446: Advanced Database Systems - Demetris Zeinalipour (University of Cyprus)

9-3

When Does a DBMS Sort Data?


( . ;)
When Does a DBMS Sort Data? (~30% of oper.)
Data requested in sorted order
e.g., SELECT * FROM Students ORDER BY gpa DESC;
Sorting is first step in bulk loading a B+ tree index.
i.e., CREATE INDEX StuAge ON Students(age) USING
BTREE;
Recall how leaf nodes of the B+tree are ordered.
Useful for eliminating duplicate copies in a collection of
records.
SELECT DISTINCT age FROM Students;
i.e., to eliminate duplicates in a sorted list requires only
the comparison of each element to its previous element
so this yields a linear order elimination algorithm.
9-4
EPL446: Advanced Database Systems - Demetris Zeinalipour (University of Cyprus)

Two-Way External Merge-Sort


( )
Let us consider the simplest idea for external sorting
Assumption: Only 3 Buffer pages are available
Idea: Divide and Conquer (similarly to MergeSort, Quicksort)

Idea Outline
Pass 0 (Sort Lists):For every page, read it, sort it, write it out

Only one buffer page is used!


Now we need to merge them hierarchically

Pass 1, 2, , etc. (Merge Lists): see next page for merging concept

For this step we need three buffer pages!


Main Memory Sort
INPUT 1

Buffer

Disk

Main
memory

OUTPUT

INPUT 2

Disk

Disk

Main memory
buffers

Passes 1,2,

Sorted Run or simply Run




EPL446: Advanced Database Systems - Demetris Zeinalipour (University of Cyprus)

Pass 0

Disk

9-5

Merging Lists
( 2 )
Merging Lists Outline (Phase 1,2,)
1. Load the next sorted runs R1 and R2 into main memory buffers B1 and B2 a
page-at-a-time (i.e., initially first page from each run) (see left figure)

Obviously R1>=B1 and R2>=B2 (a Run might be larger than a Buffer)


The rest pages will be loaded to main memory during subsequent steps.

2. Initialize indices i, j to the head of each list (i.e., i=j=0)


3. Compare B1[i] with B2[j] and move smallest item to OUTPUT buffer.

If B1[i] was smallest item then i++ else j++ (see right figure)
If OUTPUT gets full, it is appended to the end of a file on DISK and cleared in RAM.

4. Repeat the above until either index i or j reaches the end of its buffer.

At this point write the remaining records to OUTPUT, flush it to disk and finish.

5. Repeat procedure from 1-4 until all runs have been traversed.

B1

B1

OUTPUT

16

30
B2

min(B1[i],B2[j])

B2

Disk

Main memory buffers

Disk

OUTPUT

15

28 60

EPL446: Advanced Database Systems - Demetris Zeinalipour (University of Cyprus)

9-6

Two-Way External Merge Sort Example


( )

Example
Execution

3,4

6,2

9,4

8,7

5,6

3,1

Input file

PASS 0

3,4

2,6

4,9

7,8

5,6

1,3

1-page runs

PASS 1
4,7
8,9

2,3
4,6

1,3
5,6

2-page runs

PASS 2
2,3
4,4
6,7
8,9

1,2
3,5
6

4-page runs

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

PASS 4

EPL446: Advanced Database Systems - Demetris Zeinalipour (University of Cyprus)

9-7

Cost of Two-Way External Merge Sort


( )
Each pass we read + write
each f N pages in file. Pass 0
Number of passes:

log 2 N 1

e.g., for N=7, N=5 and N=4

log10 7
1 2.8 1 4
log10 2

log 2 7 1

log 2 5 1 2.3 1 4
log 2 4 1 2 1 3

Total (I/O) cost is:

2 N * (# passes)

3,4
3,4

6,2
2,6

9,4
4,9

8,7
7,8

5,6
5,6

3,1
1,3

Input file

PASS 0
1-page runs

PASS 1
4,7
8,9

2,3
4,6

1,3
5,6

2-page runs
2
PASS 2

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

1,2
3,5
6

4-page runs

log 2 N
PASS 3

1,2
2,3
2 * 7 * (log 2 7 1) 2 * 7 * 4 56
3,4
8-page runs
i.e., (read+write) * 7 pages * 4 passes
4,5
That can be validated on the right figure
6,6
Pass#0=2*7
Pass#1=2*7
7,8
Pass#2=2*7
Pass$3=2*7
9
9-8
EPL446: Advanced Database Systems - Demetris Zeinalipour (University of Cyprus)

e.g., for N=7

Two-Way External Merge-Sort


( )
Two-Way External MergeSort Pseudocode

Phase 0

We
want to
end up
with 1 Merging
Lists
final run

EPL446: Advanced Database Systems - Demetris Zeinalipour (University of Cyprus)

9-9

Internal Sorting Remarks


( . )
External Sorting Algorithms utilize some Internal Sorting Algorithm to
sort records in main memory.
Where Does In-Memory Sorting happens?
Dedicated Sorting Region (e.g., IBM, Informix, Oracle)
In general purpose Buffer Manager (e.g., Microsoft and Sybase IQ
for data warehousing)
Elsewhere, e.g., cache of recently executed stored procedures
(e.g., Sybase ASE for OnLine Transaction Processing )
What In-Memory Algorithms do real DBMSes use?
Oracle uses InsertionSort
Microsoft and Sybase ASE use MergeSort
IBM and Sybase IQ use RadixSort (quite similar to BucketSort)
All systems utilize Asynchronous () I/O and Prefetching
() we shall see the benefits of these techniques in the
subsequent slides.
EPL446: Advanced Database Systems - Demetris Zeinalipour (University of Cyprus)

9-10

General External Merge Sort


( )
Lets turn the 2-way Merge-sort Algorithm into a Practical Alg.
Assumption: B Buffer pages are available
Idea: Merge (B-1) pages in each step rather than only 2 (faster!)

Idea Outline
Pass 0 (Sort): Sort the N pages using B buffer pages

Use B buffer pages for input


That generates N1 N / B sorted runs e.g., N=8 and B=4 => N1= 8 / 4 2

Pass 1, 2, , etc. (Merge): Perform a (B-1)-way merge of runs


Use (B-1) buffer pages for input + 1 page for output
Number of passes will be reduced dramatically! (see slide 13)

Pass 0

Passes 1,2,

EPL446: Advanced Database Systems - Demetris Zeinalipour (University of Cyprus)

9-11

Cost of External Merge Sort


( )
(B-1)-way Sort

Number of passes: 1 log B 1 N / B

2-way Sort

1log 2 N

I/O Cost = 2N * (# of passes)


2N * (# of passes)
Example: N=108 pages and B=5 Buffer pages
#Passes= 1 log 4 108 / 5 4
IO Cost = Read+Write 108 pages for 4 passes => 2*108*4 = 864 IOs
Pass 0:108 / 5 = 22 sorted runs of 5 pages each (last run is
only 3 pages)
Pass 1: 22 / 4 = 6 sorted runs (5 runs x 20 pages each and
last run is only 8 pages)
Pass 2: 6 / 4= 2 sorted runs, 80 pages (4*20 pages) and 28
2
2
2
8
pages
0
0
0
2
0

2
0

Pass
3: Database
Sorted file
of 108
pages
EPL446:
Advanced
Systems
- Demetris
Zeinalipour (University of Cyprus)

9-12

Number of Passes of External Sort


External Merge Sort is quite efficient!
With only B=257 (~1MB) Buffer Pages we can sort N=1 Billion
records with four (4) passes in practice B will be larger
9
Two-Way Mergesort would require log 2 10 1 30 1 passes!

B=3
100
7
1,000
10
10,000
13
100,000
17
1,000,000
20
10,000,000
23
100,000,000
26
1,000,000,000 30

B=5
4
5
7
9
10
12
14
15

B=9
3
4
5
6
7
8
9
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

1 log
/ B of Cyprus)
EPL446:
Advanced
Database
Zeinalipour
(University
* Results
generated
withSystems
formula:- Demetris
B 1 N

9-13

External Merge-Sort
( )
External MergeSort Pseudocode

Phase 0

EPL446: Advanced Database Systems - Demetris Zeinalipour (University of Cyprus)

9-14

Double Buffering
( )
An I/O request takes time to complete. Only think about all the
involved layers and delays (Disk Delays, Buffer Manager, etc)
To reduce wait time of CPU for I/O request to complete, can prefetch
() into `shadow block ( )
Main Idea: When all tuples of INPUTi have been consumed, the CPU
can process INPUTi which is prefetched into main memory instead of
waiting for INPUTi to refill. same idea applies to OUTPUT.
INPUT 1
INPUT 1'
INPUT 2

INPUT 2'

OUTPUT
OUTPUT'

Disk

INPUT k

block size

Disk

INPUT k'

EPL446: Advanced Database


Systems
Zeinalipour
(B main
memory- Demetris
buffers, k-way
merge)(University of Cyprus)

9-15

Using B+ Trees for Sorting


( B+Tree )
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

Always Good idea!

B+ tree is not clustered

Could be a very bad idea!

EPL446: Advanced Database Systems - Demetris Zeinalipour (University of Cyprus)

9-16

Clustered B+ Tree Used for Sorting


( )
Idea: From root go to the
left-most leaf, then retrieve
all next leaf pages
(Alternative 1)
Cost: logFN, where F is
the branching factor.
If Alternative 2 is used?
Additional cost of retrieving
data records: however
each page is only fetched
once (and not not 3-4
rounds like external
sorting)

Index
(Directs search)

Data Entries
("Sequence set"

Data Records

* Always better than external sorting!

EPL446: Advanced Database Systems - Demetris Zeinalipour (University of Cyprus)

9-17

Unclustered B+ Tree Used for Sorting


( - )
How about an Unclustered Index? i.e.,
Altern.(2) or Altern.(3) (note: Altern.(1) is always
Clustered)
Cost: In general, one I/O per data record!
This solution is really bad!
Index
(Directs search)

Data Entries
("Sequence set")

EPL446: Advanced Database Systems


- Demetris
Zeinalipour (University of Cyprus)
Data
Records

9-18

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