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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)
3
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
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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
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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
11
Can We Do Better ?• We have more main memory• Should use it to improve performance
12
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
13
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
14
Phase Two
• Merge M/B – 1 runs into a new run• Result: runs have now M/R (M/B – 1) records
M bytes of main memory
DiskDisk
. .
.. . .
Input M/B
Input 1
Input 2. . . .
Output
15
Phase Three
• Merge M/B – 1 runs into a new run• Result: runs have now M/R (M/B – 1)2 records
M bytes of main memory
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
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 18
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?
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 19
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
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 20
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
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 21
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?
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 22
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/
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 23
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
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 24
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
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 25
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
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 26
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.
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 27
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.
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 28
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
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 29
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?
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 30
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!
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 31
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")
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 32
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")
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 33
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.
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 34
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.
Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 35
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.