External Sorting

Chapter 13 1

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 collection of records Sort-merge join copies in a algorithm involves sorting.

 Problem: sort 1Gb of data with 1Mb of RAM.

 why not virtual memory?

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Using secondary storage effectively

 General Wisdom :   I/O costs dominate Design algorithms to reduce I/O 3

2-Way Sort: Requires 3 Buffers

  Phase 1: PREPARE.   Read a page, sort it, write it.

only one buffer page is used Phase 2, 3, …, etc.: MERGE:  three buffer pages used.

INPUT 1 OUTPUT INPUT 2 Main memory buffers Disk Disk

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Two-Way External Merge Sort

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



Idea: Divide

and conquer:

sort subfiles and merge into larger sorts

3,4 2,6 4,9 7,8 5,6 1,3 2,3 4,6 2,3 4,4 6,7 8,9 4,7 8,9 1,3 5,6 1,2 3,5 6 2 2 Input file PASS 0 1-page runs PASS 1 2-page runs PASS 2 4-page runs PASS 3 1,2 2,3 3,4 4,5 6,6 7,8 9 8-page runs

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Two-Way External Merge Sort

 Costs for pass : all pages

3,4 6,2 9,4 8,7 5,6 3,1 2 3,4 2,6 4,9 7,8 5,6 1,3 2 2,3 4,6 4,7 8,9 1,3 5,6 2 Input file PASS 0 1-page runs PASS 1 2-page runs PASS 2

 # of passes : height of tree

2,3 4,4 6,7 8,9 1,2 3,5 6 4-page runs PASS 3

 Total cost : product of above

1,2 2,3 3,4 4,5 6,6 7,8 9 8-page runs

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Two-Way External Merge Sort

2

  Each pass we read + write each page in file.

N pages in file => 2N

3,4 3,4 2,3 4,6 6,2 2,6

 Number of passes   log 2

N

  1

2,3 4,4 6,7 8,9 9,4 4,9 4,7 8,9 8,7 7,8 5,6 5,6 1,3 5,6 3,1 1,3 1,2 3,5 6 2 2 Input file PASS 0 1-page runs PASS 1 2-page runs PASS 2 4-page runs PASS 3

 So total cost is: 2

N

  log 2

N

  1 

1,2 2,3 3,4 4,5 6,6 7,8 9 8-page runs

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External Merge Sort

  What if we had more buffer pages?

How do we utilize them wisely ?

-



Two main ideas !

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Phase 1 : Prepare

. . .

Disk INPUT 1 INPUT 2

. . .

INPUT B B Main memory buffers Disk

• Construct as large as possible starter lists.

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Phase 2 : Merge

. . .

Disk INPUT 1 INPUT 2

. . .

OUTPUT INPUT B-1 B Main memory buffers Disk Compose as many sorted sublists into one long sorted list.

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*

General External Merge Sort

How can we utilize more than 3 buffer pages?

 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 1, 2, …, etc.: merge B-1 runs .

INPUT 1

. . .

Disk INPUT 2

. . .

OUTPUT INPUT B-1 B Main memory buffers

. . .

Disk

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Cost of External Merge Sort

  Number of passes: 1  Cost = 2N * (# of passes)  log

B

 1    12

Example

  Buffer : with 5 buffer pages File to sort : 108 pages  Pass 0: • • Size of each run?

Number of runs?

 Pass 1: • • Size of each run?

Number of runs?

 Pass 2: ???

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Example

  Buffer : with 5 buffer pages File to sort : 108 pages   /  (last run is only 3 pages)    (last run is only 8 pages)   Pass 2: 2 sorted runs, 80 pages and 28 pages Pass 3: Sorted file of 108 pages • Total I/O costs: ?

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Example

  Buffer : with 5 buffer pages File to sort : 108 pages      /  (last run is only 3 pages)   (last run is only 8 pages) Pass 2: 2 sorted runs, 80 pages and 28 pages Pass 3: Sorted file of 108 pages • Total I/O costs: 2*N * (4 passes) 15

Number of Passes of External Sort

- gain of utilizing all available buffers - importance of a high fan-in during merging N 100 1,000 10,000 100,000 1,000,000 10,000,000 B=3 B=5 B=9 7 10 13 17 20 23 100,000,000 26 1,000,000,000 30 4 5 7 9 10 12 14 15 3 4 5 6 7 8 9 10 B=17 B=129 B=257 2 1 1 3 4 5 5 6 7 8 2 2 3 3 4 4 5 2 2 3 3 3 4 4 17

Optimizing External Sorting

 Cost metric ?

  I/O only (till now) CPU is nontrivial, worth reducing 18

Internal Algorithm : Heap Sort

  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

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Internal Sort Algorithm

12 4

. . .

2 8 10 3 5 INPUT CURRENT SET OUTPUT  1 input, 1 output, B-2 current set  Main idea: repeatedly pick tuple in current set with smallest k value that is still greater than largest k value in output buffer and append it to output buffer 20

Internal Sort Algorithm

12 4

. . .

2 8 10 3 5 INPUT CURRENT SET OUTPUT  Input & Output?

new input page is read in if it is consumed, output is written out when it is full  When terminate current run?

When all tuples in current set are smaller than largest tuple in output buffer.

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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?

B  Quicksort is faster, but ...

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Optimizing External Sorting

 Further optimization for external sorting.

  Blocked I/O Double buffering 23

I/O for External Merge Sort

  Thus far : do 1 I/O a page at a time But cost also includes real page read/write time.

 Reading a

block

of pages sequentially is cheaper!

 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 .

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I/O for External Merge sort

 Example buffer blocks = b pages set one buffer block for input, one buffer block for output merge |B-b/b| runs in each pass e.g., 10 buffer pages 9 runs at a time with one-page input and output buffer blocks 4 runs at a time with two-page input and output buffer block 25

Double Buffering – Overlap CPU and I/O

 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 .

Disk INPUT 1 INPUT 1' INPUT 2 INPUT 2' OUTPUT OUTPUT' b block size INPUT k INPUT k' B main memory buffers, k-way merge Disk

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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?

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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!

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Clustered B+ Tree Used for Sorting

  Cost: root to left-most leaf, then retrieve all leaf pages (Alternative 1) For Alternative 2, additional cost of retrieving data records: each page fetched just once.

Index (Directs search) Data Records Data Entries

("Sequence set") *

Always better than external sorting!

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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) Data Entries

("Sequence set")

Data Records

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External Sorting vs. Unclustered Index

N Sorting p=1 p=10 p=100 100 1,000 10,000 100,000 200 2,000 40,000 600,000 100 1,000 10,000 100,000 1,000 10,000 100,000 1,000,000 10,000 100,000 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.

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Summary

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

External merge sort minimizes disk I/O costs :  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.

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Summary, cont.

   Choice of internal sort algorithm may matter.

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.

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