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13. External Sorting Motivation 2-way External Sort: Memory, passes,cost General External Sort: Memory, passes, cost Optimizations

Snowplow Double Buffering Forecasting Using a B+ tree index

Bucket Sort Intergalactic Standard Reference

Graefe, Implementing Sorting in Database Systems http://portal.acm.org/citation.cfm?id=1132964

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Learning Objectives

Derive formula for cost of external merge sort

Derive amount of memory needed to sort a file in 2 passes, using merge or bucket sort

Describe algorithm for generating longer initial runs and identify its best and worst cases

Describe forecasting and why it is useful Identify when indexes should be used for

sorting Identify the pros and cons of external bucket

vs merge sort.

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Why sort?

A classic problem in computer science! Exercises many software and hardware features

Data is often 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 some query processing algoritms (Chapter 14)

Problem: sort 1Gb of data with 1Mb of RAM. why not virtual memory?

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Sort algorithms? If the data can fit in memory, which sort

algorithm is best? But most DBMS files will not fit in available memory

If the data is larger than memory, try the same alg. Suppose

• for this data, your sort alg. requires 220 random memory accesses

• Memory access takes 1 microsec, disk takes 10 millisecs.

How much time is required to do your sort algorithm’s memory accesses?

• If there is enough memory to hold the data?• If the data is four times the size of memory?

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External Sorts

Definition: When data is larger than memory. An aside: What is “Memory”?

Physical memory? The DBMS is not the only player We concluded that most in-memory sort algs

won’t be effective for external sorting. What sort algorithms are best for external sort?

Sort-based Hash-based

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2-Way External Merge Sort: Memory?

Pass 0: Read a page, sort it, write it. How many buffer pages needed?

Pass 1, 2, 3, …, etc.: How many buffer pages needed?

Main memory buffers

INPUT 1

INPUT 2

OUTPUT

Result ofPass k+1

Result ofPass k

13. Sorting

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2-Way External Merge Sort: Passes?

Assume file is N pages Run = sorted subfile What happens in pass Zero?

How many runs are produced? What is the cost in I/Os?

What happens in pass 1? What happens in pass i? How many passes are required? What is the total cost?

04/18/23 8PSU’s CS 587

Two-Way External Merge Sort: Cost

Each pass we read + write each page in file.

N pages in the file => the number of passes

So total 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

13. Sorting

I/Os

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General External Merge Sort

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

runs of B pages each are produced? Cost?

Pass 1,2, …, etc.: merge B-1 runs. • How many runs are created after pass i? Cost

of pass i?• How many passes? Total Cost?

B Main memory buffers

INPUT 1

INPUT B-1

OUTPUT

Disk

INPUT 2

. . .

Disk

. .

.. . .

Suppose we have more than 3 buffer pages.

13. Sorting

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

file Number of passes is (1 + log4 108/5 ) =

4 Cost is 2(108)*4

Pass 0: Output is = 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/

13. Sorting

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How much memory is needed to sort a file in one or two passes?

N = number of data pages, B = memory pages available

To sort in One pass: N B To Sort in Two passes: 1+logB-1N/B 2

N/B (B-1)1

Approximating B-1 by B, this yields N B For example, if pages are 4KBytes, a 4GByte

file can be sorted in Two Passes with ? buffers.

04/18/23 12PSU’s CS 587

Sorting in 2 passes: graphical proof

File is B pages wide Each run is B pages

File is x pages high Merge x runs in pass

1

xB since we must merge x runs in B pages of memory

So N=xB BB or NB

The File, N pages

Each run, 1xB pages

B

x

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Memory required for 2-pass sort

N = # Pages in File[File size in Bytes]

B = # Buffer Pages [ #Bytes]to sort in Two passes

2**10 [ 4Meg ] 2**5 [128K]

2**20 [4 Gig] 2**10 [ 4 Meg]

2**26 [256 Gig ] 2**13 [32 Meg]

Assuming page size of 4K

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Can we always sort in 1 or 2 passes?

Assume only one query is active, and there is at least 1 gig of physical RAM.

Yes: DB2,Oracle, SQLServer, MySQL They allocate all available memory to queries Tricky to manage memory allocation as

queries need more memory during execution No: Postgres

Memory allocated to each query, for sort and other purposes, is fixed by a config parameter.

Sort memory is typically a few meg, in case there may be many queries executing.

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Extremes of SortingOne Pass: N =B, 1-pass sort, cost = N

N BOriginal Data

Sorted Data

Quick-sort

Two Pass: N = B2, 2-pass sort, cost = 3N

N

Original Data

B

Quicksort into B runs, length B

12..B

B

B

B

Runs

B

Merge

Sorted Data

N

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Extreme Problems

Most sorting of large data falls between these two extremes

If we apply the intergalactic standard sort-merge algorithm, in every textbook, the cost for any dataset with B<N<B2 will be 3M.

Must we always pay that large price? Might there be an algorithm that is a

hybrid of the two extremes?

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Hybrid Sort when N 3B The key idea of hybrid sort is don’t waste memory. Here is an example of the hybrid sort algorithm

when N is approximately 3B.

One+ Pass: M 3B, 2-pass sort, cost = 3M – 2B

Original Data

B

Quicksort into runs, length B, leaving the last run in memory

B B

Runs

B

Merge the runs on disk with the run in memory

Sorted Data

N

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Hybrid Sort in general Let k = N/B Arrange R so the last run is B-k pages Cost is N + 2(N – (B-k)) = 3N -2B + 2(N/B) = 3N – 2(

B-N/B) When N=B, cost is N+1. When N=B2, cost is 3N

Original Data

B

Quicksort into runs, length B, leaving the last run in memory

B B

Runs

B

Merge the runs on disk with the run in memory

Sorted Data

N12..K-1

12..K

B-k pages

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13.3.1 Maximizing initial runs

Defn: making initial runs as long as possible.

Why is it helpful to maximize initial runs?

If initial run size is doubled, what is the time savings?

How can you maximize initial runs? What algorithm is best?

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Replacement Selection Initialize empty priority queues CUR,

NEXT Read B buffer pages of data into CUR Do

Pop record s with smallest key from CUR to current run

// key of s is now highest key in current run If key of next input r >= key of s

• //Can put in current run• insert r into CUR

else• insert r into NEXT

If CUR is empty• interchange NEXT and CUR and start a new run

Until (input is empty)

13. Sorting

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More on Replacement Selection

Cf. Knuth, vol 3 [442], page 255. Theorem: average length of a run in

replacement sort is 2B 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 ... 2B

13. Sorting

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How can we prove the Theorem?

Begin with some modeling assumptions Data to be sorted are real numbers between 0 and

1 Data appear at a uniform rate and distribution A snowplow picks up one datum as one falls Picking up a datum == pop( ) off the queue Each datum is infinitesimal Each run begins when the plow passes zero

13. Sorting

B

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0 1

B

0 1

B

0 1

B

0 1

B

0 1

B

0 1

B

CUR NEXT

13. Sorting

2B 2B

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Snowplow: Conclusion

The figures on the previous page show that At any time after run 0, the amount of snow

= size of memory = B. After the first run, the volume of snow

removed in one circuit is 2B. Cf. Larson and Graefe [471]

In spite of memory management problems, the snowplow optimization is very effective.

04/18/23 25PSU’s CS 587

13.4 I/O for External Merge Sort

What else can we do to improve performance? We have assumed I/O is done a page at a time Text suggests reading a block of pages

sequentially. Pass 0: No problem Pass 1,2,…: lowers fanin Sometimes a win

13. Sorting

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External Sort’s jerky behavior

Recall that each input is one page What happens after the last record on a

page is output?

B Main memory buffers

INPUT 1

INPUT B-1

OUTPUT

Disk

INPUT 2

. . .

Disk

. .

.. . .

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

complete, can prefetch into `shadow block’. This could increase the number of passes In practice, most files still sorted in 1-2

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

13. Sorting

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Forecasting

Cf. Knuth, vol 3, pg 324-7 Double Buffering requires

Doubling memory What a huge waste! Most shadow buffers lie

idle, unused, wasted. How can we forecast

which shadow buffers will be needed first?

Forecasting can achieve performance of double buffering with little memory

351434

1334555

4679

50657483

56575859

88919399

A B C

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Sorting Records!

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

www.research.microsoft.com/barc/SortBenchmark

13. Sorting

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

idea!

13. Sorting

04/18/23 31PSU’s CS 587

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")

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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!

(Directs search)

Data Records

Index

Data Entries("Sequence set")

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Bucket Sort Suppose search key values are 0-K B pages in memory, N pages in the file Pass 0: Partition the file into B-1 intervals

Inervals are not runs! If the interval fits in one page, sort it

B Main memory buffers

OUTPUT 1

OUTPUT B-1

Disk

OUTPUT 2

. . .

Disk

. .

.

[0,K/(B-1))

[K/(B-1),2K/(B-1))

[(B-2)K/(B-1),K)

INPUT

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Bucket sort cost What happens after pass 0

? intervals, each ? long After pass 1?

? intervals, each ? long How much memory is required to sort in

two passes? Each interval is at most one page, or ? Same as for external merge sort

04/18/23 35PSU’s CS 587

External Merge Sort vs External Bucket Sort

Approximately the same I/O cost Same memory requirement for two passes Same number of passes required to sort

Bucket sort has less CPU cost Bucketizing is much cheaper than

sorting/merging But bucket sort is subject to skew Thus merge sort is used in practice