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Stupid Columnsort Tricks

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... slabpose columnsort, the size of the dirty area is 2s3/2 entering the last ... With the divisibility restriction, the dirty rows after the transpose step have ... – PowerPoint PPT presentation

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Title: Stupid Columnsort Tricks


1
Stupid Columnsort Tricks
  • Geeta Chaudhry
  • Tom Cormen
  • Dartmouth College
  • Department of Computer Science

2
What Do We Know About Columnsort?
  • Sorts N values on an r s mesh
  • Uses 8 steps
  • Each step either sorts each column or performs a
    fixed permutation
  • Divisibility restriction s divides r
  • Height restriction r 2s2
  • 4s3/2
  • Exponent of s goes from 2 to 3/2
  • Mesh need not be quite so tall and skinny
  • Cost 2 additional steps
  • Can simultaneously remove the divisibility
    restriction and relax the height restriction tor
    6s3/2

3
Why Relax the Conditions?
  • Columnsort applies in more circumstances
  • Our motivation out-of-core sorting
  • Column height r is limited by amount of memory
  • Either per processor or in entire system
  • N rs, r 2s2 N r3/2/21/2
  • N rs, r 4s3/2 N r5/3/42/3
  • Reducing the exponent of s in the bound for r
    allows us to sort more values with a given amount
    of memory
  • A similar technique works for applying columnsort
    to in-core sorting

4
This Talk
  • Slabpose columnsort
  • r 4s3/2
  • Requires divisibility restriction
  • Also in the paper
  • Subblock columnsort
  • r 4s3/2 with divisibility restriction
  • r 6s3/2 without divisibility restriction
  • Proof that the divisibility restriction is
    unnecessary in the basic columnsort algorithm

5
Columnsort Steps
  • Sort each column
  • Transpose entire mesh
  • Sort each column
  • Untranspose entire mesh
  • Sort each column
  • Shift down by half a column
  • Sort each column
  • Shift up by half a column

6
Slabpose Columnsort Steps
  • Sort each column
  • Slabpose transpose within vertical slabs
  • Sort each column
  • Shuffle columns
  • Slabpose
  • Sort each column
  • Untranspose entire mesh
  • Sort each column
  • Shift down by half a column
  • Sort each column
  • Shift up by half a column
  • Sort each column
  • Slabpose transpose within vertical slabs
  • Sort each column
  • Shuffle columns
  • Slabpose
  • Sort each column
  • Untranspose entire mesh
  • Sort each column
  • Shift down by half a column
  • Sort each column
  • Shift up by half a column
  • Sort each column
  • Slabpose transpose within vertical slabs
  • Sort each column
  • Shuffle columns
  • Slabpose
  • Sort each column
  • Untranspose entire mesh
  • Sort each column
  • Shift down by half a column
  • Sort each column
  • Shift up by half a column

Oblivious!
7
Slabpose Columnsort Steps
  • Sort each column
  • Slabpose transpose within vertical slabs
  • Sort each column
  • Shuffle columns slabpose
  • Sort each column
  • Untranspose entire mesh
  • Sort each column
  • Shift down by half a column
  • Sort each column
  • Shift up by half a column

Oblivious!
8
Why Work With Vertical Slabs?
  • In regular columnsort, the matrix needs to be
    tall and skinny
  • Working with vertical slabs allows us to change
    the aspect ratio to use tall and skinny slabs
  • Well use slabs that are s columns wide
  • The mesh will have s slabs

9
0-1 Principle
  • If an oblivious algorithm sorts all input sets
    consisting solely of 0s and 1s, then it sorts all
    input sets with arbitrary values
  • Use the 0-1 Principle by looking at portions of
    the r s mesh
  • Clean all 0s or all 1s
  • Dirty may be mixed 0s and 1s

10
Step 1 Sort Each Column
0
dirty
r
1
s
11
Step 2 Slabpose
s-slab
column
s
s slabs
12
Step 3 Sort Each Column
s rows
13
Step 4 Shuffle
s-slab
s-slab
s rows
s slabs
s slabs
14
Step 5 Slabpose
s-slab
s-slab
r/ s rows
2 rows
s slabs
s slabs
s sets of dirty rows
15
Step 6 Sort Each Column
2 s rows 2s3/2 elements
16
Step 7 Untranspose Entire Mesh
2s3/2 elements
r 4s3/2 2s3/2 r/2 dirty area half
a column
Once the size of the dirty area is at most half a
column, the last four steps will finish up
17
Step 8 Sort Each Column
dirty area resides in one column done
18
Step 8 Sort Each Column
dirty area resides in two columns no change
19
Step 9 Shift Down by Half a Column
dirty area resides in one column
20
Step 10 Sort Each Column
dirty area resides in one column
21
Step 11 Shift Up by Half a Column
sorted
22
Subblock Columnsort
  • Adds two steps to columnsort
  • Sort each column
  • A fixed permutation
  • The permutation is any one that distributes all
    elements of each s s subblock to alls
    columns
  • Like slabpose columnsort, the size of the dirty
    area is 2s3/2 entering the last four steps
  • As long as 2s3/2 r/2 (half a column), the last
    four steps complete the sorting

23
Removing the Divisibility Restrictionfrom
Columnsort
  • With the divisibility restriction, the dirty rows
    after the transpose step have only 0-1
    transitions
  • Without the divisibility restriction, there may
    also be 1-0 transitions
  • The proof shows that even with the 1-0
    transitions, the size of the dirty area entering
    the last four steps does not increase
  • Thus r 2s2 suffices, even without the
    divisibility restriction

24
Conclusion
  • We can get around the restrictions of columnsort
  • Reduce the exponent in the height restriction
    from 2 to 3/2
  • The mesh need not be quite so tall and skinny
  • Cost Two extra steps
  • In out-of-core implementation, slabpose
    columnsort requires no additional I/O
  • The divisibility restriction is unnecessary
  • Open question Can we reduce the exponent further
    within the columnsort framework?
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