A Simple Algorithm for Stable Minimum Storage

Merging

Pok-Son KimKookmin University, Department of Mathematics,

Seoul 135-702, Korea

Arne KutznerSeokyeong University, Department of E-Business,

Seoul 136-704, Korea

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 2

Merging

• Make one sorted array out of two consecutive sorted arrays

4 91 3 92

3, 4 91, 92

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 3

Lower Bounds for Merging

• Number of comparisons

– Argumentation over the decision tree (see Knuth)

• Number of assignments

– Each element can change its position in the final sequence

)log(m

nm nm for

nm

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 4

Notions

• An algorithm merges two adjacent sequences with “minimum storage” when it needsbits additional space at most.

• Stability:Merging algorithm preserves the initial ordering of elements with equal value.

))((log2 nmO

We present .....

…a simple, asymptotically optimal (regarding

comparisons), stable, minimum storage merging algorithm

calledSPLITMERGE

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 6

SPLITMERGE: Step 1 - Splitting

• We compute boundaries in our inputs u and v so that we get the following property:

u1 u2 v1 v2

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 7

SPLITMERGE: Step 2 – Rotation

• We rotate u2v1 to v1u2

u1 u2 v1 v2

u1 u2v1 v2

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 8

SPLITMERGE: Step 3 – Recursive Application

• We have:1. u1 u2 and u1 v2

2. v1 u2 and v1 v2

u1 u2v1 v2

left recursion right recursion

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 9

Stability

• Lemma: SPLITMERGE is stable.

• Proof:

u1 u2 v1 v2

u1 u2v1 v2

>

rotation

!!!!

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 10

How do we split?

• Repeated halving of two search spaces until these are reduced to single points (the borders of a rotation).

• For asymmetric inputs one element of the smaller side “freezes” after some time

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 11

Symmetric Splitting – Graphical Description

u21 x u22 v21 y v22

search spaces

x y u22 v21

x y

u21 v22

x yx y

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 12

Splitting Example

98651 655201 8 80

98651 655201 8 80

98651 655201 8 80

98651 655201 8 80

?

?

?

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 13

Symmetric Splitting – Asymmetric Inputs (1)

xu1 u3 v1 v21 y v22 v3

single search space

• After some time one element “freezes” (here x)

• x can “belong” to u1, u3 or both

• x belongs to u1 means u1x v3

• x belongs to u3 means xu3 v1

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 14

Symmetric Splitting – Asymmetric Inputs (2)

xu1 u3 v1 v3

• Rotation – we distinguish two different cases

if x belongs to u3 :

xu1 u3v1 v3

otherwise :

xu1 u3v1 v3

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 15

Symmetric Splitting – Asymmetric Inputs (3)

• Recursion – we distinguish several cases

if x belongs to u1

xu1 u3v1 v3

otherwise

xu1 u3v1 v3

left Recursion right Recursion

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 16

Algorithm in Pseudo code

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 17

Some Remarks …• The “belongs to” information can be followed

from the variables l, r and m

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 18

Worst Case ComplexityLemma: The recursion depth is limited by

min , where

Corollary: Minimum Storage Algorithm

}1,loglog{ mnm

|||,| vnum

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 19

Worst Case Complexity (Comparisons)Special case m→ m/2 (1)

m10

n32m1

2 m22 m3

2 m42

m11 m2

1

n42n2

2n12

n11 n2

1

n10m

m/2 m/2

m/4 m/4 m/4 m/4

mk log1 1 1 1

recursion depth

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 20

Worst Case Complexity (Comparisons)Special case m→ m/2 (2)

m10

n32m1

2 m22 m3

2 m42

m11 m2

1

n42n2

2n12

n11 n2

1

n10 1)log( nm

1)log(1)log( 1

2

1

2

1

1

1

1 nmnm

4

1

2 )log(2j

ij

ij nm

0

1

2

i

j

ij

ij

i nm2

1)log(2

k

jk

j

k

j

k nm21 )log(2

i

k

recursion level

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 21

Worst Case Complexity (Comparisons)Special case m→ m/2 (3)

k

i j

ij

ij

ii

nm0

2

1))log(2(

Overall number of comparisons:

Mathematical Machinery

))1/log(( mnmO

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 22

Worst Case ComplexityHow to treat the other cases? (1)

Example: Maximum spanning case

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 23

Worst Case ComplexityHow to treat other cases? (2)

• Trick: Mapping into a corresponding“m → m/2 case” by the introduction of recursion groups– A recursion group consists of one or several

recursion levels – Grouping of recursion levels into

recursion groups so that each recursion group holds at most 2i-1 recursion levels.

– Altogether k+1 recursion groups

),2,1,0( kii

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 24

Worst Case ComplexityHow to treat other cases? (3)

• Mapping for the maximum spanning case:

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 25

Assignments / Worst Case Complexity

• Theorem:The SPLITMERGE algorithm needs assignments.

Sketch of Proof:At most k+1 recursion groups with O(m+n) many assignments for every group

)log)(( mnmO

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 26

Benchmarks

• n,m : lengths of input sequences (m=n)• i : number of different input elements• te : execution time in ms• #comp : number of comparisons

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 27

Summary

• Simply structured algorithm using a novel binary partition technique

• Doesn’t rely on any third algorithm as subpart

• Short definition in Pseudo Code• Asymptotically optimal regarding the

number of comparisons• Nice Textbook algorithm ?

SOFSEM 2007 A Simple Algorithm for Stable Minimum Storage Merging 28

Thank you very much foryour attention