Brief Announcement: Sorting on Skip Chains

Ajoy K. Datta, Lawrence L. Larmore, and Stéphane Devismes

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Skip Chain

October, 11, 2011

Left Right

Major nodes

Relay nodes

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Skip Chain Sorting

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3 6 2

2 3 6

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Contribution

• Skip Chain Sorting Algorithm– Self-stabilizing– Silent– Locally shared memory model

• Unfair demon• O(b) space, b = number of bits to encode a value• O(md) rounds

– m : number of major nodes– d : maximum number of relay between two major nodes– md = O(n) if the spacing between major processes is roughtly

equal

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Overview

• Idea : distributed bubble sort

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Arbitrary Configurations

Normal Configurations

Legitimate Configurations

Error Correction

Sorting

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Data Structure

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3 26

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Swap

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26

6 6 6 266

2 2 2 622

Synchronization between swaps : 4 colors

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Colors

• A value moves to the left at the crest of wave 0

• A value moves to the right at the crest of wave 1

• Colors 2 and 3 to avoid ambiguïty and to synchronize

• Color E: error color

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Example

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X Y z

33 03 20

Compare and swap

V(y)V(x)

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Example

October, 11, 2011

X Y z

33 10 20

V(y)’V(x)’

0, V(x)’ 1, V(y)’

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Example

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X Y z

03 10 21

Compare and swap

V(y)’V(x)’ V(y)’V(u) V(x)’

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Example

October, 11, 2011

X Y z

10 10 32

Compare and swap

V(y)’V(x)’V(x)’’ V(y)’V(u)’

21, V(x)’’

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Example

October, 11, 2011

X Y z

10 21 32

V(y)’V(x)’’V(x)’’ V(y)’V(u)’

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Example

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X Y z

10 32 32

V(y)’V(x)’’V(x)’’ V(y)’V(u)’

2 3

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Example

October, 11, 2011

X Y z

20 32 33

V(y)’V(x)’’V(x)’’ V(y)’V(u)’

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Example

October, 11, 2011

X Y z

21 32 03

V(y)’V(x)’’V(x)’’ V(y)’ V(z)V(u)’

Compare and swap

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Example

October, 11, 2011

X Y z

32 32 10

V(y)’V(x)’’V(x)’’ V(y)’’ V(z)’V(u)’

0, V(y)’’ 3

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Example

October, 11, 2011

X Y z

32 03 10

V(y)’’V(x)’’V(x)’’ V(y)’’ V(z)’V(u)’

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Example

October, 11, 2011

X Y z

33 03 20

V(y)’’V(x)’’V(x)’’ V(y)’’ V(z)’V(u)’

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Error correction

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Silence

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2 2 3 3 3 3 62 32

Done

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Perspective

• Can we enhance the round complexity to O(n) rounds ?

• Step complexity ?

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Thank youOctober, 11, 2011

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Min-Max Search Tree

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10

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95

84 7

62

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