Wave Algorithm

Wave Algorithm

Why waves ?

■ Basic Bloc : Generic subroutine in the design of distributed algorithms

■ Examples of use :

- Broadcast, Routing, Communication Protocols

- Leader Election

- Resource Sharing, Mutual Exclusion

- Graph Algorithms for Search, Spanning Trees, etc

- Synchronization

- Wake-up

- Deadlock, Termination Detection

- Reset Protocols

- Resource Allocation

- Distributed Sorting, Ranking, etc.

Specification

A wave algorithm is a distributed algorithm that satisfies the following requirements:

■Termination : Each computation is finite

■Decision : Each computation contains at least one decide event

■Dependence : In each computation every decide event is causally preceded by an event in each process

Algorithms

■ Sequential Design

■ Token Ring

■ Depth-First Token Circulation

■ DFTC over Tree Networks

■ Parallel Design

■ Propagation of Information with Feedback

■ Wave Algorithm for Tree Networks

■ Another BFS Algorithm

Token Ring

Token Ring

Base of Protocol IEEE 802.5

Token Ring

var !"#$_% : port label //Process % is an initiator

upon external initialization do send <&'(> to !"#$_%;

done

upon receipt of <tok> from q do decide

done

//Process % is not an initiator

upon receipt of <tok> from q do send <&'(> to !"#$_%;

done

Decide

Token Ring

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Depth-First Token

Circulation

//Process % is an initiator

upon external initialization do Parent% = ⊤;

Next();

//For any Process %

upon receipt of <tok> from q If Parent% = ⊥

then Parent% = q;

fi

If L% = used% then Decide

else if q ≠ Current% ⋀ q ≠ Parent% then used% = used% ∪ ^{Parent_%^}

send <&'(> to q;

else Next();

fi fi

var Parent%, Current% : port label ∈ L% ∪ {⊥,⊤}, init ⊥; 


used% : set of port labels, init ∅;

DFTC

Procedure Next()

If ∃ q’ ∈ L% \ (used% ∪ ^{Parent_%^})

then Current% = q’ ∈ L% \ (used% ∪ ^{Parent_%^})

used% = used% ∪ ^{Current_%^}

send <&'(> to Current%; else used% = used% ∪ ^{Parent_%^}

send <&'(> to Parent%;

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DFTC

DFTC over trees

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We are given two distributed algorithms:

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One for constructing a tree in a general network

■

DFTC

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DFTC over trees

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DFTC over trees

Propagation of Information with

Feedback

Synchronous ^r

//Process % is an initiator

upon external initialization do Parent% = ⊤;

∀ q ∈ L% : send <&'(> to q;

var Parent% : port label ∈ L% ∪ {⊥,⊤}, init ⊥; 


used% : set of port labels, init ∅;

PIF

//For any Process %

upon receipt of <tok> from q used% = used% ∪ {q};

If Parent% = ⊥ then Parent% = q;

∀ q’ ∈ L% \ {q}: send <&'(> to q’;

fi

If used% = L%

then If Parent% = ⊤ then decide;

else send <&'(> to Parent%; fi

fi

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PIF

Waves over Trees

WA over Tree Networks

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Décide Décide

//Process % is a leaf/Initiator upon external initialization do

Let q ∈ L%;

send <&'(> to q;

//For any Process %

upon receipt of <tok> from q used% = used% ∪ {q};

If ∃ q ∈ L_p : q ∉ used% ⋀ used% ∪ {q} = L_p ⋀ ¬ Sent_p then send <&'(> to q;

Sent% = true;

fi

If used% = L_p then Decide;

fi var Sent% : boolean, init false; 


used% : set of port labels, init ∅;

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WA over Tree Networks

Another BFS Algorithm

var Parent% : port label ∈ L% ∪ {⊥}, init ⊥; 


Diff% : array[q ∈ L% ] of {-1,0,1}, init 0;

BFS

loop

If ∀ q ∈ L% : Diff% [q] ≥ 0 then for all q ∈ L% do

Diff% [q] = Diff% [q] - 1;

done

Parent% = q| Diff% [q] = -1;

∀ q ∈ L% : send <INCR> to q;

fi end loop

upon receipt of <INCR> from q Diff% [q] = Diff% [q] + 1;

Parent% = q| Diff% [q] = -1;

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BFS