Randomised Distributed MIS and Colouring
Algorithms for√
Rings with Oriented Edges in
O log n bit rounds
Yves Métivier, John Michael Robson, Akka Zemmari
LaBRI - Université de Bordeaux
GT AlgoDist - LaBRI
30/05/2016
Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Content
Context
Model
Contribution
Maximal Independent Set
From the MIS Algorithm to the Colouring Algorithm
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Content
Context
Model
Contribution
Maximal Independent Set
From the MIS Algorithm to the Colouring Algorithm
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Rings
Rings
v2
Case study for many problems [AW04]
v3
v1
v4
“rings are a convenient structure for message-passing systems
and correspond to physical communication systems, for
example, token rings.”
v5
Classical Problems
vertex colouring
maximal independent set
vertex cover
maximal matching
graph decomposition
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
The Problems
Maximal Independent Set (MIS)
v1
v4
v5
v2
v3
v4
v5
v2
v3
v6
Colouring
v1
Y. Métivier, J.M. Robson, A. Zemmari
v6
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Content
Context
Model
Contribution
Maximal Independent Set
From the MIS Algorithm to the Colouring Algorithm
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
The Network
Simple Oriented Ring Graph
v1
Channel
v3
Processor
v2
Model [Tel00]
Message passing communication
FIFO channels
Reliable system
Synchronous system
Y. Métivier, J.M. Robson, A. Zemmari
Hypotheses
Anonymous network
No knowledge about the size of the
ring, position or distance information
Processors distinguish channels
Edges are (arbitrary) oriented.
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
The Network
Simple Oriented Ring Graph
v1
Channel
v3
Processor
v2
Model [Tel00]
Message passing communication
FIFO channels
Reliable system
Synchronous system
Y. Métivier, J.M. Robson, A. Zemmari
Hypotheses
Anonymous network
No knowledge about the size of the
ring, position or distance information
Processors distinguish channels
Edges are (arbitrary) oriented.
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
The Network
Simple Oriented Ring Graph
v1
Channel
v3
Processor
v2
Model [Tel00]
Message passing communication
FIFO channels
Reliable system
Synchronous system
Y. Métivier, J.M. Robson, A. Zemmari
Hypotheses
Anonymous network
No knowledge about the size of the
ring, position or distance information
Processors distinguish channels
Edges are (arbitrary) oriented.
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Distributed Probabilistic Algorithm
Theorem [Pel00]
There is no deterministic distributed algorithm for anonymous ring graphs for solving the
MIS problem or the maximal matching problem assuming all vertices wake up
simultaneously.
Definitions
Probabilistic algorithm : random choices.
Distributed probabilistic algorithm : collection of local probabilistic algorithms.
Las Vegas Algorithm : probabilistic algorithm which terminates with a positive
probability and which produces a correct result
Remark
Anonymous ring ⇒ Identical local probabilistic algorithms
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Complexity
Time Complexity [Pel00]
Round for each processor :
1
2
3
Send messages to (some) the neighbours,
Receive messages from (some) the neighbours,
Perform some local computations.
Time complexity : maximum possible number of rounds needed until every node has
completed its computation.
Bit complexity[KOSS06]
Bit round : round such that each process can send/receive at most 1 bit to/from each
neighbour.
Bit complexity for A : number of bit rounds to complete A.
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Content
Context
Model
Contribution
Maximal Independent Set
From the MIS Algorithm to the Colouring Algorithm
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Contribution
Contribution
Las Vegas distributed algorithms which compute a MIS or colouring for anonymous
arbitrary oriented rings.
√
Bit complexity and time complexity in O( log n) with high probability (1 − o(n−1 )).
Optimal algorithms ([FMRZ]).
Related Works
A. Israeli and A. Itai. A fast and simple randomized parallel algorithm for maximal
matching. Inf. Process. Lett. 1986.
K. √
Kothapalli, M. Onus, C. Scheideler, and C. Schindelhauer. Distributed coloring in
Õ( log n) bit rounds. IPDPS 2006.
Y. Métivier, J.-M. Robson, N. Saheb-Djahromi, and A. Zemmari. An optimal bit
complexity randomized distributed mis algorithm. Distributed Computing 2011.
A. Fontaine, Y. Métivier, J.-M. Robson, and A. Zemmari. Optimal bit complexity
randomized distributed MIS and maximal matching algorithms for anonymous rings.
Information and Computation 2013.
A. Fontaine, Y. Métivier, J.-M. Robson, and A. Zemmari. On lower bounds for the
time and the bit complexity of some probabilistic distributed graph algorithms.
SOFSEM 2014.
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Contribution
Contribution
Las Vegas distributed algorithms which compute a MIS or colouring for anonymous
arbitrary oriented rings.
√
Bit complexity and time complexity in O( log n) with high probability (1 − o(n−1 )).
Optimal algorithms ([FMRZ]).
Related Works
A. Israeli and A. Itai. A fast and simple randomized parallel algorithm for maximal
matching. Inf. Process. Lett. 1986.
K. √
Kothapalli, M. Onus, C. Scheideler, and C. Schindelhauer. Distributed coloring in
Õ( log n) bit rounds. IPDPS 2006.
Y. Métivier, J.-M. Robson, N. Saheb-Djahromi, and A. Zemmari. An optimal bit
complexity randomized distributed mis algorithm. Distributed Computing 2011.
A. Fontaine, Y. Métivier, J.-M. Robson, and A. Zemmari. Optimal bit complexity
randomized distributed MIS and maximal matching algorithms for anonymous rings.
Information and Computation 2013.
A. Fontaine, Y. Métivier, J.-M. Robson, and A. Zemmari. On lower bounds for the
time and the bit complexity of some probabilistic distributed graph algorithms.
SOFSEM 2014.
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
12 / 26
Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Contribution
Contribution
Las Vegas distributed algorithms which compute a MIS or a colouring for anonymous
arbitrary oriented rings.
√
Bit complexity and time complexity in O( log n) with high probability (1 − o(n−1 )).
Optimal algorithms ([FMRZ]).
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
13 / 26
Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Content
Context
Model
Contribution
Maximal Independent Set
From the MIS Algorithm to the Colouring Algorithm
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Algorithm RingMIS
Local State for a vertex v : η(v)
1 : in the MIS (final state)
0 : not in the MIS (final state)
? : undetermined state
Algorithm
Initial State : every vertex is in state ?.
Sequence of phases DEF denoted (DEF )∗ :
◮
◮
◮
Drawing (D),
Expansion (E),
Filling (F ).
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Algorithm RingMIS
Drawing
?v1
?v2
?v3
0
1
0
Remarks
v1 , v2 and v3 are in a final state with probability at least (1/3)3 .
No possible ambiguity.
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Algorithm RingMIS
Drawing - Final area
?
?
0
1
0
?
0
1
...
1
0
?
Remarks
The final area can be described by the following regular expression : 01((0 + 00)1)∗ 0.
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Algorithm RingMIS
Expansion
0
1
0
?
?
?
?
0
1
0
1
0
−
−
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Algorithm RingMIS
Filling
...
1
0
?
0
1 ...
...
1
0
1
0
1 ...
0 1 ...
F1
...
1
0
?
?
...
1
0
1
0
F2
...
1
0
?
?
?
0
1 ...
1 ...
...
1
0
0
1
0
0
1 ...
0
F3
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Algorithm RingMIS
Remarks
Invariant I :
At the start of each stage, the ring is divided as :
?
?
0
1
0
?
0
1
...
1
0
?
until each vertex is in final state.
Once a final area is formed, the number of vertices in final state is increased, at each
phase, by at least 4.
The algorithm is available only for graphs with at least 3 vertices.
For fewer than 3 vertices one can apply an alternative O(1) algorithm.
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Analysis
Lemma
Let v be a vertex that enters the MIS or its complement at time t0 > 0.
Then, for any t ≥ 0, after t phases, all the vertices at distance at most 2t ≥ 0 from v are in
the MIS or in its complement.
Theorem
Algorithm RingMIS computes a MIS in a ring of size n w.h.p. in O
√
log n rounds.
Corollary
Algorithm RingMIS computes a MIS in a ring of size n on average in O
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
√
log n rounds.
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Content
Context
Model
Contribution
Maximal Independent Set
From the MIS Algorithm to the Colouring Algorithm
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Algorithm RingColouring
MIS → Colouring
1
0
0
1
1
2
0
1
Theorem
A 3−colouring
in a ring of size n having edges with an orientation can be obtained w.h.p. in
√
log n rounds.
O
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Conclusion & Open Questions
1
If IDs are allowed to the nodes (or knowledge of an upper-bound of the size of the ring
is given) how significantly will such hypotheses help the design of MIS/clouring
algorithms ?
2
Can we apply the same schema to design algorithms :
◮
◮
3
to solve the same problems in other classes of graphs ?
to solve other problems ?
Can we do the same in other models, e.g., the beeping model ?
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Bibliography I
Y. Métivier, J.-M. Robson, and A. Zemmari.
Randomised Distributed MIS and Colouring Algorithms for Rings with Oriented Edges in O
rounds.
Information and Computation, to appear.
√
log n bit
H. Attiya and J. Welch.
Distributed Computing.
Wiley, 2004.
D. Peleg.
Distributed computing - A Locality-sensitive approach.
SIAM Monographs on discrete mathematics and applications, 2000.
K. Kothapalli, M. Onus, C. Scheideler, and C. Schindelhauer.
√
Distributed coloring in O( log n) bit rounds.
In 20th International Parallel and Distributed Processing Symposium (IPDPS 2006), Proceedings, 25-29
April 2006, Rhodes Island, Greece. IEEE, 2006.
A. Fontaine, Y. Métivier, J.-M. Robson, and A. Zemmari.
On lower bounds for the time and the bit complexity of some probabilistic distributed graph algorithms.
In 40th International Conference on Current Trends in Theory and Practice of Computer Science,
Slovakia (2014).
Y. Métivier, J.-M. Robson, N. Saheb-Djahromi, and A. Zemmari.
An optimal bit complexity randomized distributed mis algorithm.
Distributed Computing, 23(5-6) :331–340, 2011.
A. Israeli and A. Itai.
A fast and simple randomized parallel algorithm for maximal matching.
Inf. Process. Lett., 22(2) :77–80, 1986.
G. Tel.
Introduction to distributed algorithms.
Cambridge University Press, 2000.
Y. Métivier, J.M. Robson, A. Zemmari
MIS, Colouring, Anonymous Rings
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Context Model Contribution Maximal Independent Set From the MIS Algorithm to the Colouring Algorithm Bibliog
Thank You.
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MIS, Colouring, Anonymous Rings
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