Optimal Exploration of Small Rings
Stéphane Devismes
VERIMAG UMR 5104
Univ. Joseph Fourier
Grenoble, France
Talk by
Franck Petit, Univ. Pierre et Marie Curie - Paris 6, France
Context
o
A team of k “weak” robots evolving into a
ring of n nodes
o
o
o
o
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Autonomous
Anonymous
Oblivious
Disoriented
: No central authority
: Undistinguishable
: No mean to know the past
: No mean to agree on a common
direction or orientation
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Context
o
A team of k “weak” robots evolving into a
ring of n nodes
o
o
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Atomicity : In every configuration, each robot is
located at exactly one node
: In every configuration, each node may
Weak
Multiplicity contain some robots
(a robot cannot detect the exact
number of robots located at each node
but it is able to detect if there are zero,
one, or more)
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Context
o
A team of k “weak” robots evolving into a
ring of n nodes
o
SSM
o
The k’ activated robots execute the cycle:
1.
2.
3.
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: In every configuration, k’ robots are
activated (0 < k’ ≤ k)
Look
: Instantaneous snapshot with
multiplicity
detection
on this
observation, decides
Compute: Based
to either
stay its
idledestination
or move to one of
Move
: Move
toward
the neighboring nodes
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Problem: Exploration
Starting from a configuration where no two
robots are located at the same node:
o Exploration:
Each node must be visited by at least one robot
o Termination:
Eventually, every robot stays idle
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Performance: Number of robots (k<n)
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Related works (Deterministic)
o
Tree networks
Ω(n) robots are necessary in general
A deterministic algorithm with O(log n/log log n)
robots, assuming that Δ ≤ 3
[Flocchini, Ilcinkas, Pelc, Santoro, SIROCCO 08]
o
Ring networks
Θ(log n) robots are necessary and sufficient,
provided that n and k are coprime
A deterministic algorithm for k ≥ 17
[Flocchini, Ilcinkas, Pelc, Santoro, OPODIS 07]
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Related works (Probabilistic)
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Ring networks [Devismes, Petit, Tixeuil, SIROCCO 2010]
 4 robots are necessary
 For ring of size n>8, 4 robots are sufficient to
solve the problem
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Contribution
Question.
Are 4 probabilistic robots necessary and sufficient to explore any
ring of any size n ?
Remark.
• The problem is not defined for n < 4
• For n = 4, no algorithm required
Contribution.
• Algorithm for 5 ≤ n ≤ 8
• Corollary: 4 probabilistic robots are necessary and
sufficient to explore any ring of any size n
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Definitions
Tower.
A node with at least two robots.
k≥2
F. Petit – SIROCCO 2009
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Definitions
Segment.
A maximal non-empty elementary path of occupied nodes.
A 1-segment
F. Petit – SIROCCO 2009
a 2-segment
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Definitions
Hole.
A maximal non-empty elementary path of free nodes.
F. Petit – SIROCCO 2009
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Definitions
Arrow. A 1-segment, followed by a hole, a tower, and a
1-segment.
Head
Tail
1 arrow
F. Petit – SIROCCO 2009
of length 2
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Definitions
Arrow. A 1-segment, followed by a hole, a tower, and a
1-segment.
Primary arrow
F. Petit – SIROCCO 2009
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Definitions
Arrow. A 1-segment, followed by a hole, a tower, and a
1-segment.
final arrow
F. Petit – SIROCCO 2009
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Algorithm: Overview
o
3 main steps:

Phase I: Initial configuration 4-segment


Phase II: 4-segment primary arrow


Invariant: 4-segment or primary arrow
Phase III: Primary arrow final arrow

o
Invariant: no arrow
Invariant: increasing arrow
(2 special cases)
Let start with phase II and III, it’s easier …
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Algorithm: Phase II
o
Phase II: 4-segment primary arrow

Invariant: 4-segment or primary arrow
Probabilistic moves
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Algorithm: Phase II
o
Phase II: 4-segment primary arrow

Invariant: 4-segment or primary arrow
Primary arrow
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Algorithm: Phase III
o
Phase III: Primary arrow final arrow

Invariant: increasing arrow
Deterministic move
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Algorithm: Phase III
o
Phase III: Primary arrow final arrow

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Invariant: increasing arrow
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Algorithm: Phase III
o
Phase III: Primary arrow final arrow

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Invariant: increasing arrow
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Algorithm: Phase III
o
Phase III: Primary arrow final arrow

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Invariant: increasing arrow
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Algorithm: Phase III
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Phase III: Primary arrow final arrow

Invariant: increasing arrow
Termination
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Algorithm: Back to Phase I
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Phase I: Initial configuration 4-segment

o
Principle:


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Invariant: no arrow
No symmetry: Deterministic moves
Symmetry: Probabilistic or deterministic moves
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Phase I: no symmetry
o
There exists a unique largest segment S:

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move toward S following the shortest neighboring hole
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Phase I: no symmetry
o
There exists a unique largest segment S:

move toward S following the shortest neighboring hole
Ambiguity:
Decision taken
by an adversary
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Phase I: no symmetry
o
There exists a unique largest segment S:

move toward S following the shortest neighboring hole
Ambiguity:
Decision taken
by an adversary
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Phase I: no symmetry
o
There exists a unique largest segment S:

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move toward S following the shortest neighboring hole
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Phase I: symmetry
Case by Case Study
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Phase I: n = 5
o
No symmetry


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Initial configuration: a 4-segment
Phase I & II
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Phase I: n = 6
o
Only one symmetry is initially possible
The 2 special
cases
Stop
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Stop
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Phase I: n = 7
o
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Only one symmetry is initially possible
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Phase I: n = 8
o
Three symmetries are initially possible:
(a)
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(b)
(c)
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Phase I: n = 8, Case (a)
Case (c)
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Phase I: n = 8, Case (b)
Case (c)
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Phase I: n = 8, Case (c)
(c)
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Really complex!!!
See the paper…
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Conclusion
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General Result:

o
Future works:



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4 probabilistic robots are necessary and sufficient to
solve the exploration of any anonymous ring
Convergence time (experimental result:O(n) moves)
Full asynchronous model
Other (regular) topologies
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Conclusion
Thank you.
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