stéphane devismes verimag umr 5104 univ. joseph fourier grenoble, france optimal exploration of...
TRANSCRIPT
Stéphane DevismesVERIMAG UMR 5104Univ. Joseph Fourier
Grenoble, France
Optimal Exploration of Small Rings
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
2WRAS 2010
o Autonomous
: No central authorityo Anonymous: Undistinguishable o Oblivious : No mean to know the past o Disoriented : No mean to agree on a
common direction or orientation
Context
o A team of k “weak” robots evolving into a ring of n nodes
3WRAS 2010
o Atomicity
: In every configuration, each robot is located at exactly one node o Weak
Multiplicity
: In every configuration, each node may 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)
Context
o A team of k “weak” robots evolving into a ring of n nodes
4WRAS 2010
o SSM : In every configuration, k’ robots are activated (0 < k’ ≤ k)
1. Look : Instantaneous snapshot with multiplicity detection
o The k’ activated robots execute the cycle:
2. Compute
: Based on this observation, decides to either stay idle or move to one of the neighboring nodes
3. Move : Move toward its destination
Problem: Exploration
o Exploration:Each node must be visited by at least one robot
o Termination:Eventually, every robot stays idle
5WRAS 2010
o Performance: Number of robots (k<n)
Starting from a configuration where no two robots are located at the same node:
Related works (Deterministic)
o Tree networks Ω(n) robots are necessary in generalA 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 coprimeA deterministic algorithm for k ≥ 17[Flocchini, Ilcinkas, Pelc, Santoro, OPODIS 07] 6WRAS 2010
Related works (Probabilistic)
o 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
7WRAS 2010
Contribution
8WRAS 2010
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
Definitions
9F. Petit – SIROCCO 2009
Tower. A node with at least two robots.
k ≥ 2
Definitions
10F. Petit – SIROCCO 2009
Segment. A maximal non-empty elementary path of occupied nodes.
A 1-segment a 2-segment
Definitions
11F. Petit – SIROCCO 2009
Hole. A maximal non-empty elementary path of free nodes.
1 hole of length 3
A 1-hole
Definitions
12F. Petit – SIROCCO 2009
Arrow. A 1-segment, followed by a hole, a tower, and a 1-segment.
1 arrow
Head
of length 2
Tail
Definitions
13F. Petit – SIROCCO 2009
Arrow. A 1-segment, followed by a hole, a tower, and a 1-segment.
Primary arrow
Definitions
14F. Petit – SIROCCO 2009
Arrow. A 1-segment, followed by a hole, a tower, and a 1-segment.
final arrow
WRAS 2010
Algorithm: Overview
o 3 main steps: Phase I: Initial configuration 4-segment
Invariant: no arrow Phase II: 4-segment primary arrow
Invariant: 4-segment or primary arrow Phase III: Primary arrow final arrow
Invariant: increasing arrow
o (2 special cases)
15
Let start with phase II and III, it’s easier …
WRAS 2010
Algorithm: Phase II
o Phase II: 4-segment primary arrow Invariant: 4-segment or primary arrow
16
Probabilistic moves
WRAS 2010
Algorithm: Phase II
o Phase II: 4-segment primary arrow Invariant: 4-segment or primary arrow
17
Primary arrow
WRAS 2010
Algorithm: Phase III
o Phase III: Primary arrow final arrow Invariant: increasing arrow
18
Deterministic move
WRAS 2010
Algorithm: Phase III
o Phase III: Primary arrow final arrow Invariant: increasing arrow
19
WRAS 2010
Algorithm: Phase III
o Phase III: Primary arrow final arrow Invariant: increasing arrow
20
WRAS 2010
Algorithm: Phase III
o Phase III: Primary arrow final arrow Invariant: increasing arrow
21
WRAS 2010
Algorithm: Phase III
o Phase III: Primary arrow final arrow Invariant: increasing arrow
22
Termination
WRAS 2010
Algorithm: Back to Phase I
o Phase I: Initial configuration 4-segment Invariant: no arrow
o Principle: No symmetry: Deterministic moves Symmetry: Probabilistic or deterministic
moves
23
WRAS 2010
Phase I: no symmetry
o There exists a unique largest segment S: move toward S following the shortest
neighboring hole
24
WRAS 2010
Phase I: no symmetry
o There exists a unique largest segment S: move toward S following the shortest
neighboring hole
25
Ambiguity: Decision taken by an adversary
WRAS 2010
Phase I: no symmetry
o There exists a unique largest segment S: move toward S following the shortest
neighboring hole
26
Ambiguity: Decision taken by an adversary
WRAS 2010
Phase I: no symmetry
o There exists a unique largest segment S: move toward S following the shortest
neighboring hole
27
WRAS 2010
Phase I: symmetry
28
Case by Case Study
WRAS 2010
Phase I: n = 5
o No symmetry Initial configuration: a 4-segment Phase I & II
29
WRAS 2010
Phase I: n = 6
o Only one symmetry is initially possible
30
StopStop
The 2 special cases
WRAS 2010
Phase I: n = 7
o Only one symmetry is initially possible
31
WRAS 2010
Phase I: n = 8
o Three symmetries are initially possible:
32
(a) (c)(b)
WRAS 2010
Phase I: n = 8, Case (a)
33
Case (c)
WRAS 2010
Phase I: n = 8, Case (b)
34
Case (c)
WRAS 2010
Phase I: n = 8, Case (c)
35
(c)
o Really complex!!!
o See the paper…
WRAS 2010
Conclusion
o General Result: 4 probabilistic robots are necessary and
sufficient to solve the exploration of any anonymous ring
o Future works: Convergence time (experimental
result:O(n) moves) Full asynchronous model Other (regular) topologies
36
Conclusion
37WRAS 2010
Thank you.