Tell me where I am so I can meet you sooner
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Transcript of Tell me where I am so I can meet you sooner
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TELL ME WHERE I AM SO I CAN MEET YOU SOONER
Andrew Collins1, Jurek Czyżowicz2, Leszek Gąsieniec1 & Arnaud Labourel3
1University of Liverpool2Université du Québec en Outaouais3LaBRI, University Bordeaux
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THE PROBLEMTwo mobile agents are aware of their own location and are
required to meet locally in an asynchronous manner
Definitions: Network: undirected graph, G = (V, E) (infinite 2D grids) Mobile Agents: entities traversing the vertices of V via the
edges of E Rendezvous: agents are allowed to meet on a vertex or an edge Cost: length of the agent trajectories until rendezvous
Related topics: Rendezvous Problem Graph Exploration Search Games Space-filling curves, traversal sequences
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CONSIDER... 2D GRID
(0, 0)
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CONSIDER... 2D GRID WITH 2 AGENTS
d
(0, 0)(x1, y1)
(x2, y2)
d = ((x1- x2)2 + (y1 – y2)2)½
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TWO AGENTS ATTEMPT RENDEZVOUS
(0, 0)
(x1, y1)
(x1, y1)
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BACKGROUND Finite/countable graphs
Labelled agents can always rendezvous in a finite graphs as well as in any connected countable infinite graph. [1]
2D Euclidean space Asynchronous rendezvous is unfeasible for
agents starting at arbitrary positions in the plane, unless the agents have an > 0 visibility range. [1]
[1] J. Czyzowicz, A. Pelc, and A. Labourel, How to meet asynchronously (almost) everywhere, In Proc. SODA 2010, 22-30.
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THE MODEL The Network:
An infinite 2D grid Each agent knows its own location (x, y) in the
grid, however it is neither aware of the distance d to nor the location of the other agent
The agents do not share a common knowledge of time, i.e., the rendezvous is performed asynchronously
The Goal: Agents are expected to meet locally with a cost
proportional to (polynomial in) d
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NOW IT’S TRIVIAL...
(0, 0)
zZzZz
264+1264
264
264+1
zZzZz
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PERHAPS SPACE-FILLING CURVES? An infinite space-filling curve with fixed
precision provides a route on which the agents can rendezvous
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THE RENDEZVOUS ROUTE COULD BE LONG Gotsman and Lindenbaum pointed out in [2]
that space-filling curves fail in preserving the locality in the worst case. They show that for any space-filling curve there will always be some close points in 2D-space that are arbitrarily far apart on the space-filling curve.
[2] C. Gotsman and M. Lindenbaum, On the metric properties of discrete space-filling curves, IEEE Transactions on Image Processing 5(5), 794-797, 1996.
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MAYBE THIRD TIME LUCKY... So we can rendezvous eventually however
at a possibly huge (unjustified) cost
Can we design a method that will lead to a more efficient rendezvous which will guide the agents to stay local?
More importantly, can we find a solution as close as possible to the lower bound of Ω(d2)
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LOWER BOUND EXPLAINED
Ω(d2)
d
zZzZz
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EXPANDING NEIGHBOURHOODS
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EXPANDING NEIGHBOURHOODS
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EXPANDING NEIGHBOURHOODS
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EXPANDING NEIGHBOURHOODS
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EXPANDING NEIGHBOURHOODS
A1
A1
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EXPANDING NEIGHBOURHOODS
p
BBA1
A2 A3
A4 A5A6
A7 A8A9 A1 A2 A3 A4 A5 A6 A7 A8 A9
The overlapping areas in consecutive layers induce an infinite tree-likestructure
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FORMATION OF THE ROUTE
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ASCENDING SEQUENCE OF NEIGHBOURHOODS WITH ASSOCIATED SEQUENCES Si(P)
S0(p)
S1(p)
S2(p)
S3(p)
S4(p)
S5(p) Si(p) is the area at layer i that contains point p
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THE RENDEZVOUS ALGORITHMAlgorithm RV (point p in 2D-space)1. i = 0;2. repeat3. Go along the route and:
a) visit the left end of Si(p);b) visit the right end of Si(p);c) go back to the location of p
4. i = i + 1;5. until rendezvous is reached;
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INFINITE QUAD TREE
x
y
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INFINITE CENTRAL SQUARES
x
y
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RESULTS Alternating sequence of central squares and
infinite quad tree trimmed appropriately leads to the cost O(d2+ε), for any constant ε > 0. [3]
[3] A. Collins, J. Czyżowicz, L. Gąsieniec & A Labourel. ICALP ’10.
Surprisingly a properly trimmed structure with central squares suffice leading to O(d2· log7 d). [4]
[4] F.Bampas, J. Czyżowicz, L. Gąsieniec, D. Ilcinkas & A Labourel. DISC ‘10.
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FURTHER RESEARCH Construction of more cost-efficient covering
sequences o(d2· log7 d)?
Lower bound on the length of a covering sequences connecting agents at distance d
Ω(d2· log d)?
Local asynchronous rendezvous in other types of graphs
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THANK YOU!