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

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

CONSIDER... 2D GRID

(0, 0)

CONSIDER... 2D GRID WITH 2 AGENTS

d

(0, 0)(x1, y1)

(x2, y2)

d = ((x1- x2)2 + (y1 – y2)2)½

TWO AGENTS ATTEMPT RENDEZVOUS

(0, 0)

(x1, y1)

(x1, y1)

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.

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

NOW IT’S TRIVIAL...

(0, 0)

zZzZz

264+1264

264

264+1

zZzZz

PERHAPS SPACE-FILLING CURVES? An infinite space-filling curve with fixed

precision provides a route on which the agents can rendezvous

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.

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)

LOWER BOUND EXPLAINED

Ω(d2)

d

zZzZz

EXPANDING NEIGHBOURHOODS

EXPANDING NEIGHBOURHOODS

EXPANDING NEIGHBOURHOODS

EXPANDING NEIGHBOURHOODS

EXPANDING NEIGHBOURHOODS

A1

A1

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

FORMATION OF THE ROUTE

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

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;

INFINITE QUAD TREE

x

y

INFINITE CENTRAL SQUARES

x

y

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.

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

THANK YOU!