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Tell me where I am so I can meet you sooner

Tell me where I am so I can meet you sooner. Andrew Collins 1 , Jurek Czyżowicz 2 , Leszek Gąsieniec 1 & Arnaud Labourel 3 1 University of Liverpool 2 Université du Québec en Outaouais 3 LaBRI , University Bordeaux. The Problem.

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Tell me where I am so I can meet you sooner

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  1. Tell me where I am so I can meet you sooner Andrew Collins1, Jurek Czyżowicz2, Leszek Gąsieniec1 & Arnaud Labourel3 1University of Liverpool 2Université du Québec en Outaouais 3LaBRI, University Bordeaux

  2. The Problem Two 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

  3. Consider... 2D Grid (0, 0)

  4. Consider... 2D Grid with 2 agents (x2, y2) d (x1, y1) (0, 0) d = ((x1- x2)2 + (y1 – y2)2)½

  5. Two agents attempt rendezvous (x1, y1) (x1, y1) (0, 0)

  6. 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.

  7. 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 tonor 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

  8. Now it’s trivial... zZzZz 264+1 264 zZzZz (0, 0) 264 264+1

  9. Perhaps Space-filling curves? • An infinite space-filling curve with fixed precision provides a route on which the agents can rendezvous

  10. 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.

  11. 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)

  12. Lower bound explained zZzZz d Ω(d2)

  13. Expanding Neighbourhoods

  14. Expanding Neighbourhoods

  15. Expanding Neighbourhoods

  16. Expanding Neighbourhoods

  17. Expanding Neighbourhoods A1 A1

  18. Expanding Neighbourhoods p The overlapping areas in consecutive layers induce an infinite tree-like structure A2 A3 A1 B B A6 A4 A5 A7 A9 A8 A1 A2 A3 A4 A5 A6 A7 A8 A9

  19. Formation of the route

  20. Ascending sequence of neighbourhoods with associated sequences Si(p) Si(p) is the area at layer i that contains point p S5(p) S4(p) S3(p) S2(p) S1(p) S0(p)

  21. The Rendezvous Algorithm Algorithm RV (point p in 2D-space) • i = 0; • repeat • Go along the route and: • visit the left end of Si(p); • visit the right end of Si(p); • go back to the location of p • i = i + 1; • until rendezvous is reached;

  22. Infinite quad tree y x

  23. Infinite Central Squares y x

  24. 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.

  25. 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

  26. Thank you!

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