Abstract
We consider the rendezvous problem for two robots on an arbitrary connected graph with n vertices and all its edges of length one. Two robots are initially located on two different vertices of the graph and can traverse its edges with different but constant speeds. The robots do not know their own speed. During their movement they are allowed to meet on either vertices or edges of the graph. Depending on certain conditions reflecting the knowledge of the robots we show that a rendezvous algorithm is always possible on a general connected graph.
More specifically, we give new rendezvous algorithms for two robots as follows. (1) In unknown topologies. We provide a polynomial time rendezvous algorithm based on universal exploration sequences, assuming that n is known to the robots. (2) In known topologies. In this case we prove the existence of more efficient rendezvous algorithms by considering the special case of the two-dimensional torus.
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Notes
- 1.
In fact they are (deterministically) constructible in polylogarithmic space, but to date it is unknown whether a universal traversal (or exploration) sequence of length \(O(n^3d^2 \log n)\) can be constructed in polynomial time.
- 2.
This algorithm builds on an idea proposed (without its analysis) by an unknown reviewer based on an algorithm appearing in an earlier version of this paper.
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Kranakis, E., Krizanc, D., Markou, E., Pagourtzis, A., RamÃrez, F. (2017). Different Speeds Suffice for Rendezvous of Two Agents on Arbitrary Graphs. In: Steffen, B., Baier, C., van den Brand, M., Eder, J., Hinchey, M., Margaria, T. (eds) SOFSEM 2017: Theory and Practice of Computer Science. SOFSEM 2017. Lecture Notes in Computer Science(), vol 10139. Springer, Cham. https://doi.org/10.1007/978-3-319-51963-0_7
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