Abstract
We give lower bounds for various natural node- and edge-based local strategies for exploring a graph. We consider this problem both in the setting of an arbitrary graph as well as the abstraction of a geometric exploration of a space by a robot, both of which have been extensively studied. We consider local exploration policies that use time-of-last-visit or alternatively least-frequently-visited local greedy strategies to select the next step in the exploration path. Both of these strategies were previously considered by Cooper et al. (2011) for a scenario in which counters for the last visit or visit frequency are attached to the edges. In this work we consider the case in which the counters are associated with the nodes, which for the case of dual graphs of geometric spaces could be argued to be intuitively more natural and likely more efficient. Surprisingly, these alternate strategies give worst-case superpolynomial/exponential time for exploration, whereas the least-frequently-visited strategy for edges has a polynomially bounded exploration time, as shown by Cooper et al. (2011).
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Notes
- 1.
In general, this property holds whenever there are several neighboring nodes with the lowest time stamp. For example, in a star starting from the center we visit all neighboring nodes in arbitrary order until all of them have time stamp 1. At this point we can once again choose an arbitrary order to visit the neighbors anew.
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Akash, A.K., Fekete, S.P., Lee, S.K., López-Ortiz, A., Maftuleac, D., McLurkin, J. (2016). Lower Bounds for Graph Exploration Using Local Policies. In: Kaykobad, M., Petreschi, R. (eds) WALCOM: Algorithms and Computation. WALCOM 2016. Lecture Notes in Computer Science(), vol 9627. Springer, Cham. https://doi.org/10.1007/978-3-319-30139-6_5
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DOI: https://doi.org/10.1007/978-3-319-30139-6_5
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