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Lower bounds for graph exploration using local policies. (English) Zbl 1358.05269

Summary: 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 C. Cooper et al. [Distrib. Comput. 24, No. 2, 91–99 (2011; Zbl 1231.68178)] 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. [loc. cit.].

MSC:

05C85 Graph algorithms (graph-theoretic aspects)
05C81 Random walks on graphs
05C99 Graph theory

Citations:

Zbl 1231.68178
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