Sequential metric dimension for random graphs
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In the localization game, the goal is to find a fixed but unknown target node v^ with the least number of distance queries possible. In the j^th step of the game, the player queries a single node v_j and receives the distance between the nodes v_j and v^. The sequential metric dimension (SMD) is the minimal number of queries that the player needs to guess the target with absolute certainty, no matter where the target is. The term SMD originates from the related notion of metric dimension (MD), which can be defined the same way as the SMD, except that the player must choose all queries before any of the distances are revealed. In this work, we extend the results of arXiv:1208.3801 on the MD of Erdős-Rényi graphs to the SMD. We find that, in fully connected Erdős-Rényi graphs, the MD and the SMD are a constant factor apart. For the lower bound we present a clean analysis by combining tools developed for the MD and a novel coupling argument. For the upper bound we show that a strategy that greedily minimizes the number of candidate targets in each step uses asymptotically optimal queries in Erdős-Rényi graphs. Connections with source localization, binary search on graphs and the birthday paradox are discussed.
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