Online Adversarial Coverage: Proofs
نویسنده
چکیده
Proof. We first prove that every free cell accessible from s becomes a frontier at least once. We prove this by induction on the Manhattan distance d of the cells from s. As the base case, consider d = 0. The only cell with d = 0 is s itself, and it is added to the frontier set F in line 3 of OnlineCoverage. We now assume correctness for d = k and prove for d = k + 1. Consider a cell c with distance d = k+ 1 from s. This cell belongs to the Von Neumann neighborhood of some cell c′ with distance k from s. By the induction hypothesis, c′ becomes a frontier at some point of the algorithm. Thus, at some point c′ is selected by ChooseNextFrontier and then the function UpdateFrontier is called with it (in line 12 of OnlineCoverage). UpdateFrontier iterates over all the neighbors of c′, including c, which can be in one of the following states:
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