On the optimal consensus of crab submarines in one dimension

We consider the problem of computing the optimal meeting point of a set of N crab submarines. First, we analyze the case where the submarines are allowed any position on the real line: we provide a constructive proof of optimality and we use it to provide a linear-time algorithm to find the optimal meeting point. Second, we use the results for the continuous case to solve the case where the crab submarines are restricted to integer locations: we show that, given the solution of the corresponding continuous problem, we can find the optimal integer solution in linear time.