On the optimal consensus of crab submarines in one dimension

TL;DR

This paper investigates the problem of finding the optimal meeting point for crab submarines on a line, providing efficient algorithms for both continuous and integer-restricted scenarios, with proofs of optimality and linear-time solutions.

Contribution

It introduces a linear-time algorithm for the continuous case and extends it to the integer case, with constructive proofs of optimality.

Findings

Linear-time algorithm for continuous case

Extension to integer locations with linear-time solution

Constructive proof of optimality for both cases

Abstract

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 for the case of sorted starting points. 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.