Optimal Pheromone Utilization

The minimal amount of pheromones necessary and sufficient to deterministically find a treasure (food) by a colony of ants, each modeled by either a Finite State Machine or a Turing Machine, is considered. The k mobile agents (ants), initially located at the origin (nest) of an infinite grid, communicate only through pheromones to perform a collaborative search for an adversarially hidden treasure at an unknown distance D. We begin by proving a tight lower bound of Ω(D) on the number of pheromones required by a FSM to even complete the search, and continue to reduce the lower bound to Ω(k) for the stronger ants modeled as TM. Matching optimal algorithms, in terms of run time as well as pheromone usage, are provided both for synchronous and asynchronous models in each case.