Optimal and Resilient Pheromone Utilization in Ant Foraging

Pheromones are a chemical substance produced and released by ants as means of communication. In this work we present the minimum amount of pheromones necessary and sufficient for a colony of ants (identical mobile agents) to deterministically find a food source (treasure), assuming that each ant has the computational capabilities of either a Finite State Machine (FSM) or a Turing Machine (TM). In addition, we provide pheromone-based foraging algorithms capable of handling fail-stop faults. In more detail, we consider the case where k identical ants, initially located at the center (nest) of an infinite two-dimensional grid and communicate only through pheromones, perform a collaborative search for an adversarially hidden treasure placed at an unknown distance D. We begin by proving a tight lower bound of Ω(D) on the amount of pheromones required by any number of FSM based ants to complete the search, and continue to reduce the lower bound to Ω(k) for the stronger ants modeled as TM. We provide algorithms which match the aforementioned lower bounds, and still terminate in optimal O(D + D/k) time, under both the synchronous and asynchronous models. Furthermore, we consider a more realistic setting, where an unknown number f < k of ants may fail-stop at any time; we provide fault-tolerant FSM algorithms (synchronous and asynchronous), that terminate in O(D+D/(k− f) +Df) rounds and emit no more than the same asymptotic minimum number of O(D) pheromones overall.