Optimal Patrolling of High Priority Segments While Visiting the Unit Interval with a Set of Mobile Robots
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Consider a region that requires to be protected from unauthorized penetrations. The border of the region, modeled as a unit line segment, consists of high priority segments that require the highest level of protection separated by low priority segments that require to be visited infinitely often. We study the problem of patrolling the border with a set of k robots. The goal is to obtain a strategy that minimizes the maximum idle time (the time that a point is left unattended) of the high priority points while visiting the low priority points infinitely often. We use the concept of single lid cover (segments of fixed length) where each high priority point is covered with at least one lid, and then we extend it to strong double-lid cover where each high priority point is covered with at least two lids, and the unit line segment is fully covered. Let λ_k-1 be the minimum lid length that accepts a single λ_k-1-lid cover with k-1 lids and Λ_2k be the minimum lid length that accepts a strong double Λ_2k-lid cover with 2k lids. We show that 2min(Λ_2k, λ_k-1) is the lower bound of the idle time when the max speed of the robots is one. To compute Λ_2k and λ_k-1, we present an algorithm with time complexity O(max(k, n)logn) where n is the number of high priority sections. For the upper bound, first we present a strategy with idle time λ_k-1 where one robot covers the unit line, and the remaining robots cover the lids of a single λ_k-1-lid cover with k-1 lids. Then, we present a simple strategy with idle time 3Λ_2k that splits the unit line into not-disjoint k segments of equal length that robots synchronously cover. Then, we present a complex strategy that split the unit line into k non-disjoint segments that robots asynchronously cover.
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