Time-Energy Tradeoffs for Evacuation by Two Robots in the Wireless Model

Abstract

Two robots stand at the origin of the infinite line and are tasked with searching collaboratively for an exit at an unknown location on the line. They can travel at maximum speed b and can change speed or direction at any time. The two robots can communicate with each other at any distance and at any time. The task is completed when the last robot arrives at the exit and evacuates. We study time-energy tradeoffs for the above evacuation problem. The evacuation time is the time it takes the last robot to reach the exit. The energy it takes for a robot to travel a distance x at speed s is measured as xs 2 . The total and makespan evacuation energies are respectively the sum and maximum of the energy consumption of the two robots while executing the evacuation algorithm.Assuming that the maximum speed is b, and the evacuation time is at most cd, where d is the distance of the exit from the origin, we study the problem of minimizing the total energy consumption of the robots. We prove that the problem is solvable only for bc ≥ 3. For the case bc = 3, we give an optimal algorithm, and give upper bounds on the energy for the case bc > 3.We also consider the problem of minimizing the evacuation time when the available energy is bounded by ∆. Surprisingly, when ∆ is a constant, independent of the distance d of the exit from the origin, we prove that evacuation is possible in time O(d 3/2 log d), and this is optimal up to a logarithmic factor. When ∆ is linear in d, we give upper bounds on the evacuation time.

Evacuating the ℓp Disc

“…Searchers in other problems suffer a cost penalty for every turn that is made, or instead require the searcher to accelerate to its maximum velocity after completing a turn [27]. Even the cost function itself varies in some problems where the cost represents energy expenditure instead of elapsed search time [20], [21]. Another interesting variation is where mobile agents are faulty and provide unreliable readings [9].…”

Evacuating the ℓp Disc

“…Searchers in other problems suffer a cost penalty for every turn that is made, or instead require the searcher to accelerate to its maximum velocity after completing a turn [27]. Even the cost function itself varies in some problems where the cost represents energy expenditure instead of elapsed search time [20], [21]. Another interesting variation is where mobile agents are faulty and provide unreliable readings [9].…”

A Multi-Objective Optimization Problem on Evacuating 2 Robots from the Disk in the Face-to-Face Model; Trade-Offs between Worst-Case and Average-Case Analysis

“…Since the inception of 2 EVAC F2F in [1], numerous search and evacuation-type variants have emerged that studied different robot specs and/or number of searchers, different communication models, and different domains. Some notable examples include search and/or evacuation in the disk with more than 1 exits [29,30], in triangles [31,32], on multiple rays [33], in graphs [34,35], on a line with at least two robots [36,37] (generalizing the seminal result of [38]), with faulty robots [39][40][41] or with probabilistically faulty robots [42], with advice (information) [43], with priority specification on the searchers [44,45], with immobile agents [46,47], with time/energy trade-off requirements [48,49], with speed bounds [50], and with terrain dependent speeds [51], just to name some. The interested reader may also see the recent survey [52] that elaborates more on some selected topics.…”

“…-To the best of our knowledge, the current paper is the first attempt to study multi-objective optimization search-type problems. It was followed by [48,49] who considered time energy trade-offs for a search problems on the line. This line of research admits many future directions based on any combination of multiple objectives, e.g., worst-case, average-case and competitive cost, time, energy and any other efficiency measure, or even trade-offs involving number of faults or even complexity resources, e.g., memory, communication or randomness.…”