Energy Consumption of Group Search on a Line

Abstract

Consider two robots that start at the origin of the infinite line in search of an exit at an unknown location on the line. The robots can collaborate in the search, but can only communicate if they arrive at the same location at exactly the same time, i.e. they use the so-called face-to-face communication model. The group search time is defined as the worst-case time as a function of d, the distance of the exit from the origin, when both robots can reach the exit. It has long been known that for a single robot traveling at unit speed, the search time is at least 9d - o(d); a simple doubling strategy achieves this time bound. It was shown recently in [Chrobak et al., 2015] that k >= 2 robots traveling at unit speed also require at least 9d group search time. 
We investigate energy-time trade-offs in group search by two robots, where the energy loss experienced by a robot traveling a distance x at constant speed s is given by s^2 x, as motivated by energy consumption models in physics and engineering. Specifically, we consider the problem of minimizing the total energy used by the robots, under the constraints that the search time is at most a multiple c of the distance d and the speed of the robots is bounded by b. Motivation for this study is that for the case when robots must complete the search in 9d time with maximum speed one (b=1; c=9), a single robot requires at least 9d energy, while for two robots, all previously proposed algorithms consume at least 28d/3 energy.
When the robots have bounded memory and can use only a constant number of fixed speeds, we generalize an algorithm described in [Baeza-Yates and Schott, 1995; Chrobak et al., 2015] to obtain a family of algorithms parametrized by pairs of b,c values that can solve the problem for the entire spectrum of these pairs for which the problem is solvable. In particular, for each such pair, we determine optimal (and in some cases nearly optimal) algorithms inducing the lowest possible energy consumption. 
We also propose a novel search algorithm that simultaneously achieves search time 9d and consumes energy 8.42588d. Our result shows that two robots can search on the line in optimal time 9d while consuming less total energy than a single robot within the same search time. Our algorithm uses robots that have unbounded memory, and a finite number of dynamically computed speeds. It can be generalized for any c, b with cb=9, and consumes energy 8.42588b^2d.