Group Search on the Line.

In this paper we consider the group search problem, or evacuation problem, in which k mobile entities (MEs) located on the line perform search for a specific destination. The MEs are initially placed at the same origin on the line L and the target is located at an unknown distance d, either to the left or to the right from the origin. All MEs must simultaneously occupy the destination, and the goal is to minimize the time necessary for this to happen. The problem with k = 1 is known as the cow-path problem, and the time required for this problem is known to be 9d - o(d) in the worst case (when the cow moves at unit speed); it is also known that this is the case for k >= 1 unit-speed MEs. In this paper we present a clear argument for this claim by showing a rather counter-intuitive result. Namely, independent of the number of MEs, group search cannot be performed faster than in time 9d - o(d). We also examine the case of k = 2 MEs with different speeds, showing a surprising result that the bound of 9d can be achieved when one ME has unit speed, and the other ME moves with speed at least 1/3.