The minimum firing time of the generalized firing squad synchronization problem for squares

Abstract

For almost all variations of the firing squad synchronization problem for elementary geometric figures such as lines, rings, squares, rectangles, and cubes, minimal-time solutions are known. However, in 2012 Umeo and Kubo introduced a very simple variation of this type and pointed out that its minimal-time solutions are unknown. In that variation, a problem instance is a square array of n columns and n rows and the position of the general is arbitrary. For this variation they constructed a solution that fires at time 2n−2 for any position of the general and wrote that it is not known whether this solution is minimal-time or not. We determine the exact value of the minimum firing time of this variation. For some problem instances this value is smaller than 2n−2 and hence the 2n−2 time solution is not minimal-time. Our result does not solve the problem of existence or non-existence of minimal-time solutions of the variation. However the result gives one necessary condition for solutions to be minimal-time.