On minimum and maximum spanning trees of linearly moving points

The authors investigate the upper bounds on the numbers of transitions of minimum and maximum spanning trees (MinST and MaxST for short) for linearly moving points. Suppose that one is given a set of n points in general d-dimensional space, S={p₁,p₂, . . ., p_n}, and that all points move along different straight lines at different but fixed speeds, i.e., the position of p_i is a linear function of a real parameter. They investigate the numbers of transitions of MinST and MaxST when t increases from -∞ to +∞. They assume that the dimension d is a fixed constant. Since there are O(n²) distances among n points, there are naively O(n⁴) transitions of MinST and MaxST. They improve these trivial upper bounds for L₁ and L_∞ distance metrics. Let c_p(n, min) (resp. c_p(n, max)) be the number of maximum possible transitions of MinST (resp. MaxST) in L_p metric for n linearly moving points. They give the following results; c₁(n, min)=O(n^5/2a(n)), c_∞ (n, min)=O(n^5/2a(n)), c₁(n, max)=O(nⁿ ) and c_∞(n, max)=O(n²) where O(n) is the inverse Ackermann function. They also investigate two restricted cases