On minimum and maximum spanning trees of linearly moving points

In this paper we investigate the upper bounds on the numbers of transitions of minimum and maximum spanning trees (MinST and MaxST for short) for linearly moving points. Here, a transition means a change on the combinatorial structure of the spanning trees. Suppose that we are given a set of n points in d-dimensional space, S={p ₁, p ₂, ... 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 t. We investigate the numbers of transitions of MinST and MaxST when t increases from-∞ to +∞. We assume that the dimension d is a fixed constant. Since there are O(n ²) distances among n points, there are naively O(n ⁴) transitions of MinST and MaxST. We improve these trivial upper bounds for L ₁ and L _∞ distance metrics. Let k _p (n) (resp.[Figure not available: see fulltext.]) be the number of maximum possible transitions of MinST (resp. MaxST) in L _p metric for n linearly moving points. We give the following results in this paper: κ₁(n)=O(n ^5/2 α(n)), κ _∞(n)=O(n ^5/2 α(n)),[Figure not available: see fulltext.], and[Figure not available: see fulltext.] where α(n) is the inverse Ackermann's function. We also investigate two restricted cases, i.e., the c-oriented case in which there are only c distinct velocity vectors for moving n points, and the case in which only k points move.