Reducing the Steiner problem in four uniform orientations

We studied the Steiner tree problem in four uniform orientations where any line, half-line, or line segment must be on a line which makes an angle of (iπ)/4 with the positive x-axis, for some i ∈ {0,1,2,3}, and the distance between two points is measured as the length of the shortest polygonal path connecting them. We show that for any set P of n terminal points there exists a Steiner minimum tree interconnecting P such that all Steiner points are in script G sign^[2n/3]-1(P), the ([(2n)/3] - 1)^sl-generation grid points of P. Our result improves the previous best result which guarantees that for any set P of n terminal points there is a Steiner minimum tree in which all Steiner points are in script G sign^n-2(P).