alpha-Gap Greedy Spanner

In this paper, we have introduced a new geometric spanner called $alpha$-Gap greedy spanner, which is a parametric approximation of the well-known Gap-greedy spanner. We will show theoretically and experimentally that this spanner is similar to the Gap-greedy spanner in terms of qualitative features such as weight and maximum degree of vertices. %Wehave shown that this spanner can be computed in $O(n^2 log n)$ time with$O(n)$ space, and $O(n log n)$ expected time on the set of points placedrandomly in a unit square.Two algorithms have been proposed with running time $O(n^2 log n)$ for constructing the $alpha$-Gap greedy spanner. Space complexity of the first algorithm is $O(n^2)$, but it is reduced to $O(n)$ in the second one. %The proposed algorithms have a parameter, called $alpha$, by which the similarity of the $alpha$-Gap greedy spanner to the Gap-greedy spanner, in terms of quality features mentioned above, can be determined. Also, we have shown on the points placed randomly in a unit square, the $alpha$-Gap greedy spanner can be constructed in the expected $O(n log n)$ time.