# $\alpha$-Gap Greedy Spanner

Document Type : Research Paper

Authors

1 Department of Computer Engineering, Ferdowsi University of Mashhad

2 Department of Computer Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

Abstract

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.

Keywords