Improved Algorithms for the Point-Set Embeddability problem for Plane 3-Trees

In the point set embeddability problem, we are given a plane graph $G$ with $n$ vertices and a point set $S$ with $n$ points. Now the goal is to answer the question whether there exists a straight-line drawing of $G$ such that each vertex is represented as a distinct point of $S$ as well as to provide an embedding if one does exist. Recently, in \cite{DBLP:conf/gd/NishatMR10}, a complete characterization for this problem on a special class of graphs known as the plane 3-trees was presented along with an efficient algorithm to solve the problem. In this paper, we use the same characterization to devise an improved algorithm for the same problem. Much of the efficiency we achieve comes from clever uses of the triangular range search technique. We also study a generalized version of the problem and present improved algorithms for this version of the problem as well