Connecting the Maximum Number of Nodes in the Grid to the Boundary with Nonintersecting Line Segments

Abstract

Given a finite setSof nodes in a rectangular grid, we consider the problem of finding the maximum size subset ofSsuch that the nodes in the subset can be connected to the boundary of the grid by means of nonintersecting line segments parallel to the grid axes. The work is motivated from the VLSI/WSI array processor technology, and in particular, the single-track switch model for configurable array processors (S. Y. Kinget al.,Fault-tolerant array processors using single-track switches,IEEE Trans. Comput.38(1989), 501–514). The problem has been investigated by Bruck and Roychowdhury, who described an algorithm to find the maximum number of compatible connections ofngiven nodes in the grid inO(n3) time andO(n2) space (J. Bruck and V. P. Roychowdhury, How to play bowling in parallel on the grid,J. Algorithms12(1991), 516–529). In this paper, we improve their result by describing anO(n2logn) time andO(n2) space algorithm; instrumental in this improvement is the introduction of a new type of priority search trees which is of interest in its own right. Finally, we extend the algorithm to handle the additional constraint thatnear-missesare disallowed; this is the first algorithm to resolve this case, and, like the general algorithm, it runs inO(n2logn) time and requiresO(n2) space.