A combinatorial theorem on labeling squares with points and its application

Abstract

In this paper, we present a combinatorial theorem on labeling disjoint axis-parallel squares of edge length two using points. Given an arbitrary set of disjoint axis-parallel squares of edge length two, we show that if we label points on the boundary of all squares (one for each square) and define a distance label graph such that there is an edge between any two labeling points if and only if their L∞-distance is at most 1 − e (0 < e < 1), then the maximum connected component of the graph contains Θ(1/e) vertices, which is tight. With this theorem we present a new and simple factor-(3 + e) approximation for labeling points with axis-parallel squares under the slider model.