The Firm Gap Property and Its Applications

Let S be a set of n points in Rd, and let t > 1 be a real number. A (directed) geometric graph G with vertex set S is called a (directed) t-spanner for S if for each two vertices p and q in G, there exists a (directed) path between p and q in G of length of at most t times Euclidean distance between p and q. In this paper, we introduce a property on the edges of a geometric graph, denoted by firm gap property, and then we prove that the weight of the edge set of any geometric graph satisfies this property is O(wt(MST(S)), where wt(MST(S)) is the weight of the minimum spanning tree of S. Moreover, we present a quadratic-time algorithm based on the firm gap property that constructs a directed t-spanner for S that is asymptomatically optimal in terms of its edge count, maximum degree and weight