The Firm Gap Property and Its Applications

‎Let $S$ be a set of $n$ points in $\mathbb{R}^d$‎, ‎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 {\it 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