Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let (S,d) be a finite metric space, where each element p∈S has a non-negative ight w (p).  study spanners for the set S with respect to the following ighted distance function: $$\mathbf{d}_{\omega}(p,q)=\left\{\begin{array}{ll}0&\mbox{ if $p=q$,}\\ \operatorname {w}(p)+\mathbf{d}(p,q)+ \operatorname {w}(q)&\mbox{ if $p\neq q$.}\end{array}\right.$$  present a general method for turning spanners with respect to the d-metric into spanners with respect to the d ω -metric. For any given e>0,  can apply our method to obtain (5+e)-spanners with a linear number of edges for three cases: points in Euclidean space ℝd , points in spaces of bounded doubling dimension, and points on the boundary of a convex body in ℝd where d is the geodesic distance function.#R##N##R##N# also describe an alternative method that leads to (2+e)-spanners for ighted point points in ℝd and for points on the boundary of a convex body in ℝd . The number of edges in these spanners is O(nlog n). This bound on the stretch factor is nearly optimal: in any finite metric space and for any e>0, it is possible to assign ights to the elements such that any non-complete graph has stretch factor larger than 2−e.
['Mohammad Ali Abam', 'Mark de Berg', 'Mohammad Farshi', 'Joachim Gudmundsson', 'Michiel H. M. Smid']
I am writing a research paper on a idea Geometric Spanners for Weighted Point Sets
[{"from": "human", "value": "I am writing a research paper on a idea Geometric Spanners for Weighted Point Sets"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let (S,d) be a finite metric space, where each element p\u2208S has a non-negative ight w\u2009(p).  study spanners for the set S with respect to the following ighted distance function: $$\\mathbf{d}_{\\omega}(p,q)=\\left\\{\\begin{array}{ll}0&\\mbox{ if $p=q$,}\\\\ \\operatorname {w}(p)+\\mathbf{d}(p,q)+ \\operatorname {w}(q)&\\mbox{ if $p\\neq q$.}\\end{array}\\right.$$  present a general method for turning spanners with respect to the d-metric into spanners with respect to the d \u03c9 -metric. For any given e>0,  can apply our method to obtain (5+e)-spanners with a linear number of edges for three cases: points in Euclidean space \u211dd , points in spaces of bounded doubling dimension, and points on the boundary of a convex body in \u211dd where d is the geodesic distance function.#R##N##R##N# also describe an alternative method that leads to (2+e)-spanners for ighted point points in \u211dd and for points on the boundary of a convex body in\u00a0\u211dd . The number of edges in these spanners is O(nlog\u2009n). This bound on the stretch factor is nearly optimal: in any finite metric space and for any e>0, it is possible to assign ights to the elements such that any non-complete graph has stretch factor larger than 2\u2212e."}]