Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let $P \subset \mathbb{R}^2$ be a planar $n$-point set such that each point $p \in P$ has an associated radius $r_p > 0$. The transmission graph $G$ for $P$ is the directed graph with vertex set $P$ such that for any $p, q \in P$, there is an edge from $p$ to $q$ if and only if $d(p, q) \leq r_p$. #R##N#Let $t > 1$ be a constant. A $t$-spanner for $G$ is a subgraph $H \subseteq G$ with vertex set $P$ so that for any two vertices $p,q \in P$,  have $d_H(p, q) \leq t d_G(p, q)$, where $d_H$ and $d_G$ denote the shortest path distance in $H$ and $G$, respectively (with Euclidean edge lengths).  show how to compute a $t$-spanner for $G$ with $O(n)$ edges in $O(n (\log n + \log \Psi))$ time, where $\Psi$ is the ratio of the largest and smallest radius of a point in $P$. Using more advanced data structures,  obtain a construction that runs in $O(n \log^6 n)$ time, independent of $\Psi$. #R##N# give two applications for our spanners. First,  show how to use our spanner to find a BFS tree from any given start vertex in $O(n \log n)$ time (in addition to the time it takes to build the spanner). Second,  show how to use our spanner to extend a reachability oracle to ansr geometric reachability queries. In a geometric reachability query  ask whether a vertex $p$ in $G$ can "reach" a target $q$ which is an arbitrary point the plane (rather than restricted to be another vertex $q$ of $G$ in a standard reachability query). Our spanner allows the reachability oracle to ansr geometric reachability queries with an additive overhead of $O(\log n\log \Psi)$ to the query time and $O(n \log \Psi)$ to the space.
['Haim Kaplan', 'Wolfgang Mulzer', 'Liam Roditty', 'Paul Seiferth']
I am writing a research paper on a idea Spanners for Directed Transmission Graphs
[{"from": "human", "value": "I am writing a research paper on a idea Spanners for Directed Transmission Graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let $P \\subset \\mathbb{R}^2$ be a planar $n$-point set such that each point $p \\in P$ has an associated radius $r_p > 0$. The transmission graph $G$ for $P$ is the directed graph with vertex set $P$ such that for any $p, q \\in P$, there is an edge from $p$ to $q$ if and only if $d(p, q) \\leq r_p$. #R##N#Let $t > 1$ be a constant. A $t$-spanner for $G$ is a subgraph $H \\subseteq G$ with vertex set $P$ so that for any two vertices $p,q \\in P$,  have $d_H(p, q) \\leq t d_G(p, q)$, where $d_H$ and $d_G$ denote the shortest path distance in $H$ and $G$, respectively (with Euclidean edge lengths).  show how to compute a $t$-spanner for $G$ with $O(n)$ edges in $O(n (\\log n + \\log \\Psi))$ time, where $\\Psi$ is the ratio of the largest and smallest radius of a point in $P$. Using more advanced data structures,  obtain a construction that runs in $O(n \\log^6 n)$ time, independent of $\\Psi$. #R##N# give two applications for our spanners. First,  show how to use our spanner to find a BFS tree from any given start vertex in $O(n \\log n)$ time (in addition to the time it takes to build the spanner). Second,  show how to use our spanner to extend a reachability oracle to ansr geometric reachability queries. In a geometric reachability query  ask whether a vertex $p$ in $G$ can \"reach\" a target $q$ which is an arbitrary point the plane (rather than restricted to be another vertex $q$ of $G$ in a standard reachability query). Our spanner allows the reachability oracle to ansr geometric reachability queries with an additive overhead of $O(\\log n\\log \\Psi)$ to the query time and $O(n \\log \\Psi)$ to the space."}]