Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let \(S \subset \mathbb {R}^2\) be a set of n sites. The unit disk graph \({{\mathrm{UD}}}(S)\) on S has vertex set S and an edge beten two distinct sites \(s,t \in S\) if and only if s and t have Euclidean distance \(|st| \le 1\). A routing scheme R for \({{\mathrm{UD}}}(S)\) assigns to each site \(s \in S\) a label\(\ell (s)\) and a routing table\(\rho (s)\). For any two sites \(s, t \in S\), the scheme R must be able to route a packet from s to t in the following way: given a current siter (initially, \(r = s\)), a headerh (initially empty), and the label\(\ell (t)\) of the target, the scheme R consults the routing table \(\rho (r)\) to compute a neighbor \(r'\) of r, a new header \(h'\), and the label \(\ell (t')\) of an intermediate target \(t'\). (The label of the original target may be stored at the header \(h'\).) The packet is then routed to \(r'\), and the procedure is repeated until the packet reaches t. The resulting sequence of sites is called the routing path. The stretch of R is the maximum ratio of the (Euclidean) length of the routing path produced by R and the shortest path in \({{\mathrm{UD}}}(S)\), over all pairs of distinct sites in S. For any given \(\varepsilon > 0\),  show how to construct a routing scheme for \({{\mathrm{UD}}}(S)\) with stretch \(1+\varepsilon \) using labels of \(O(\log n)\) bits and routing tables of \(O(\varepsilon ^{-5}\log ^2 n \log ^2 D)\) bits, where D is the (Euclidean) diameter of \({{\mathrm{UD}}}(S)\). The header size is \(O(\log n \log D)\) bits.
['Haim Kaplan', 'Wolfgang Mulzer', 'Liam Roditty', 'Paul Seiferth']
I am writing a research paper on a idea Routing in Unit Disk Graphs
[{"from": "human", "value": "I am writing a research paper on a idea Routing in Unit Disk 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 \\(S \\subset \\mathbb {R}^2\\) be a set of n sites. The unit disk graph \\({{\\mathrm{UD}}}(S)\\) on S has vertex set S and an edge beten two distinct sites \\(s,t \\in S\\) if and only if s and t have Euclidean distance \\(|st| \\le 1\\). A routing scheme R for \\({{\\mathrm{UD}}}(S)\\) assigns to each site \\(s \\in S\\) a label\\(\\ell (s)\\) and a routing table\\(\\rho (s)\\). For any two sites \\(s, t \\in S\\), the scheme R must be able to route a packet from s to t in the following way: given a current siter (initially, \\(r = s\\)), a headerh (initially empty), and the label\\(\\ell (t)\\) of the target, the scheme R consults the routing table \\(\\rho (r)\\) to compute a neighbor \\(r'\\) of r, a new header \\(h'\\), and the label \\(\\ell (t')\\) of an intermediate target \\(t'\\). (The label of the original target may be stored at the header \\(h'\\).) The packet is then routed to \\(r'\\), and the procedure is repeated until the packet reaches t. The resulting sequence of sites is called the routing path. The stretch of R is the maximum ratio of the (Euclidean) length of the routing path produced by R and the shortest path in \\({{\\mathrm{UD}}}(S)\\), over all pairs of distinct sites in S. For any given \\(\\varepsilon > 0\\),  show how to construct a routing scheme for \\({{\\mathrm{UD}}}(S)\\) with stretch \\(1+\\varepsilon \\) using labels of \\(O(\\log n)\\) bits and routing tables of \\(O(\\varepsilon ^{-5}\\log ^2 n \\log ^2 D)\\) bits, where D is the (Euclidean) diameter of \\({{\\mathrm{UD}}}(S)\\). The header size is \\(O(\\log n \\log D)\\) bits."}]