Amplified Pruning and Path Selectors for Deterministic Radio Gossiping

We study deterministic gossiping in unknown strongly connected directed radio networks with $n$ vertices, $m$ arcs, and distinct identifiers in $\{1,\dots ,N\}$, where $N=n^{\mathcal{O}(1)}$. Messages are unrestricted; collision detection is unavailable. Our protocol combines amplified reception certificates, adaptive pruning, and permutation selectors. Each pruning iteration costs $\mathcal{O}(n{\log }^{2}n\log \log n)$ rounds. A two-step quadratic contraction limits each receiver and each class of identical closed in-neighborhoods to $\mathcal{O}(\log \log n)$ qualifying selections. Under an explicit degree-concentration condition, our protocol takes $\mathcal{O}(m{\log }^{2}n(\log \log n{)}^2)$ rounds. For directed regular graphs with common in- and out-degree $\mathrm{\Delta }$, our strategy achieves $\tilde{\mathcal{O}}(\min \{n\sqrt{\mathrm{\Delta }},n^2\mathrm{/}\mathrm{\Delta }\})$, extending the identifier range of Wu and Chrobak and improving their high-degree bound. At completion entry, let $a\ge 1$ count original sources not yet credited as globally disseminated. Permutation selectors reduce our balanced completion budget by a factor of ${\log }^{2\mathrm{/}3}(a+1)$ relative to interleaved path selectors. Our geometric protocol attains $\tilde{\mathcal{O}}(k^2+n\tau +\min \{u^2,(nu{)}^{2\mathrm{/}3}\})$ for an original path cover of scale $k$, weight $\tau$, and $u$ exceptions. Every graph also admits $\tilde{\mathcal{O}}(\min \{n^{4\mathrm{/}3},N+(mn{)}^{3\mathrm{/}5}\})$ rounds. Only $n,N$ are inputs. Within the stated graph classes and identifier ranges, these results answer the polynomial-identifier and average-degree questions of Wu and Chrobak. The unrestricted general-graph cases remain open.