We study deterministic gossiping in unknown strongly connected directed radio networks with $n$ vertices, $m$ arcs, and distinct identifiers in $\{1,\ldots,N\}$, 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 $\mathcal O(n\log^2n\log\log n)$ 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 $\mathcal O(m\log^2n(\log\log n)^2)$ rounds. For directed regular graphs with common in- and out-degree $Δ$, our strategy achieves $\widetilde{\mathcal O}(\min\{n\sqrtΔ,n^2/Δ\})$, extending the identifier range of Wu and Chrobak and improving their high-degree bound. At completion entry, let $a\ge1$ count original sources not yet credited as globally disseminated. Permutation selectors reduce our balanced completion budget by a factor of $\log^{2/3}(a+1)$ relative to interleaved path selectors. Our geometric protocol attains $\widetilde{\mathcal O}(k^2+nτ+\min\{u^2,(nu)^{2/3}\})$ for an original path cover of scale $k$, weight $τ$, and $u$ exceptions. Every graph also admits $\widetilde{\mathcal O}(\min\{n^{4/3},N+(mn)^{3/5}\})$ 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.

翻译：暂无翻译