Amplified Pruning and Path Selectors for Deterministic Radio Gossiping
Qin Xin
We study deterministic gossiping in unknown strongly connected directed radio networks with $n$ vertices, $m$ arcs, and distinct identifiers in $[N]$, where $N=n^{\mathcal O(1)}$. Messages are unrestricted and collision detection is unavailable. Our neighborhood-clearing procedure reuses an elected receiver to secure its entire closed in-neighborhood in $\mathcal O(n\log^2n\log\log n)$ rounds. Charging each distinct closed neighborhood once yields $\mathcal O(m\log^2n\log\log n)$ rounds under an explicit degree-concentration condition, for every polynomial identifier range. A complementary adaptive-pruning procedure combines quadratic contraction with completion sensitive to the number of unsecured sources. If $a\ge1$ sources remain at completion entry, permutation selectors reduce the balanced completion budget by a factor of $\log^{2/3}(a+1)$ relative to interleaved path selectors. A geometric execution attains $\widetilde{\mathcal O}(k^2+nτ+\min\{u^2,(nu)^{2/3}\})$ for a path cover of scale $k$, weight $τ$, and $u$ exceptions. For directed graphs with common in- and out-degree $Δ$, the portfolio achieves $\widetilde{\mathcal O}(\min\{n\sqrtΔ,n^2/Δ\})$. Every graph also admits $\widetilde{\mathcal O}(\min\{n^{4/3},N+(mn)^{3/5}\})$ rounds. Only $n,N$ are inputs; degrees, covers, and structural constants are used solely in the analysis. The unrestricted polynomial-identifier density extension remains open.
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