Adding O(n^{1-ε/α}) edges to a tree yields an n-node graph with broadcast time (1+ε)log₂n, improving the prior O(n^{1-ε}) bound, with a matching Ω(n) lower bound at ε→0.
On the com- plexity of telephone broadcasting from cacti to bounded pathwidth graphs
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Sparse Relaxed Broadcast Graphs
Adding O(n^{1-ε/α}) edges to a tree yields an n-node graph with broadcast time (1+ε)log₂n, improving the prior O(n^{1-ε}) bound, with a matching Ω(n) lower bound at ε→0.