  We consider the local broadcasting problem in the SINR model, which is a
basic primitive for gathering initial information among $n$ wireless nodes.
Assuming that nodes can measure received power, we achieve an essentially
optimal constant approximate algorithm (with a $\log^2 n$ additive term). This
improves upon the previous best $O(\log n)$-approximate algorithm. Without
power measurement, our algorithm achieves $O(\log n)$-approximation, matching
the previous best result, but with a simpler approach that works under harsher
conditions, such as arbitrary node failures. We give complementary lower bounds
under reasonable assumptions.
