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We consider a wireless network composed of a set of n wireless nodes distributed in a two dimensional plane. The signal sent by every node can be received by all nodes within its transmission range, which is uniform and normalized to one unit. We present the first distributed method to construct a bounded degree planar connected structure LRNG, whose total link length is within a constant factor of the minimum spanning tree1 using total O(n) messages under the broadcast communication model. Moreover, in our method, every node only uses its two-hop information to construct such structure, i.e., it is localized method. We show that some two-hop information is necessary to construct any low-weighted structure. We also study the application of this structure in efficient broadcasting in wireless ad hoc networks. We prove that, for broadcasting, the relative neighborhood graph (RNG), which is the previously best-known sparse structure that can be constructed locally, could use energy O(n) times the total energy used by our structure LRNG. Our simulations show that the broadcasting based on LRNG consumes energy about 36% more than that by MST, and broadcasting based on RNG consumes energy about 64% more than that by MST. We also show that no localized method can construct a structure for broadcasting with total power consumption asymptotically better than LRNG.