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We solve a neighbor discovery problem for a wireless ad hoc network using directional antennae, in order to achieve k-connectivity. Highly connected network enables each node to transmit its data to any other nodes. In high density network, high connectivity is easily achieved since each node has sufficiently many neighbors. However, radio transmission range should be increased for high connectivity in low density network. To increase transmission range, we use directional antennae without increasing transmission power in this paper. We derive optimal beam width for the directional antennae to achieve a given connectivity in a random directional neighbor discovery algorithm. The required time to achieve a given connectivity is also considered. Results in this paper could be used to derive solutions that minimize the required time while achieving a given connectivity in the network.