This paper derives the analytical solution of a novel distributed node-specific block-diagonal linearly constrained minimum variance beamformer from the centralized linearly constrained minimum variance (LCMV) beamformer when considering that the noise covariance matrix is block-diagonal. To further reduce the computational complexity of the proposed beamformer, the ShermanMorrison-Woodbury formula is introduced to compute the inversion of noise sample covariance matrix. By doing so, the exchanged signals can be computed with lower dimensions between nodes, where the optimal LCMV beamformer is still available at each node as if each node is to transmit its all raw sensor signal observations. The proposed beamformer is fully distributable without imposing restrictions on the underlying network topology or scaling computational complexity, i.e., there is no increase in the per-node complexity when new nodes are added to the networks. Compared with state-of-the-art distributed node-specific algorithms that are often time-recursive, the proposed beamformer exactly solves the LCMV beamformer optimally frame by frame, which has much lower computational complexity and is more robust to acoustic transfer function estimation error and voice activity detector error. Numerous experimental results are presented to validate the effectiveness of the proposed beamformer.