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Water-Filling (WF) is widely applied in power allocation in multichannel wireless communications. By mathematically linearising the optimal WF expression, the authors observe an intrinsic parallel-shift property of WF, based on which a fast and efficient WF algorithm is proposed. Compared with the conventional WF algorithms, it greatly simplifies WF execution by removing the Lagrange multiplier (or water-level) searching process. The authors further apply the proposed parallel-shift WF to solve the power allocation problem in orthogonal frequency division multiplexing (OFDM)-based underlay cognitive radios (CRs), with the objective of maximising the secondary user's throughput over all OFDM sub-channels under the transmit power and the interference constraints. To this end, the existing algorithm adopts an iterative binary searching process to find the solution, where the conventional WF algorithm with Lagrange multiplier searching is invoked in each iteration. In contrast, the authors propose a new power allocation algorithm to remove the iterative binary searching process. It runs the simplified parallel-shift WF only once, and then directly calculates the final solution using a power adjustment process (with the parallel-shift property as the underlying enabling mechanism). Numerical results show that both the proposed parallel-shift WF and the OFDM-based CR power allocation algorithms can run multiple times faster than the existing counterparts, and the gap on the running time increases with the total number of OFDM sub-channels in the system.