Essential proteins are indispensable to cells' survival and development. Prediction and analysis of essential proteins are crucial for uncovering the mechanisms of cells. With the help of computer science and high-throughput technologies, forecasting essential proteins by protein-protein interaction (PPI) networks has become more efficient than traditional approaches (expensive experimental methods are generally used). Many computational algorithms were employed to predict the essential proteins; however, they have various restrictions. To improve the prediction accuracy, by introducing the Local Fuzzy Fractal Dimension (LFFD) of complex networks into the analysis of the PPI network, we propose a novel algorithm named LDS, which combines the LFFD of the PPI network with the protein subcellular location information. By testing the proposed LDS algorithm on three different yeast PPI networks, the experimental results show that LDS outperforms some state-of-the-art essential protein-prediction techniques.