Tandem mass spectrometry has emerged to be one of the most powerful high-throughput techniques for protein identification. Tandem mass spectrometry selects and fragments peptides of interest into N-terminal ions and C-terminal ions, and it measures the mass/charge ratios of these ions. The de novo peptide sequencing problem is to derive the peptide sequences from given tandem mass spectral data of k ion peaks without searching against protein databases. By transforming the spectral data into a matrix spectrum graph G = (V, E), where |V| = O(k(2)) and |E| = O(k(3)), we give the first polynomial time suboptimal algorithm that finds all the suboptimal solutions (peptides) in O(p|E|) time, where p is the number of solutions. The algorithm has been implemented and tested on experimental data. The program is available at http://hto-c.usc.edu:8000/msms/menu/denovo.htm.