Sequence assembly is a difficult problem whose importance has grown again recently as the cost of sequencing has dramatically dropped. Most new sequence assembly software has started by building a de Bruijn graph, avoiding the overlap-based methods used previously because of the computational cost and complexity of these with very large numbers of short reads. Here, we show how to use suffix array-based methods that have formed the basis of recent very fast sequence mapping algorithms to find overlaps and generate assembly string graphs asymptotically faster than previously described algorithms.