Next-generation de novo short reads assemblers typically use the following strategy: (1) assemble unpaired reads using heuristics leading to contigs; (2) order contigs from paired reads information to produce scaffolds. We propose to unify these two steps by introducing localized assembly: direct construction of scaffolds from reads. To this end, the paired string graph structure is introduced, along with a formal framework for building scaffolds as paths of reads. This framework leads to the design of a novel greedy algorithm for memory-efficient, parallel assembly of paired reads. A prototype implementation of the algorithm has been developed and applied to the assembly of simulated and experimental short reads. Our experiments show that our methods yields longer scaffolds than recent assemblers, and is capable of assembling diploid genomes significantly better than other greedy methods.