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A generalized rewriting model is defined for flash memory that represents stored data and permitted rewrite operations by a directed graph. This model is a generalization of previously introduced rewriting models of codes, including floating codes, write-once memory codes, and buffer codes. This model is used to design a new rewriting code for flash memories. The new code, referred to as trajectory code, allows stored data to be rewritten as many times as possible without block erasures. It is proved that the trajectory codes are asymptotically optimal for a wide range of scenarios. In addition, rewriting codes that use a randomized rewriting scheme are presented that obtain good performance with high probability for all possible rewrite sequences.