Using compression to improve edit distance computation
Crochemore, Landau and Ziv-Ukelson were, I believe, the first to consider (LZ) compression explicitly in this context. As mentioned in the comments above, the earlier technique of Masek and Paterson is also in effect a form of compression. Considering general grammar-compression (SLP) is, I believe, a relatively new idea. Using it for string comparison speedup is original to the paper you cite, although comparing grammar-compressed strings as such has been considered before. You will find references and discussion in the compression chapter of http://arxiv.org/abs/0707.3619 .