We consider lossy (Hamming distortion) representation of memoryless input. In any lossy compression codebook, we typically assign to each codeword the set of input sequences within the given distortion budget. The codewords of the codebook may themselves be constructed with well defined structure, as in Lempel Ziv like algorithms. In this talk, we consider matching input to codewords in a more structured way. We demand not just that the input and its codeword match within the distortion budget, but also that all prefixes of the two strings match within the distortion budget as well. Using the celebrated Cycle Lemma of Dvoretsky and Motzkin, we prove that the coderate for such constructions does not increase and that we obtain computational speedups owing to better codebook structure.