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Optimal fractal coding consists of finding, in a finite set of contractive affine mappings, one whose unique fixed point is closest to the original image. Optimal fractal coding is an NP-hard combinatorial optimization problem. Conventional coding is based on a greedy suboptimal algorithm known as collage coding. In a previous study, we proposed a local search algorithm that significantly improves on collage coding. However the algorithm, which requires the computation of many fixed points, is computationally expensive. In this paper we provide techniques that drastically reduce the time complexity of the algorithm.