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Turbo codes are the most recent breakthrough in coding theory. However, the decoder's implementation cost limits their incorporation in commercial systems. Although the decoding algorithm is highly data dominated, no true memory optimization study has been performed yet. We have extensively and systematically investigated different memory optimizations for the maximum a posteriori (MAP) class of decoding algorithms. It turns out that it is not possible to present one decoder structure as being optimal. In fact, there are several tradeoffs, which depend on the specific turbo code, the implementation target (hardware or software), and the selected cost function. We therefore end up with a parametric family of new optimized algorithms out of which the designer can choose. The impact of our optimizations is illustrated by a representative example, which shows a significant decrease in both decoding energy (factor 2.5) and delay (factor 1.7).