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We review unnormalized and normalized information distances based on incomputable notions of Kolmogorov complexity and discuss how Kolmogorov complexity can be approximated by data compression algorithms. We argue that optimal algorithms for data compression with side information can be successfully used to approximate the normalized distance. Next, we discuss an alternative information distance, which is based on relative entropy rate (also known as Kullback-Leibler divergence), and compression-based algorithms for its estimation. We conjecture that in bioinformatics and computational linguistics this alternative distance is more relevant and important than the ones based on Kolmogorov complexity.