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We present two stable mergesort variants, "peeksort" and "powersort", thatexploit existing runs and find nearly-optimal merging orders with practicallynegligible overhead. Previous methods either require substantial effort fordetermining the merging order (Takaoka 2009; Barbay & Navarro 2013) or do nothave a constant-factor optimal worst-case guarantee (Peters 2001; Auger, Nicaud& Pivoteau 2015; Buss & Knop 2018). We demonstrate that our methods arecompetitive in terms of running time with state-of-the-art implementations ofstable sorting methods.

Nearly-Optimal Mergesorts: Fast, Practical Sorting Methods That Optimally Adapt to Existing Runs