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This work seeks to leverage Processing-with-storage-technology (PWST) to accelerate a key bioinformatics kernel calledk-mer counting, which involves processing large files of sequence data on the disk to build a histogram of fixed-size genome sequence substrings and thereby entails prohibitively high I/O overhead. In particular, this work proposes a set of accelerator designs called Abakus that offer varying degrees of tradeoffs in terms of performance, efficiency, and hardware implementation complexity. The key to these designs is a set of domain-specific hardware extensions to accelerate the key operations fork-mer counting at various levels of the SSD hierarchy, with the goal of enhancing the limited computing capabilities of conventional SSDs, while exploiting the parallelism of the multi-channel, multi-way SSDs. Our evaluation suggests that Abakus can achieve 8.42×, 6.91×, and 2.32× speedup over the CPU-, GPU-, and near-data processing solutions.