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Abstract Background Alignment-free methods for sequence comparisons have become popular in many bioinformatics applications, specifically in the estimation of sequence similarity measures to construct phylogenetic trees. Recently, the average common substring measure, ACS , and its k -mismatch counterpart, ACS k , have been shown to produce results as effective as multiple-sequence alignment based methods for reconstruction of phylogeny trees. Since computing ACS k takes O ( n log k n ) time and hence impractical for large datasets, multiple heuristics that can approximate ACS k have been introduced. Results In this paper, we present a novel linear-time heuristic to approximate ACS k , which is faster than computing the exact ACS k while being closer to the exact ACS k values compared to previously published linear-time greedy heuristics. Using four real datasets, containing both DNA and protein sequences, we evaluate our algorithm in terms of accuracy, runtime and demonstrate its applicability for phylogeny reconstruction. Our algorithm provides better accuracy than previously published heuristic methods, while being comparable in its applications to phylogeny reconstruction. Conclusions Our method produces a better approximation for ACS k and is applicable for the alignment-free comparison of biological sequences at highly competitive speed. The algorithm is implemented in Rust programming language and the source code is available at https://github.com/srirampc/adyar-rs .