We construct a compact vector representation on $\mathbb{R}24$ of a DNA sequence of arbitrary length. Each component of this vector is obtained from a representative sequence, the elements of which are the values realized by a function $\Gamma$. The function $\Gamma$, so defined, acts on neighborhoods of arbitrary radius that are located at strategic positions within the DNA sequence. $\Gamma$ carries complete information about the local multiplicity of the nucleotides as a consequence of the uniqueness of prime factorisation of integer. The two parameters characterizing the radius and location of the neighbourhoods are fixed by comparing the phylogenetic tree we find through our algorithm with standard results for the $\beta$ -globin gene sequences of eleven different species. Remarkably, the time complexity for this similarity analysis turns out to be $\mathcal{O}(n)$. Using the values of the two fitting parameters so obtained, the method is further applied to analyze mitochondrial genome sequences.

翻译：我们构建了一个DNA序列（任意长度）在$\mathbb{R}24$上的紧凑向量表示。该向量的每个分量均源自一条代表性序列，其元素由函数$\Gamma$实现的数值构成。所定义的函数$\Gamma$作用于DNA序列中位于战略位置的任意半径邻域。由于整数质因数分解的唯一性，$\Gamma$携带了关于核苷酸局部多样性的完整信息。通过将我们的算法构建的系统发育树与十一个不同物种$\beta$-珠蛋白基因序列的标准结果进行比较，我们确定了表征邻域半径和位置的两个参数。值得注意的是，该相似性分析的时间复杂度为$\mathcal{O}(n)$。利用通过这种方式获得的拟合参数值，该方法进一步应用于分析线粒体基因组序列。