Fig. 1.

Examples of the Jaccard and the minimizer Jaccard estimator. Each example shows the k-mers of a sequence A on top, the k-mers of a sequence B on the bottom and lines connecting k-mers show the k-mer-matching between A and B. Each k-mer is labeled by its hash value. In Example 1, $J(A,B)=1/3$. The minimizers for w = 3 are circled in bold red. Here, $\widehat{I}(A,B;3)=1,\hspace{0.17em}\widehat{U}(A,B;3)=4$, and $\widehat{J}(A,B;3)=1/4$. Examples 2a and 2b give intuition for why the minimizer Jaccard estimator is biased. Here, a_i refers to the hash value assigned to position i and x and y are k-mers shared between A and B. The expected minimizer Jaccard for w = 2 is different in the two examples but the Jaccard is not (J = 0.2); hence the expected minimizer Jaccard cannot be equal to the true Jaccard. (A color version of this figure appears in the online version of this article.)