Suppose we have a tool for finding super-maximal exact matches (SMEMs) and we want to use it to find all the long SMEMs between a noisy long read $P$ and a highly repetitive pangenomic reference $T$. Notice that if $L \geq k$ and the $k$-mer $P [i..i + k - 1]$ does not occur in $T$ then no SMEM of length at least $L$ contains $P [i..i + k - 1]$. Therefore, if we have a Bloom filter for the distinct $k$-mers in $T$ and we want to find only SMEMs of length $L \geq k$, then when given $P$ we can break it into maximal substrings consisting only of $k$-mers the filter says occur in $T$ -- which we call pseudo-SMEMs -- and search only the ones of length at least $L$. If $L$ is reasonably large and we can choose $k$ well then the Bloom filter should be small (because $T$ is highly repetitive) but the total length of the pseudo-SMEMs we search should also be small (because $P$ is noisy). Now suppose we are interested only in the longest $t$ SMEMs of length at least $L$ between $P$ and $T$. Notice that once we have found $t$ SMEMs of length at least $\ell$ then we need only search for SMEMs of length greater than $\ell$. Therefore, if we sort the pseudo-SMEMs into non-increasing order by length, then we can stop searching once we have found $t$ SMEMs at least as long as the next pseudo-SMEM we would search. Our preliminary experiments indicate that these two admissible heuristics may significantly speed up SMEM-finding in practice.

翻译：假设我们拥有一个用于寻找超最大精确匹配（SMEM）的工具，并希望利用该工具在含噪声的长读序列$P$与高度重复的泛基因组参考序列$T$之间寻找所有长SMEM。注意到若$L \geq k$且$k$-mer $P [i..i + k - 1]$未出现在$T$中，则任何长度不小于$L$的SMEM都不会包含$P [i..i + k - 1]$。因此，若我们构建了$T$中不同$k$-mer的布隆过滤器，且仅需寻找长度$L \geq k$的SMEM，则在给定$P$时，可将其划分为仅由过滤器判定存在于$T$中的$k$-mer构成的最大子串（我们称之为伪SMEM），并仅对长度不小于$L$的伪SMEM进行搜索。若$L$取值合理且能恰当选择$k$值，则布隆过滤器可保持较小规模（因为$T$具有高度重复性），同时待搜索伪SMEM的总长度也会较小（因为$P$含有噪声）。现假设我们仅关注$P$与$T$之间长度不小于$L$的最长$t$个SMEM。注意到当已找到$t$个长度不小于$\ell$的SMEM后，仅需继续搜索长度大于$\ell$的SMEM即可。因此，若将伪SMEM按长度非递增顺序排序，则当已找到的SMEM数量达到$t$且其长度不小于下一个待搜索伪SMEM的长度时，即可终止搜索。我们的初步实验表明，这两种可采纳启发式策略在实践中可能显著提升SMEM查找速度。