# BWT Construction and K-mer Counting

Over the last few weeks, we have been going through the algorithms of various BWT construction and k-mer counting methods, and came to realize that they are two sides of the same coin. That means the two communities can mutually benefit from the improved algorithm/tools developed by each other. Our insight is not novel, because Tallymer, a k-mer counting program from earlier generation used suffix arrays and LCP arrays to perform k-mer counting in a genome.

Let us quickly explain the equivalence. Suppose you like to construct the BTW of a small word - JAMES\$ . That is very easy to do, because you have go through the increasing order of letters and then pick the letter right in front of them. For example, \$ is the smallest letter. So, our BWT will start with S, which precedes \$ immediately. A is the next small letter. So, S in BWT will be followed by J, which precedes A. Following in that order, you get to SJM\$AE .

How about a more complicated word - ONION\$ ? In this case, we have two Os and cannot proceed in the same way by ordering letters. However, we can take the 3-mers and follow the same formula, because all 3-mers are unique. Smallest two-mer is \$ON. Therefore, the BWT starts with N, which precedes \$. The next 2-mer is ION. So, N will be followed by N, which precedes N. Continuing in that manner, we can compute the BWT as NNOOI\$.

The same method can be used for any large string as long we can define a k-mer size so that all k-words become independent. But is it necessary for all k-mers to be independent? What if that k-mer size is 10,000, because a genome has a 10Kb block that is duplicate? Actually, one can find the BWT from a much smaller k-mer size and infer the positions of letters preceding all unique k-mers based on k-mer count. Then a small set of duplicate k-mers need to be resolved by going to higher k-values.

Jared Simpson’s SGA algorithm uses BWT/FM construction and string graph traversal as two major steps. Given the equivalence of BWT construction with k-mer counting and string graph with de Bruijn graph, SGA algorithm is essentially similar to any dBG algorithm despite its apparent dissimilarity.

Written by M. //