Fig. 2

Available blocks. Left: an example of a haplotype matrix up to column 6 with the two arrays \(a_6\) and \(a_6^{-1}\) on the right. Center: the colexicographically sorted rows and the array \(d_6\) listed on the right. Right: the trie of the reverses of the rows of the matrix. For example, the block \((\{1,2,4,5\},5,6)\) is available because \(a_6^{-1}(1) = 3\), \(a_6^{-1}(2) = 1\), \(a_6^{-1}(4) = 2\), \(a_6^{-1}(5) = 4\) is the consecutive range \([x,y] = [1,4]\), we have \(d_6[r] \le 5\) for all \(r \in [1+1,4]\) with \(d_6[3] = 5\), and we have \(x=1\) and \(d_6[4+1] = 6 > 5\). The repeat in the block is 00, and we see it is a branching node in the trie on the right