Fig. 6

Construction of the overlap of two long reads \(r_1\) and \(r_2\) (long black arrows) from all unitigs \({\mathscr {S}}_{r_1r_2}:=\{s_1,...,s_5\}\) (short black bars) that match to both \(r_1\) and \(r_2\). A significant match (s, r) of \(s\in {\mathscr {S}}_{r_1r_2}\) on \(r\in \{r_1,r_2\}\) is illustrated by blue and green thick arrows on r. The relative orientation of (s, r) is indicated by the direction of its arrow, that is, \(\theta (s,r)=+1\) (resp. \(\theta (s,r)=-1\)) if its arrow points to the right (resp. left). The subsets \({\mathscr {S}}^1_{r_1r_2}:=\{s_1,s_3,s_5\}\) (unitigs with blue significant matches) and \({\mathscr {S}}^2_{r_1r_2}:=\{s_2,s_4\}\) (unitigs with green significant matches) of \({\mathscr {S}}_{r_1r_2}\) are both inclusion-maximal and consists of pairwise consistent unitigs. The set \({\mathscr {S}}^1_{r_1r_2}\) maximizes \(\Omega (r_1,r_2)\) and thus determines the overlap. It implies \(\theta (r_1,r_2)=+1\). Moreover, \(i_{r_1}\) (resp. \(j_{r_1}\)) is the minimal (resp. maximal) coordinate of significant matches of unitigs from \({\mathscr {S}}^1_{r_1r_2}\) on \(r_1\). The corresponding coordinates on \(r_2\) are \(k_{r_2}\) and \(l_{r_2}\), respectively. The spanning intervals \([i_{r_1},j_{r_1}]\) and \([k_{r_2},l_{r_2}]\) define the overlap of \(r_1\) and \(r_2\). In this example we have \(i_{r_1}>k_{r_2}\) and \(|r_1|-j_{r_1}>|r_2|-l_{r_2}\), implying that \(r_2\) extends \(r_1\) neither to the left or right and thus, edge \(r_1r_2\) is contracted in G