Table 1.

Summary of theoretical results for the second phase of different algorithms for generating a set of $k$-mers hitting all $L$-long sequences. PDOCKS is DOCKS with the improved hitting number calculation, i.e. greedy removal of one vertex at each iteration. $p_D$, $p_{DA}$ denote the total number of picked vertices for DOCKS/PDOCKS and DOCKSany, respectively. $m$ denotes the number of parallel threads used, $T_{max}$ the maximum vertex hitting number, and $ϵ$ and $\delta$ PASHA’s randomization parameters.

Algorithm	DOCKS	PDOCKS	DOCKSany	PASHA
Theoretical runtime	$O\left(\left(1+p_D\right)|\Sigma {|}^{k+1}\cdot L\right)$	$O\left(\left(1+p_D\right)|\Sigma {|}^{k+1}\cdot L/m\right)$	$O\left(\left(1+p_{DA}\right)|\Sigma {|}^{k+1}\right)$	$O\left(\left(L^2\cdot |\Sigma {|}^{k+1}\cdot {\text{log}}^2\left(|\Sigma {|}^k\right)\right)/\left(\epsilon {\delta }^{3}m\right)\right)$
Approximation ratio	$1+\text{log}{T}_{max}$	$1+\text{log}{T}_{max}$	N/A	$\left(1+\text{log}{T}_{max}\right)/(1-4\delta -2\epsilon )$