Runtime results on seed-chain-extend between for two sequences of length n and $k=C\log n={\displaystyle \frac{2}{1-2\alpha }\log \left(n\right)=9,10,\dots ,19}$, where θ = 0.10 and α = −log ₄(1 − θ). (A) The y-axis shows the runtime ratios (with 95% confidence intervals) for iteration k + 1 divided by runtime for iteration k, where the sequence length n_k is plotted on the x-axis. As n grows, both the sketched and nonsketched extension ratios asymptotically approach the predicted ratio of $n^{1+\frac{2}{1-2\alpha }\alpha }\log \left(n\right)$ runtimes. (B) Multiplicative speed-up for sketched versus nonsketched alignment when sketching with density ${\displaystyle \frac{1}{c}={\displaystyle \frac{1}{k-7}}}$, where k = C log n. The slow-down for extension flattens out, whereas the chaining speed-up is almost linear on the log-scaled x-axis; chaining speed-up dominates extension slow-down as predicted by our theory.