Incorporating indel channels into average-case analysis of seed-chain-extend

Given a sequence $s_1$ of $n$ letters drawn i.i.d. from an alphabet of size $\sigma$ and a mutated substring $s_2$ of length $m < n$, we often want to recover the mutation history that generated $s_2$ from $s_1$. Modern sequence aligners are widely used for this task, and many employ the seed-chain-extend heuristic with $k$-mer seeds. Previously, Shaw and Yu showed that optimal linear-gap cost chaining can produce a chain with $1 - O\left(\frac{1}{\sqrt{m}}\right)$ recoverability, the proportion of the mutation history that is recovered, in $O\left(mn^{2.43\theta} \log n\right)$ expected time, where $\theta < 0.206$ is the mutation rate under a substitution-only channel and $s_1$ is assumed to be uniformly random. However, a gap remains between theory and practice, since real genomic data includes insertions and deletions (indels), and yet seed-chain-extend remains effective. In this paper, we generalize those prior results by introducing mathematical machinery to deal with the two new obstacles introduced by indel channels: the dependence of neighboring anchors and the presence of anchors that are only partially correct. We are thus able to prove that the expected recoverability of an optimal chain is $\ge 1 - O\Bigl(\frac{1}{\sqrt{m}}\Bigr)$ and the expected runtime is $O(mn^{3.15 \cdot \theta_T}\log n)$, when the total mutation rate given by the sum of the substitution, insertion, and deletion mutation rates ($\theta_T = \theta_i + \theta_d + \theta_s$) is less than $0.159$.