  An elastic-degenerate string is a sequence of $n$ finite sets of strings of
total length $N$, introduced to represent a set of related DNA sequences, also
known as a pangenome. The ED string matching (EDSM) problem consists in
reporting all occurrences of a pattern of length $m$ in an ED text. This
problem has recently received some attention by the combinatorial pattern
matching community, culminating in an
$\tilde{\mathcal{O}}(nm^{\omega-1})+\mathcal{O}(N)$-time algorithm [Bernardini
et al., SIAM J. Comput. 2022], where $\omega$ denotes the matrix multiplication
exponent and the $\tilde{\mathcal{O}}(\cdot)$ notation suppresses polylog
factors. In the $k$-EDSM problem, the approximate version of EDSM, we are asked
to report all pattern occurrences with at most $k$ errors. $k$-EDSM can be
solved in $\mathcal{O}(k^2mG+kN)$ time, under edit distance, or
$\mathcal{O}(kmG+kN)$ time, under Hamming distance, where $G$ denotes the total
number of strings in the ED text [Bernardini et al., Theor. Comput. Sci. 2020].
Unfortunately, $G$ is only bounded by $N$, and so even for $k=1$, the existing
algorithms run in $\Omega(mN)$ time in the worst case. In this paper we show
that $1$-EDSM can be solved in $\mathcal{O}((nm^2 + N)\log m)$ or
$\mathcal{O}(nm^3 + N)$ time under edit distance. For the decision version, we
present a faster $\mathcal{O}(nm^2\sqrt{\log m} + N\log\log m)$-time algorithm.
We also show that $1$-EDSM can be solved in $\mathcal{O}(nm^2 + N\log m)$ time
under Hamming distance. Our algorithms for edit distance rely on non-trivial
reductions from $1$-EDSM to special instances of classic computational geometry
problems (2d rectangle stabbing or 2d range emptiness), which we show how to
solve efficiently. In order to obtain an even faster algorithm for Hamming
distance, we rely on employing and adapting the $k$-errata trees for indexing
with errors [Cole et al., STOC 2004].
