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The edit distance is a fundamental measure of sequence similarity, defined asthe minimum number of character insertions, deletions, and substitutions neededto transform one string into the other. Given two strings of length at most$n$, simple dynamic programming computes their edit distance exactly in$O(n^2)$ time, which is also the best possible (up to subpolynomial factors)assuming the Strong Exponential Time Hypothesis (SETH). The last few decadeshave seen tremendous progress in edit distance approximation, where the runtimehas been brought down to subquadratic, near-linear, and even sublinear at thecost of approximation.  In this paper, we study the dynamic edit distance problem, where the stringschange dynamically as the characters are substituted, inserted, or deleted overtime. Each change may happen at any location of either of the two strings. Thegoal is to maintain the (exact or approximate) edit distance of such dynamicstrings while minimizing the update time. The exact edit distance can bemaintained in $\tilde{O}(n)$ time per update (Charalampopoulos, Kociumaka,Mozes; 2020), which is again tight assuming SETH. Unfortunately, even with theunprecedented progress in edit distance approximation in the static setting,strikingly little is known regarding dynamic edit distance approximation.Utilizing the off-the-shelf tools, it is possible to achieve an$O(n^{c})$-approximation in $n^{0.5-c+o(1)}$ update time for any constant $c\in[0,\frac16]$. Improving upon this trade-off remains open.  The contribution of this work is a dynamic $n^{o(1)}$-approximation algorithmwith amortized expected update time of $n^{o(1)}$. In other words, we bring theapproximation-ratio and update-time product down to $n^{o(1)}$. Our solutionutilizes an elegant framework of precision sampling tree for edit distanceapproximation (Andoni, Krauthgamer, Onak; 2010).

Approximating Edit Distance in the Fully Dynamic Model