Let G = (V, E, w ) be a weighted directed graph without negative cycles. For two vertices s,t ∈ V, we let d≤h(s, t ) be the minimum, according to the weight function w, of a path from s to t that uses at most h edges, or hops. We consider algorithms for computing d<h(s,t ) for every 1 ≤ h ≤ n, where n = |V|, in various settings. We consider the singlepair, single-source and all-pairs versions of the problem. We also consider a distance oracle version of the problem in which we are not required to explicitly compute all distances d

STOC ’25, Prague, Czechia

Dynamic Time Warping (DTW) is a widely used similarity measure for comparing strings that encode time series data, with applications to areas including bioinformatics, signature verification, and speech recognition. The standard dynamic-programming algorithm for DTW takes $O(n^2)$ time, and there are conditional lower bounds showing that no algorithm can do substantially better. In many applications, however, the strings $x$ and $y$ may contain long runs of repeated letters, meaning that they can be compressed using run-length encoding. A natural question is whether the DTW-distance between these compressed strings can be computed efficiently in terms of the lengths $k$ and $\ell$ of the compressed strings. Recent work has shown how to achieve $O(k\ell^2 + \ell k^2)$ time, leaving open the question of whether a near-quadratic $\tilde{O}(k\ell)$-time algorithm might exist. We show that, if a small approximation loss is permitted, then a near-quadratic time algorithm is indeed possible: our algorithm computes a $(1 + ε)$-approximation for $DTW(x, y)$ in $\tilde{O}(k\ell / ε^3)$ time, where $k$ and $\ell$ are the number of runs in $x$ and $y$. Our algorithm allows for $DTW$ to be computed over any metric space $(Σ, δ)$ in which distances are $O(log(n))$-bit integers. Surprisingly, the algorithm also works even if $δ$ does not induce a metric space on $Σ$ (e.g., $δ$ need not satisfy the triangle inequality).