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Abstract:We present a $+2\sum_{i=1}^{k+1}{W_i}$-APASP algorithm for dense weighted graphs with runtime $\tilde O\left(n^{2+\frac{1}{3k+2}}\right)$, where $W_{i}$ is the weight of an $i^{th}$ heaviest edge on a shortest path. Dor, Halperin and Zwick [FOCS'96, SICOMP'00] had two algorithms for the commensurate unweighted $+2\cdot\left( k+1\right)$-APASP: $\tilde O\left(n^{2-\frac{1}{k+2}}m^{\frac{1}{k+2}}\right)$ runtime for sparse graphs and $\tilde O\left(n^{2+\frac{1}{3k+2}}\right)$ runtime for dense graphs. Cohen and Zwick [SODA'97, JALG'01] adapted the sparse variant to weighted graphs: $+2\sum_{i=1}^{k+1}{W_i}$-APASP algorithm in the same runtime. We show an algorithm for dense weighted graphs.
For nearly additive APASP, we present a $\left(1+\varepsilon,\min{\left\{2W_1,4W_{2}\right\}}\right)$-APASP algorithm with $\tilde O\left(\left(\frac{1}{\varepsilon}\right)^{O\left(1\right)}\cdot n^{2.15135313}\cdot\log W\right)$ runtime. This improves the $\left(1+\varepsilon,2W_1\right)$-APASP of Saha and Ye [SODA'24].
For multiplicative APASP, we show a framework of $\left(\frac{3\ell +4}{\ell + 2}+\varepsilon\right)$-APASP algorithms, reducing the runtime of Akav and Roditty [ESA'21] for dense graphs and generalizing the $\left(2+\varepsilon\right)$-APASP algorithm of Dory et al [SODA'24]. Our base case is a $\left(\frac{7}{3}+\varepsilon\right)$-APASP in $\tilde O\left(\left(\frac{1}{\varepsilon}\right)^{O\left(1\right)}\cdot n^{2.15135313}\cdot \log W\right)$ runtime, improving the $\frac{7}{3}$-APASP algorithm of Baswana and Kavitha [FOCS'06, SICOMP'10] for dense graphs.
Finally, we "bypass" an $\tilde \Omega \left(n^\omega\right)$ conditional lower bound by Dor, Halperin, and Zwick for $\alpha$-APASP with $\alpha < 2$, by allowing an additive term (e.g. $\left(\frac{6k+3}{3k+2},\sum_{i=1}^{k+1}W_{i}\right)$-APASP in $\tilde O\left(n^{2+\frac{1}{3k+2}}\right)$ runtime).

Submission history

From: Ariel Sapir [view email]
[v1] Thu, 4 Sep 2025 20:00:17 UTC (107 KB)
[v2] Mon, 16 Feb 2026 15:09:30 UTC (250 KB)
[v3] Tue, 18 Aug 2026 11:06:46 UTC (654 KB)