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 $\tilde Ω\left(n^ω\right)$ conditional lower bound by Dor, Halperin, and Zwick for $α$-APASP with $α< 2$, 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).

翻译：我们针对稠密加权图提出了一种运行时间为 $\tilde O\left(n^{2+\frac{1}{3k+2}}\right)$ 的 $+2\sum_{i=1}^{k+1}{W_i}$-APASP 算法，其中 $W_{i}$ 表示最短路径上第 $i$ 重边的权重。Dor、Halperin 和 Zwick [FOCS'96, SICOMP'00] 曾针对对应的无权图 $+2\cdot\left( k+1\right)$-APASP 问题提出两种算法：稀疏图的运行时间为 $\tilde O\left(n^{2-\frac{1}{k+2}}m^{\frac{1}{k+2}}\right)$，稠密图的运行时间为 $\tilde O\left(n^{2+\frac{1}{3k+2}}\right)$。Cohen 和 Zwick [SODA'97, JALG'01] 将稀疏图算法适配到加权图，在相同运行时间内实现了 $+2\sum_{i=1}^{k+1}{W_i}$-APASP 算法。我们则展示了针对稠密加权图的算法。对于近加性 APASP，我们提出了一种运行时间为 $\tilde O\left(\left(\frac{1}{\varepsilon}\right)^{O\left(1\right)}\cdot n^{2.15135313}\cdot\log W\right)$ 的 $\left(1+\varepsilon,\min{\left\{2W_1,4W_{2}\right\}}\right)$-APASP 算法，改进了 Saha 和 Ye [SODA'24] 的 $\left(1+\varepsilon,2W_1\right)$-APASP 算法。对于乘性 APASP，我们提出了一个 $\left(\frac{3\ell +4}{\ell + 2}+\varepsilon\right)$-APASP 算法框架，降低了 Akav 和 Roditty [ESA'21] 针对稠密图的算法运行时间，并推广了 Dory 等人 [SODA'24] 的 $\left(2+\varepsilon\right)$-APASP 算法。我们的基础案例是一个运行时间为 $\tilde O\left(\left(\frac{1}{\varepsilon}\right)^{O\left(1\right)}\cdot n^{2.15135313}\cdot \log W\right)$ 的 $\left(\frac{7}{3}+\varepsilon\right)$-APASP 算法，改进了 Baswana 和 Kavitha [FOCS'06, SICOMP'10] 针对稠密图的 $\frac{7}{3}$-APASP 算法。最后，我们通过允许一个加性项（例如在 $\tilde O\left(n^{2+\frac{1}{3k+2}}\right)$ 运行时间内实现 $\left(\frac{6k+3}{3k+2},\sum_{i=1}^{k+1}W_{i}\right)$-APASP），"绕过"了 Dor、Halperin 和 Zwick 针对 $α< 2$ 的 $α$-APASP 问题所提出的 $\tilde Ω\left(n^ω\right)$ 条件性下界。