We present implications from the known conjectures like APSP, 3SUM and ETH in a form of a negated containment of a linear-time with a non-deterministic logarithmic-bit oracle in a respective deterministic bounded-time class They are different for different conjectures and they exhibit in particular the dependency on the input range parameters. On the other hand, for the problem of verifying if an $N\times N$ matrix defines a metric and subcubic-time equivalent problems, we show that one may assume w.l.o.g. that the integer edge weights have at most $N^{2+o(1)}$-bit representation.

中文翻译：

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APSP、三角形检测和 3SUM 硬度用于区分确定性和非确定性的后果

我们以线性时间与非确定性对数位预言机在各自的确定性有界时间类中的否定包含的形式，展示了来自已知猜想（如 APSP、3SUM 和 ETH）的含义。它们对于不同的猜想是不同的，并且它们特别表现出对输入范围参数的依赖性。另一方面，对于验证 $N\times N$ 矩阵是否定义度量和亚立方时间等价问题的问题，我们表明可以假设 wlog 整数边权重至多有 $N^{2+ o(1)}$ 位表示。