Despite extensive research on distance oracles, there are still large gaps between the best constructions for spanners and distance oracles. Notably, there exist sparse spanners with a multiplicative stretch of $1+\varepsilon$ plus some additive stretch. A fundamental open problem is whether such a bound is achievable for distance oracles as well. Specifically, can we construct a distance oracle with multiplicative stretch better than 2, along with some additive stretch, while maintaining subquadratic space complexity? This question remains a crucial area of investigation, and finding a positive answer would be a significant step forward for distance oracles. Indeed, such oracles have been constructed for sparse graphs. However, in the more general case of dense graphs, it is currently unknown whether such oracles exist. In this paper, we contribute to the field by presenting the first distance oracles that achieve a multiplicative stretch of $1+\varepsilon$ along with a small additive stretch while maintaining subquadratic space complexity. Our results represent an advancement particularly for constructing efficient distance oracles for dense graphs. In addition, we present a whole family of oracles that, for any positive integer $k$, achieve a multiplicative stretch of $2k-1+\varepsilon$ using $o(n^{1+1/k})$ space.

翻译：尽管对距离查询进行了广泛研究，但在构造最优点生成树和距离查询之间仍存在巨大差距。值得注意的是，存在具有乘法伸缩因子$1+\varepsilon$加上一些加法伸缩因子的稀疏点生成树。一个基本开放问题是，这样的界限是否也能在距离查询中实现。具体来说，我们能否构造一个乘法伸缩因子优于2且带有一些加法伸缩因子的距离查询，同时保持次二次空间复杂度？这个问题仍是一个关键的研究领域，找到肯定答案将是距离查询领域的重要一步。事实上，这类查询已在稀疏图中构造出来。然而，在更一般的稠密图情况下，目前尚不清楚是否存在这样的查询。在本文中，我们通过提出首个实现乘法伸缩因子$1+\varepsilon$和小加法伸缩因子同时保持次二次空间复杂度的距离查询，为这一领域做出了贡献。我们的结果代表了在稠密图中构造高效距离查询的进展。此外，我们提出了一整个查询族，对于任意正整数$k$，它们使用$o(n^{1+1/k})$空间实现乘法伸缩因子$2k-1+\varepsilon$。