AI 中文总结
本文针对幂律图提出无需预处理的剪枝双向搜索算法，其运行时间为次线性，近似比最高1.05，实验显示比现有方法快1.84-7.76倍。
AI 中文摘要
在大型图中计算最短路径一直是且仍是一个具有实际意义的基础问题。虽然已有许多算法被提出用于高效计算顶点对之间的最短路径，但其中许多算法（基于索引的方法）需要大量预处理，而另一些算法（基于遍历的方法）时间复杂度更高。本文针对参数 $\beta\in[2,3)$ 的幂律图，提出并分析了一种简单的次线性近似算法——剪枝双向搜索（Pruned Bidirectional Search, PBS）：该算法无需任何预处理，却能达到与轻量型基于索引的算法（索引规模为线性或次线性）相当的性能，即PBS的运行时间为 $O(n^{(1-1/\log\log n)/2})$，且大概率能返回长度不超过最短路径 $\frac{41}{32}$ 的路径。此外，若允许进行 $n^{\Theta(2-1/\log\log n)}$ 时间的预处理，其查询时间可提升至 $n^{\Theta(1/\log\log n)}$。我们通过在真实世界和合成幂律图上开展实验补充了理论结果，实验表明PBS通常比现有替代方法快1.84倍至7.76倍，同时实现的近似比最高为1.05。
英文摘要
Computing shortest paths in large graphs is, and remains, a fundamental and practically motivated problem. While many algorithms were proposed to calculate shortest path between pairs of vertices efficiently, many of them (index-based methods) require substantial preprocessing, while others (traversal-based methods) have higher time complexity. In this paper, we propose and analyze Pruned Bidirectional Search (PBS), a simple sublinear approximation algorithm for power-law graphs with parameter $β\in[2,3)$: our algorithm does not require any preprocessing, yet exhibits performance comparable to light index-based algorithms (of linear or sublinear index size): that is, PBS runs in time $O(n^{(1-1/\log\log n)/2})$ and, with high probability, returns a path with length within $\frac{41}{32}$ of the shortest path. Moreover, if one does allow a $n^{Θ(2-1/\log\log n)}$-time preprocessing step, its query time improves to $n^{Θ(1/\log\log n})$. We complement our theoretical results by experiments on both real-world and synthetic power-law graphs, which show that PBS is typically $1.84\times$-$7.76\times$ times faster than existing alternatives, while achieving an approximation ratio at most 1.05.
