简单图上双向Dijkstra算法的实例最优性
Instance-Optimality of Bidirectional Dijkstra on Simple Graphs
浏览论文内容
中文总结 AI 辅助
该研究解答了简单图上双向Dijkstra算法的实例最优性问题，明确其在部分模型下最优、部分模型下非最优，并给出最优性比值下界与对数因子内最优的结论。
中文摘要 AI 辅助
我们研究具有正实值边权的图上的最短路径问题，给定源顶点$s$和目标顶点$t$，目标是计算从$s$到$t$的最短路径长度，我们尤其关注能以亚线性时间求解的实例。最近，Haeupler、Hladík、Rozhoň、Tarjan和Tětek证明，在考虑算法查询的顶点和边数量时，（某一版本的）双向Dijkstra算法在正权多重图（包括有向和无向）上是实例最优的。然而，多重图并非最短路径问题的标准设定，该问题通常在无环、无平行边的简单图上表述，因此他们留下了一个开放问题：双向Dijkstra在简单加权图上是否仍保持实例最优。我们解答了该问题，但对于简单图，答案更为复杂，取决于设定：我们证明，在入射边为随机顺序的顺序无关模型下，双向Dijkstra在简单无向加权图上仍是实例最优的；相反，在入射边有给定顺序的顺序相关模型下，双向Dijkstra并非实例最优。对于简单有向加权图，我们证明其在顺序无关和顺序相关模型下均非实例最优，两种情况下的实例最优性比值均差$\theta(m/n)$因子。我们进一步证明，在顺序相关模型下，或在$m=O(n\backslash sqrt{n})$的顺序无关模型下，不存在算法能达到$o(m/n)$的实例最优性比值。有利的一面是，上述结果表明，对于所有满足$m/n=\backslash log^{O(1)}n$的稀疏有向和无向图，双向Dijkstra在对数因子范围内是实例最优的。
英文摘要
We study the shortest-path problem on graphs with positive real-valued edge weights. Given a source vertex $s$ and a target vertex $t$, the goal is to calculate the length of the shortest path from $s$ to $t$. We are particularly interested in instances that can be solved in sublinear time. Recently, Haeupler, Hladík, Rozhoň, Tarjan, and Tětek proved that (a version of) the bidirectional Dijkstra's algorithm is instance-optimal on positively weighted multigraphs, both directed and undirected, considering the number of vertices and edges queried by the algorithm. However, multigraphs are not the canonical setting for the shortest-path problem. The problem is typically formulated on simple graphs without loops and parallel edges. They therefore left as an open problem whether bidirectional Dijkstra remains instance-optimal on simple weighted graphs. We answer this question, but for simple graphs, the answer is more complex, depending on the setting. We show that bidirectional Dijkstra is still instance-optimal on simple undirected weighted graphs under the order-oblivious model, where incident edges are given in a random order. In contrast, under the order-dependent model, where incident edges have a given order, we show that bidirectional Dijkstra is not instance-optimal. For simple directed weighted graphs, we show that bidirectional Dijkstra is not instance-optimal under either the order-oblivious or the order-dependent model, being off by a factor of $Θ(m/n)$ in both cases. We further show that no algorithm can have instance-optimality ratio $o(m/n)$ under the order-dependent model, or under the order-oblivious model when $m=O(n\sqrt{n})$. On the positive side, the above results imply that bidirectional Dijkstra is instance-optimal up to logarithmic factors on all sparse directed and undirected graphs satisfying $m/n=\log^{O(1)} n$.
