发表机构
UP FAMNIT, University of Primorska（波雷切大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对最短路径算法实例最优性问题，修正双向Dijkstra算法分析的缺陷，提出修改后的双向Dijkstra算法，证明其在加权图中实例最优，还在无权图及简单图的相关开放问题上取得进展。
AI 中文摘要
Haeupler、Hladík、Rozhon、Tarjan和Tětek近期关于最短路径算法实例最优性的研究，已得出加权与无权图中Dijkstra算法和双向Dijkstra算法的若干结果。受这些结果启发，我们重新研究标准查询模型下最短st-路径算法的实例最优性问题，发现单向和双向Dijkstra算法实例最优性分析中的若干问题并给出对应反例。随后，我们对双向Dijkstra算法提出最小的简单修改，证明该变体在加权设置下是实例最优的。此外，我们重新研究无权情况，给出下界的简化证明，表明没有算法能达到优于O(Δ)的实例最优性（Δ为图的最大度），并讨论该结果对近似算法的意义。最后，我们在简单图的实例最优性这一开放问题上取得进展：若问题实例满足n≥m/16（n为节点数，m为算法查询的边数），则该算法是常数因子最优的；特别地，当图的最大度不超过已探索边数的平方根时，该算法也表现出常数因子最优性。
英文摘要
Recent work by Haeupler, Hladík, Rozhon, Tarjan, and Tětek on the instance optimality of shortest-path algorithms established several results concerning Dijkstra's algorithm and bidirectional Dijkstra's algorithm in weighted and unweighted graphs. Motivated by these results, we revisit the question of instance optimality for shortest $st$-path algorithms in the standard query model. We identify several issues in the analysis of the instance optimality of both unidirectional and bidirectional Dijkstra's algorithms and provide corresponding counterexamples. We then propose a minimal simple modification of the bidirectional Dijkstra algorithm and prove that the resulting variant is instance optimal in the weighted setting. Furthermore, we revisit the unweighted case, provide a simplified proof of the lower bound showing that no algorithm can achieve instance optimality up to a factor better than $O(Δ)$, where $Δ$ denotes the maximum degree of the graph, and discuss the implications of this result for approximation algorithms. Finally, we make progress on the open problem of instance optimality in simple graphs. We show that if the problem instance satisfies $n\ge m/16$, where $n$ is the number of nodes and $m$ is the number of edges queried by our algorithm, then it is optimal up to a constant factor. Additionally, we show instance optimality for a broad class of instances, in particular when the largest degree in the graph is at most the square root of the number of explored edges, our algorithm exhibits optimality up to a constant factor.
Comments25 pages, 2 figures
