发表机构
Wuhan University of Technology（武汉理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探讨精确带标记有向最短路径的资源复杂度，提出带费用的加法-比较成本基准，证明存在统一最优数值策略，高效常竞争导航仍待解决。
AI 中文摘要
我们研究精确单源最短路径问题，其输出仅为已实现的带标记距离向量（DIST），而非距离排序。在完全确定性比较-加法模型中，对于每个固定有向拓扑，最坏情况下的最小加法次数恰好等于根顶点顺序上的正向非源端点类的最大数量ρ_fwd；该下界允许自适应控制、文字及任意混合和。此算术规律与有向无环图（DAG）上的比较最优值一致，其中完整资源区域为精确矩形。循环会破坏这种一致性：一个双辐条共享中心图的坐标最优值为(4,2)，但在两次加法预算下需要五次比较；其k辐条扩展在加法最优值处需要k log₂k + O(k)次比较，且存在熵紧的确定性权衡关系C_{k+r}^*(H_k)=Θ(k+Λ_{k,r})，其中Λ_{k,r}=log₂(k!/[r!(r+1)^{k-r}])，当Λ_{k,r}/k→∞时首项常数为1。由于两个坐标最小值不必属于同一程序，这些冲突引出了同一程序基准OPT_DIST=inf_A sup_w(C_A(w)+P_A(w))。精确转录锥博弈产生一个统一解释器，其带费用的加法-比较成本在每个拓扑上都等于OPT_DIST；其最优动作可在多项式空间中合成，但可能需要指数时间。最后，通过核心活性归约和当前确定性有向单源最短路径（directed-SSSP）界，得到高效统一的带费用操作界O(OPT_DIST√(log(2+OPT_DIST)log log(4+OPT_DIST)))。因此存在统一的最优数值策略，而高效的常竞争导航仍待解决。
英文摘要
We study exact single-source shortest paths when the output is only the materialized labeled distance vector ($\mathrm{DIST}$), rather than a distance order. In the full deterministic comparison-addition model, the minimum worst-case number of additions on every fixed directed topology is exactly the maximum number $ρ_{\mathrm{fwd}}$ of forward nonsource endpoint classes over rooted vertex orders; the lower bound permits adaptive control, literals, and arbitrary mixed sums. This arithmetic law aligns with the comparison optimum on DAGs, where the full resource region is an exact rectangle. Cycles destroy that alignment: a two-spoke shared-hub graph has coordinatewise optima $(4,2)$ but requires five comparisons at the two-addition budget. Its $k$-spoke extension forces $k\log_2 k+O(k)$ comparisons at the addition optimum and has an entropy-tight deterministic tradeoff $C_{k+r}^*(H_k)=Θ(k+Λ_{k,r})$, where $Λ_{k,r}=\log_2(k!/[r!(r+1)^{k-r}])$, with leading constant one when $Λ_{k,r}/k\to\infty$. Because the two coordinatewise minima need not belong to one program, these conflicts lead to the same-program benchmark $\operatorname{OPT}_{\mathrm{DIST}}=\inf_A\sup_w(C_A(w)+P_A(w))$. An exact transcript-cone game yields one uniform interpreter whose charged addition-comparison cost equals $\operatorname{OPT}_{\mathrm{DIST}}$ on every topology; its optimal actions are synthesizable in polynomial space but may require exponential time. Finally, an active-core reduction and the current deterministic directed-SSSP bound give an efficient uniform $O\!\bigl(\operatorname{OPT}_{\mathrm{DIST}}\sqrt{\log(2+\operatorname{OPT}_{\mathrm{DIST}})\log\log(4+\operatorname{OPT}_{\mathrm{DIST}})}\bigr)$ charged-operation bound. Thus optimal numerical policies exist uniformly, while efficient constant-competitive navigation remains open.
Comments45 pages, including appendices and references. One endpoint theorem is verified by an exact computer-assisted finite certificate
