发表机构
Beihang University（北京航空航天大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Dasgupta层次聚类目标，通过几何方法分析基数约束的代价，证明最坏情况线性差距，但在稀疏图上可显著缩小。
AI 中文摘要
层次聚类的代价可以用一个超度量来表示，其最低公共祖先标签为簇的基数。我们将这一已知表示与相似图的最短路径几何联系起来。对于连通的支撑图 $G$，设 $d_G$ 为其单位长度最短路径度量，而边权重仅进入目标函数。我们证明平移后的 Dasgupta 最优值恰好是支配 $d_G$ 的基数可实现超度量的最小边加权代价。连通化引理将该问题及其自由标记的支配超度量松弛置于同一类连通二叉层次结构上，分别以基数和图直径标记。作为尖锐基线，我们确定了基数可实现性的最坏情况代价：在每个 $n$ 顶点实例上，两个最优值之比至多为 $(2n-1)/3$，在未加权完全图上取等号；标准未平移目标的尖锐因子为 $2(n+1)/3$。我们的主要结构结果通过连通平衡割定义的遗传加权碎片化轮廓来界定这一差距。均匀局部控制给出 $O(\log n)$ 的差距，多项式衰减给出常数差距，且对数阶即使在最大度为 $3$ 的未加权树上也是紧的。在局部正则有界度树上，层次结构可以在 $O(n\log n)$ 时间内构造。能量分解和几何密度界提供了支持性的实例敏感估计。因此，基数标签具有不可避免的线性最坏情况，但在自然稀疏图类上允许显著更小的界限。
英文摘要
The cost of a hierarchical clustering can be represented by an ultrametric whose lowest-common-ancestor labels are cluster cardinalities. We relate this known representation to the shortest-path geometry of a similarity graph. For a connected support graph $G$, let $d_G$ be its unit-length shortest-path metric and let the edge weights enter only the objective. We prove that the shifted Dasgupta optimum is exactly the minimum edge-weighted cost of a cardinality-realizable ultrametric that dominates $d_G$. Connectedification lemmas put this problem and its freely labeled dominating-ultrametric relaxation on the same class of connected binary hierarchies, labeled respectively by cardinality and graph diameter. As a sharp baseline, we determine the exact worst-case price of cardinality realizability: on every $n$-vertex instance the ratio of the two optima is at most $(2n-1)/3$, with equality on the unweighted complete graph; the sharp factor for the standard unshifted objective is $2(n+1)/3$. Our principal structural result bounds this gap by a hereditary weighted fragmentation profile defined through connected balanced cuts. Uniform local control gives an $O(\log n)$ gap, polynomial decay gives a constant gap, and the logarithmic order is tight even for unweighted trees of maximum degree $3$. On locally regular bounded-degree trees, the hierarchy can be constructed in $O(n\log n)$ time. An energy decomposition and a geometric density bound provide supporting instance-sensitive estimates. Thus the cardinality label has an unavoidable linear worst case but admits substantially smaller bounds on natural sparse graph classes.
