AI 中文总结
该研究针对簇图的商编辑距离，推导其度量代理、多尺度嵌入与复杂性，提出相关模型与算法，证明其强NP完全性等计算性质，改进了欧氏失真上界。
AI 中文摘要
n个顶点的簇图是完全图的不交并，其同构类对应n的整数分拆，商编辑距离q^*(λ,μ)=min_{σ∈S_n}|E(G_λ)△σE(G_μ)|使该集合成为度量空间。其度量几何与计算复杂性均源于同一恒等式：q^*是行和与列和为λ、μ的列联表的Frobenius范数平方最大值的仿射函数。组合上，它产生两个显式ℓ₁模型：排序度序列上的顶点质量度量δ₁，满足½δ₁≤q^*<¾δ₁且两个常数均为最优；以及向量(λ_i选2)上的块能量度量B，通过测量对齐与块双射的偏差，满足q^*≤B≤2q^*-1，由此得c₁(𝒦ₙ)≤2，存在O(n log n)时间算法返回成本低于2q^*的对齐，且q^*∈[⌈(B+1)/2⌉,B]作为证明。该类的欧氏失真为c₂(𝒦ₙ)=Θ(n^{1/4})，针对此，我们测量Ferrers阶梯的加权二进和F^{(γ)}，其维度低于4n，可在O(n)时间计算。未加权成员的失真恰好为Θ(n^{1/4}√log n)，而临界权重γ=¼通过阶梯跳跃量化证明的逆能量不等式，无条件将其改进为O(n^{1/4}(log n)^{1/4})；消除剩余的(log n)^{1/4}简化为可实现锥上的一个逆不等式。计算上，同一恒等式给出分类：判定q^*(λ,μ)≤Q是强NP完全问题，评估是强NP难问题且除非P=NP否则不存在FPTAS，而最远对齐可在多项式时间求解。
英文摘要
The cluster graphs on $n$ vertices, the disjoint unions of complete graphs, have the integer partitions of $n$ as their isomorphism classes, and the quotient edit distance $q^*(λ,μ)=\min_{σ\in S_n}|E(G_λ)\triangleσE(G_μ)|$ makes that set a metric space. Its geometry and its complexity both issue from one identity: $q^*$ is an affine function of the maximum of $\lVert X\rVert_F^2$ over the contingency tables with margins $λ$ and $μ$. Our main result is an explicit optimal embedding. The weighted dyadic sums of the Ferrers staircase, taken at the critical exponent $\frac14$, give a map $F_n$ into $\ell_2^{\,<4n}$ that acts on a single partition and is computable in $O(n)$ time, and its distortion is $Θ(n^{1/4})$. That order is optimal, since $c_2(\mathcal K_n)=Θ(n^{1/4})$: the lower half follows from a $Θ(\sqrt n)$-dimensional Hamming cube of partitions and Enflo's theorem, so the determination needs no other external input. The analytic core is a scale-free inverse inequality for every integer sequence with $v(1)=v(N+1)=0$ and $v(s)-v(s+1)\in s\mathbb Z$: its critical dyadic energy is at least $\lVert v\rVert_1^2/(63504\sqrt{\mathrm{TV}(v)})$. Combinatorially the same identity yields two explicit $\ell_1$ models, the vertex-mass metric on sorted degree sequences with $\frac12δ_1\le q^*<\frac32δ_1$ and the block-energy metric with $q^*\le B\le2q^*-1$, both constants optimal; hence $c_1(\mathcal K_n)\le2$, and an $O(n\log n)$-time algorithm returns an alignment of cost below $2q^*$ carrying the certificate $q^*\in[\lceil(B+1)/2\rceil,B]$. Computationally, deciding $q^*(λ,μ)\le Q$ is strongly NP-complete and admits no FPTAS, while the farthest alignment is polynomial-time solvable. The best constant in the inverse inequality remains open; an exactly solvable chirp family caps it at $\frac23$.
Comments49 pages, 6 figures. v2: the critical inverse-energy conjecture of v1 is now proved, as a scale-free inverse theorem on the full cone of closed quantized integer sequences (Theorem 6.2)
