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)=minσ∈Sn∣E(Gλ)△σE(Gμ)∣
Nearby in the stack
makes that set a metric space. Its metric geometry and its computational complexity both issue from one identity:
q∗
is an affine function of the maximum of
∥X∥F2
over the contingency tables with margins
λ
and
μ
. Combinatorially, it yields two explicit
ℓ1
models: the vertex-mass metric
δ1
on sorted degree sequences, with
21δ1≤q∗<23δ1
and both constants optimal, and the block-energy metric
B
on the vectors
((2λi))i
, with
q∗≤B≤2q∗−1
by a per-table refinement measuring how far an alignment is from a block bijection. Hence
c1(Kn)≤2
, and an
O(nlogn)
-time algorithm returns an alignment of cost below
2q∗
with the certificate
q∗∈[⌈(B+1)/2⌉,B]
. The Euclidean distortion of the class is
c2(Kn)=Θ(n1/4)
; against it we measure the weighted dyadic sums
F(γ)
of the Ferrers staircase, of dimension below
4n
and computable in
O(n)
time. The unweighted member has distortion exactly
Θ(n1/4logn)
, while the critical weight
γ=41
improves this unconditionally to
O(n1/4(logn)1/4)
through an inverse energy inequality proved from the quantization of staircase jumps; removing the residual
(logn)1/4
is reduced to one inverse inequality on the realizable cone. Computationally, the same identity gives a classification: deciding
q∗(λ,μ)≤Q
is strongly NP-complete, evaluation is strongly NP-hard and admits no FPTAS unless
P=NP
, while the farthest alignment is polynomial-time solvable.