Given a graph $G=(V,E)$ with $n$ vertices and $m$ edges, let $C(d)=|\{v\in V:°(v)\ge d\}|$ denote its complementary cumulative degree histogram (CCDH), and let $h$ be the $h$-index of $G$. The goal is to compute a $(\varepsilon_D,\varepsilon_R)$-bi-criteria multiplicative approximation of $C$, where $\varepsilon_D$ and $\varepsilon_R$ control the approximation in the domain and range, respectively. In this work, we study the complexity of this problem in the streaming model and an augmented query model that provides access to uniformly random edge queries in addition to standard local queries. We design a model-agnostic algorithm that combines uniformly sampled vertices and edges to estimate the CCDH. Ignoring the dependence on the approximation parameters and polylogarithmic factors in $n$, the complexity in the streaming model is $Θ(m/h)$: the upper bound holds for a single-pass algorithm, while the lower bound holds even for multi-pass algorithms. In the query model, the complexity is $Θ(m/h)$ for adaptive algorithms. For non-adaptive algorithms, we give an upper bound of $O(n/h+m/h)$ and establish lower bounds through a reduction from estimating the number of distinct elements in a multiset when sampling without replacement. Our results resolve an open problem in sublinear graph algorithms posed in Sublinear.info (Problem 98). Finally, we validate our algorithm through extensive empirical evaluations on real-world graphs.

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