We initiate the study of approximation algorithms and computational barriers for constructing sparse \(\alpha\)-navigable graphs, an important principle underlying recent advances in graph-based nearest neighbor search. Given an \(n\)-point dataset \(P\) with an associated metric \(d\) and a parameter \(\alpha \ge 1\), the goal is to efficiently build the sparsest graph \(G = (P, E)\) that is \(\alpha\)-navigable: for every distinct \(s, t \in P\), there exists an edge \((s, u) \in E\) with \(d(u, t) \lt \texttt{d}(s, t)/\alpha\). We consider two natural sparsity objectives: minimizing the maximum out-degree and minimizing the total size (or equivalently, the average degree).

We study the space complexity of estimating the diameter of a subset of points in an arbitrary metric space in the dynamic (turnstile) streaming model. The input is given as a stream of updates to a frequency vector $x \in \mathbb{Z}_{\geq 0}^{n}$, where the support of x defines a multiset of points in a fixed metric space $\mathcal{M}=([n], \mathrm{d})$. The goal is to estimate the diameter of this multiset, defined as max $\left\{\mathrm{d}(i, j): x_{i}, x_{j} \gt \right. 0\}$, to a specified approximation factor while using as little space as possible. In insertion-only streams, a simple $O(\log n)$-space algorithm achieves a $\mathbf{2}$-approximation. In sharp contrast to this, we show that in the dynamic streaming model, any algorithm achieving a constant-factor approximation to diameter requires polynomial space. Specifically, we prove that a c-approximation to the diameter requires $n^{\Omega(1 / c)}$ space. Our lower bound relies on two conceptual contributions: (1) a new connection between dynamic streaming algorithms and linear sketches for scale-invariant functions, a class that includes diameter estimation, and (2) a connection between linear sketches for diameter and the minrank of graphs, a notion previously studied in index coding. We complement our lower bound with a nearly matching upper bound, which gives a c-approximation to the diameter in general metrics using $n^{O(1 / c)}$ space.

In the Max-k-Diameter problem, we are given a set of points in a metric space, and the goal is to partition the input points into k parts such that the maximum pairwise distance between points in the same part of the partition is minimized.

A linear circuit for computing an N × N matrix M is a circuit with N inputs corresponding to the entries of a vector x and N outputs corresponding to the entries of the transformed vector Mx, and where each gate computes a linear combination of its inputs. Each gate may have unbounded fan-in, and the size of the circuit is the number of wires. This model captures most known algorithms for computing linear transforms, and (in the constant-depth or 'synchronous' settings) is equivalent to factoring M as the product of sparse matrices. We give new, smaller constructions of constant-depth linear circuits for computing any matrix which is the Kronecker power of a fixed matrix. A standard argument (e.g., the mixed product property of Kronecker products, or a generalization of the Fast Walsh-Hadamard transform) shows that any such N × N matrix has a depth-2 circuit of size O(N1.5). We improve on this for all such matrices, and especially for some such matrices of particular interest: • For any integer q > 1 and any matrix which is the Kronecker power of a fixed q × q matrix, we construct a depth-2 circuit of size O(N1.5-aq), where aq > 0 is a positive constant depending only on q. No bound beating size O(N1.5) was previously known for any q > 2. • For the case q = 2, i.e., for any matrix which is the Kronecker power of a fixed 2 × 2 matrix, we construct a depth-2 circuit of size O(N1.446), improving the prior best size O(N1.493) [Alman, 2021]. • For the Walsh-Hadamard transform, we construct a depth-2 circuit of size O(N1.443), improving the prior best size O(N1.476) [Alman, 2021]. • For the disjointness matrix (the communication matrix of set disjointness, or equivalently, the matrix for the linear transform that evaluates a multilinear polynomial on all 0/1 inputs), we construct a depth-2 circuit of size O(N1.258), improving the prior best size O(N1.272) [Jukna and Sergeev, 2013]. Our constructions also generalize to improving the standard construction for any depth ≤ O (log N). Our main technical tool is an improved way to convert a nontrivial circuit for any matrix into a circuit for its Kronecker powers. Our new bounds provably could not be achieved using the approaches of prior work.