A (Very) Nearly Optimal Sketch for $k$-Edge Connectivity Certificates
Pachara Sawettamalya, Huacheng Yu
TL;DR
This work addresses certifying $k$-edge-connectivity in dynamic graph streams by introducing a single-pass algorithm that uses $O\bigl(n \log^2 n \cdot \max\{k, \log n \log k\}\bigr)$ bits of space with high probability. The core idea replaces $k$ independent AGM sketches with a logarithmic hierarchy of subsketches, using a $1 + \log_2 k$ structure and a $k$-support-find subroutine to fix all small cuts and elevate connectivity from $1$ to $k$. The results nearly close the space-competitive gap for sublinear $k$, achieving a tight $\Theta(k n \log^2 n)$ bound when $k = \Omega(\log n \log \log n)$ and reducing the remaining gap to $O(\log \log n)$ for smaller $k$, with a corollary in distributed sketching. The approach combines an AGM sketch for a spanning forest with a geometric hierarchy of $2^i$-support-find sketches to systematically fix cuts, enabling efficient certificates with favorable space complexity. These findings advance practical certifying tools in dynamic streaming and shed light on fundamental limits for edge-connectivity certificates in near-linear space regimes.
Abstract
In this note, we present a simple algorithm for computing a \emph{$k$-connectivity certificate} in dynamic graph streams. Our algorithm uses $O(n \log^2 n \cdot \max\{k, \log n \log k\})$ bits of space which improves upon the $O(kn \log^3 n)$-space algorithm of Ahn, Guha, and McGregor (SODA'12). For the values of $k$ that are truly sublinear, our space usage \emph{very nearly} matches the known lower bound $Ω(n \log^2 n \cdot \max\{k, \log n\})$ established by Nelson and Yu (SODA'19; implicit) and Robinson (DISC'24). In particular, our algorithm fully settles the space complexity at $Θ(kn \log^2{n})$ for $k = Ω(\log n \log \log n)$, and bridges the gap down to only a doubly-logarithmic factor of $O(\log \log n)$ for a smaller range of $k = o(\log n \log \log n)$.