Streaming Euclidean MST to a Constant Factor

We study streaming algorithms for the fundamental geometric problem of computing the cost of the Euclidean Minimum Spanning Tree (MST) on an n-point set X ⊂ R d . In the streaming model, the points in X can be added and removed arbitrarily, and the goal is to maintain an approximation in small space. In low dimensions, (1+ ) approximations are possible in sublinear space [Frahling, Indyk, Sohler, SoCG '05]. However, for high dimensional spaces the best known approximation for this problem was Õ(log n), due to [Chen, Jayaram, Levi, Waingarten, STOC '22], improving on the prior O(log 2 n) bound due to [Indyk, STOC '04] and [Andoni, Indyk, Krauthgamer, SODA '08]. In this paper, we break the logarithmic barrier, and give the first constant factor sublinear space approximation to Euclidean MST. For any ≥ 1, our algorithm achieves an Õ( -2 ) approximation in n O( ) space.

We complement this by proving that any single pass algorithm which obtains a better than 1.10-approximation must use Ω( √ n) space, demonstrating that (1 + ) approximations are not possible in high-dimensions, and that our algorithm is tight up to a constant. Nevertheless, we demonstrate that (1 + ) approximations are possible in sublinear space with O(1/ ) passes over the stream. More generally, for any α ≥ 2, we give a α-pass streaming algorithm which achieves a (1 + O( log α+1 α )) approximation in n O( ) d O(1) space. All our streaming algorithms are linear sketches, and therefore extend to the massivelyparallel computation model (MPC). Thus, our results imply the first (1 + )-approximation to Euclidean MST in a constant number of rounds in the MPC model. Previously, such a result was only known for low-dimensional space [Andoni, Nikolov, Onak, Yaroslavtsev, STOC '15], or required either O(log n) rounds or a O(log n) approximation.