An I/O Efficient Algorithm for Minimum Spanning Trees

Abstract

An $O(\mathrm{Sort}(E)\cdot \log \log _{E/V} B)$ I/Os algorithm for computing a minimum spanning tree of a graph $G=(V,E)$ is presented, where $\mathrm{Sort}(E)=(E/B)\log _{M/B}(E/B)$, M is the main memory size, and B is the block size. This improves on the previous bound of $O(\mathrm{Sort}(E) \cdot \log \log (VB/E))$ I/Os by Arge et al. for all values of V, E and B, for which I/O optimality is still open. In particular, our algorithm matches the lowerbound $\varOmega (E/V \cdot \mathrm{Sort}(V))$, when $E/V \ge B^{\epsilon }$ for a constant $\epsilon > 0$, an $O(\log \log B)$ factor improvement over the algorithm of Arge et al. Our algorithm can compute the connected components too, for the same number of I/Os, which is an improvement on the best known upper bound.