Logarithmic Embeddings into Probabilistic Spanning Trees for Distances and Capacities

We prove that every connected $n$-vertex graph admits a probabilistic embedding into its spanning trees with expected distance stretch $O(\log n)$. This settles the logarithmic stretch conjecture of Alon, Karp, Peleg, and West, whose spanning-tree game appeared at the 1991 DIMACS Workshop on On-Line Algorithms and in SIAM Journal on Computing in 1995. A weighted form of the theorem also yields a distribution over spanning trees with capacities exactly equal to their fundamental graph cuts. Every tree dominates every graph cut, every cut is approximated within $O(\log n)$ in expectation, and the same capacity distribution defines an $O(\log n)$-competitive fractional oblivious routing. The distance and capacity distributions may differ. Diamond graphs give matching logarithmic lower bounds for stretch, congestion, and cut distortion. For rational input, both distributions are constructed deterministically in polynomial time and use at most $m$ trees, where $m$ is the number of graph edges. We prove an $Ω(m/ρ)$ support lower bound for congestion $ρ$; in particular, $(1-o(1))m$ trees can be necessary for near-unit congestion, even when the optimum tends to one. Thus linear dependence on $m$ is unavoidable in this regime. The weighted theorem combines a protected-geodesic petal decomposition with a pathwise entropy bound for locally selected centers. A local averaging inequality derandomizes the radii, and an adaptive rational packing procedure constructs the distributions.