Merging costs for the additive Marcus-Lushnikov process, and Union-Find algorithms

Starting with a monodisperse configuration with n size–1 particles , an additive Marcus–Lushnikov process evolves until it reaches its final state (a unique particle with mass n). At each of the n − 1 steps of its evolution , a merging cost is incurred, that depends on the sizes of the two particles involved, and on an independent random factor. This paper studies the asymp-totic behaviour of the cumulated costs up to the kth clustering, under various regimes for (n, k), with applications to the study of Union–Find algorithms.