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 deals with the asymptotic behaviour of the cumulated costs up to the $k$th clustering, under various regimes for $(n,k)$, with applications to the study of Union--Find algorithms.Comment: 28 pages, 1 figur

Probabilistic analysis of disjoint set union algorithms

A number of open questions are settled about the expected costs of two disjoint set Union and Find algorithms raised by Knuth and Schonhage. This paper shows that the expected time of the Weighted Quick-Find (QFW) algorithm to perform (n-1) randomly chosen unions is cn+o(n/log n), where c = 2.0847.... Through an observation of Tarjan and Van Leeuwen in [J. Assoc. Comput. Mach., 22 (1975), pp. 215-225] this implies linear time bounds to perform O(n) unions and finds for a class of other union-find algorithms. It is also proved that the expected time of the Unweighted Quick-Find (QF) algorithm is n2/8+O(n(log n)2). The expected costs of QFW and QF are analyzed when fewer than (n-1) unions are performed. Among other results, for QFW it is shown that the expected cost of m = o(n) randomly chosen unions is m(1+o(1)). If m = αn/2, where α≤e-2, this cost is m(1+ε(α)+o(1)), where ε(α)→0 as α→0 and ε(e-2)≤.026. For QF, the expected cost of n/2-n2/3(log n)2/3 randomly chosen unions is O(n log n)