Consider N particles, which merge into clusters according to the rule: a cluster of size x and a cluster of size y merge at (stochastic) rate K(x; y)=N , where K is a specified rate kernel. This MarcusLushnikov model of stochastic coalescence, and the underlying deterministic approximation given by the Smoluchowski coagulation equations, have an extensive scientific literature. Some mathematical literature (Kingman's coalescent in population genetics; component sizes in random graphs) implicitly studies the special cases K(x; y) = 1 and K(x; y) = xy. We attempt a wide-ranging survey. General kernels are only now starting to be studied rigorously, so many interesting open problems appear. Keywords. branching process, coalescence, continuum tree, densitydependent Markov process, gelation, random graph, random tree, Smoluchowski coagulation equation Research supported by N.S.F. Grant DMS96-22859 1 Introduction Models, implicitly or explicitly stochastic, of coalescence (= coagulati...

This report is a contributed chapter to the Handbook of Theoretical Computer Science (North-Holland, 1990). Its aim is to describe the main mathematical methods and applications in the average-case analysis of algorithms and data structures. It comprises two parts: First, we present basic combinatorial enumerations based on symbolic methods and asymptotic methods with emphasis on complex analysis techniques (such as singularity analysis, saddle point, Mellin transforms). Next, we show how to apply these general methods to the analysis of sorting, searching, tree data structures, hashing, and dynamic algorithms. The emphasis is on algorithms for which exact "analytic models" can be derived.

Regard an element of the set \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; X i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent of Evans and Pitman (1997) is the \Delta-valued Markov process in which pairs of clusters of masses fx i ; x j g merge into a cluster of mass x i +x j at rate x i +x j . They showed that a version (X 1 (t); \Gamma1 ! t ! 1) of this process arises as a n !1 weak limit of the process started at time \Gamma 1 2 log n with n clusters of mass 1=n. We show this standard additive coalescent may be constructed from the continuum random tree of Aldous (1991,1993) by Poisson splitting along the skeleton of the tree. We describe the distribution of X 1 (t) on \Delta at a fixed time t. We show that the size of the cluster containing a given atom, as a process in t, has a simple representation in terms of the stable subordinator of index 1=2. As t ! \Gamma1, we establish a Gaussian limit for (centered and norm...

Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitude...

Grow a tree on n vertices by starting with no edges and successively adding an edge chosen uniformly from the set of possible edges whose addition would not create a cycle. This process is closely related to the classical random graph process. We describe the asymptotic structure of the tree, as seen locally from a given vertex. In particular, we give an explicit expression for the asymptotic degree distribution. Our results an be applied to study the random minimum-weight spanning tree question, when the edge-weight distribution is allowed to vary almost arbitrarily with n.

Regard an element of the set of ranked discrete distributions \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; P i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent is the \Delta-valued Markov process in which pairs of clusters of masses fx i ; x j g merge into a cluster of mass x i + x j at rate x i + x j . Aldous and Pitman (1998) showed that a version of this process starting from time \Gamma1 with infinitesimally small clusters can be constructed from the Brownian continuum random tree of Aldous (1991,1993) by Poisson splitting along the skeleton of the tree. In this paper it is shown that the general such process may be constructed analogously from a new family of inhomogeneous continuum random trees.

This paper is concerned with the problem of computing spanning tree (MST) for n points in a p-dimensional space where the "distance" between each pair of points i and j satisfies the relationship' dq max {Ixti - xtql} , where xki is the coordinate of object i along the ktti dimension. This relationship is clearly satisfied by all Minkowski metrics dq = [ Ixki - xnjl r] x/r, r > 1

This paper presents an analysis of a simple treestructured disjoint set Union-Find algorithm, and shows that this algorithm requires between n and 2n steps on the average to execute a sequence of n Union and Find instructions, assuming that each pair of existing classes is equally likely to be merged by a Union instruction. Union-Find algorithms are useful in the solution to a number of problems which require the construction of equivalence classes of a set of elements. The common parts of algorithms for constructing spanning trees of graphs, processing EQUIVALENCE statements, and determining the equivalence of finite automata are es sentially the operations of combining two equivalence classes into one new equivalence class, and of finding the equivalence class to which an element belongs. A number of algorithms for executing Union-Find instructions on a RAM are presented in ref. [1]. The fastest of these use tree structures by representing each equivalence class by a tree whose vertices represent the elements of the equivalence class. The simplest such tree-structured algorithm, which is the subject of the following analysis, executes the instruction Union (A, B) by making the root of the tree representing the equiv alence class B a son of the root of the tree representing equivalence class A. The instruction Find (x) is executed by tracing up the tree from the vertex representing x un ‘ til the root of the tree representing the class containing x is reached, and then returning the name of the equiva lence class stored there. This algorithm requires at worst

. Consider N particles, which merge into clusters according to the rule: a cluster of size x and a cluster of size y merge at (stochastic) rate K(x; y)=N , where K is a specified rate kernel. This Marcus-Lushnikov model of coalescence, and the underlying deterministic approximation provided by the Smoluchowski coagulation equations, have an extensive scientific literature. A recent reformulation is the general stochastic coalescent, whose state space is the infinite-dimensional simplex (the state x = (x i ; i 1) represents unit mass split into clusters of masses x i ), and which evolves by clusters of masses x i and x j coalescing at rate K(x i ; x j ). Existing mathematical literature (Kingman's coalescent, component sizes in random graphs, fragmentation of random trees) implicitly studies certain special cases. Recent work has uncovered deeper constructions of special cases of the stochastic coalescent in terms of Brownian-type processes. Rigorous study of general kernels has only j...

Abstract. 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 asymptotic 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.