Linear Expected Time of a Simple Union-Find Algorithm

This paper presents an analysis of a simple tree-structured 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 * e(s 2) steps to execute s instructions on s elements [I], and may in fact require this many, since a chain of length e(s) can be created with s/2 union instructions, and s/2 subsequent finds can cost e(s) each. Two improvements to this algorithm are " weighted unions " , which make the root of the tree representing the smaller of the two classes being merged a son of the root of the larger, and " collapsing finds " , which make the vertices encountered on the way up to the root sons of the root. Either of these two improvements reduces the worst-case running time to e[s in(s)] steps forsin structions on s elements [1], and Tarjan [3] has shown that using both improvements produces a e(s. a(s)) run-fling time where a(s) is related to a functional inverse of Ackermann's function. …