random mappings with constraints on coalescence ... - Project Euclid

PACIFIC JOURNAL OF MATHEMATICS Vol. 103, No. 2, 1982

RANDOM MAPPINGS WITH CONSTRAINTS ON COALESCENCE AND NUMBER OF ORIGINS JAMES ARNEY AND EDWARD A.

BENDER

In § 2 we tabulate for easy reference probability distributions associated with some functions of random mappings on large sets (e.g., number of points on cycles, size of the component containing x) when the number of immediate predecessors of each point is required to lie in some set &. Our results allow the number of origins to be restricted, a useful constraint in some shift register situations. Although limiting the number of immediate predecessors to {0,1,2} and constraining the number of origins is in some ways a poor model for random shift registers, we show in §§ 3 and 4 that most of the tabulated results fit shift register data quite well. Derivations of our results are given in §§ 5 through 9. 1* Introduction and terminology* Let X be an n element set and let φ be a mapping from X to X. We can picture φ as a directed graph with vertices X and edges (x, φ(x)), xeX. We use the graph theory terminology component and cycle. The component containing x is denoted K(x). For sufficiently large m, φm{x) lies on a cycle denoted G{x). The least m such that φm(x) e C(x) is called the tail length of x and written t(x). The six length of x, written s(x), is the length of the path from x to the first repeat; i.e., t{x) plus the size of C(x). If x lies on a cycle, the set of u e X such that φnu)(u) — x is called the tree of x and is written T{x). It consists precisely of those elements which first "hit" a cycle at x when φ is iterated. If xeT(u), we define T(x) = T(u). The elements of φ~\x) are called the immediate predecessors of x. The number of points with r immediate predecessors is denoted by nr(φ). The elements of U φ~%x), where the union ranges over all i ^ 0 are called the predecessors of x. The number of points with r predecessors is denoted by Nr(φ). A point without predecessors is called an origin. The number of origins is no(φ) — N^φ). Let Pi(φ) = n^)/n, the probability that a point chosen at random has exactly i immediate predecessors, then we call Σ* (i — ^Viiψ) = MΦ) the coefficient of coalescence of φ. Many authors have employed this concept. Among the equivalent definitions are (a) the variance of the number of predecessors of a random point, (b) (n — 1) times the probability that two distinct elements have the same immediate successor, 269

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JAMES ARNEY AND EDWARD A. BENDER

(c) the expected number of other immediate predecessors to the immediate successor of a randomly selected point. The equivalence of these four definitions is easily established by simple counting arguments. We use #(S) to denote the cardinality of S. For example, %(C(x)) is the length of the cycle of x. Let ^ be a set of nonnegative integers including zero and at least one integer greater than one. Let τ be the set of mappings from X to X such that the vertices of the associated directed graphs all have indegrees lying in 3fm In other words, if φ e τ and nt(φ) Φ 0, then i e 2$. Three special types of 3ί have been considered in the literature: (a) all nonnegative integers, (b) the integers 0 and k, (c) the integers between 0 and k inclusive. Rubin and Sitgreaves [9] and Harris [6] discuss (a). Rubin and Sitgreaves also discuss (b) and (c), and Harris also discusses (a) when 1-cycles are forbidden. We establish results for all 3f. The total number of vertices in the graph of φ can be computed by counting the immediate predecessors of each point. This gives us n = Σ* %0-P)It follows that n must be a multiple of gcd(£&), the greatest common divisor of the elements in 3ί. We can ask for the distributions of various quantities over τ; however, we must be clear just what we are asking for. One possibility is to define a function f(φ) and ask for its distribution assuming that all φ 6 τ are equally likely. This gives us information about the overall appearance of a random map. The number of cycles of a map is an example of this. Another possibility is to define a function g(x9 φ) and ask for its distribution assuming that all (x, φ) e X x τ are equally likely. This gives us information about the appearance of a random map when viewed from a radom point. The tail length of a point is an example of this. We usually write g(x) instead of g(x, φ) to emphasize the fact that we are choosing a random point. When f or g can be obtained simply by counting (e.g., the cycles of 1

7

tail length t(x)

same mean and distribution as #(C(a?))

5

six length s(x)

same mean and distribution as c(φ)

6

where the domain of j in the maximum is restricted for #(JBL(OJ)). The last column in Table II gives the section where the results for that entry are proved. The variance for nr(