Generalized Connectors

An n-connector is an acyclic directed graph having n inputs and n outputs and satisfying the following condition: given any one-to-one correspondence between inputs and distinct outputs, there exists a set of vertex-disjoint paths that join each input to the corresponding output. It is known that the minimum possible number of edges in an n-connector lies between lower and upper bounds that are asymptotic to 3n log3 n and 6n log3 n respectively. A generalized n-connector satisfies the following stronger condition: given any one-to-many correspondence between inputs and disjoint sets of outputs, there exists a set of vertex-disjoint trees that join each input to the corresponding set of outputs. It is shown that the minimum number of edges in a generalized n-connector is asymptotic to the minimum number in an n-connector. Imagine an information transmission network intended to mediate between n sources of information and n users of this information. At any time, any of the users may wish to be connected with any of the sources; a user can be connected with only one source at a time, but many users may wish to be connected with the same source. This paper deals with an idealized version of the problem of designing a network capable of providing any such pattern of simultaneous connections. An (n, m)-graph is an acyclic directed graph with a set of n distinguished vertices called inputs and a disjoint set of m distinguished vertices called outputs. An n-graph is an (n, n)-graph. An n-connector is an n-graph satisfying the following condition: given any oneto-one correspondence between inputs and distinct outputs, there exists a set of vertex-disjoint paths that join each input to the corresponding output. (A path joining an input to an output is a directed path whose origin is the input and whose destination is the output.) Let c(n) denote the minimum possible number of edges in an nconnector; it is known that 3n log3 n c(n)<= 6n log3 n + O(n) (see Pippenger and Valiant [4, Remark 2.2.6]). A generalized n-connector is an n-graph satisfying the following stronger condition: given any one-to-many correspondence between inputs and disjoint sets of outputs, there exists a set of vertex-disjoint trees that join each input to the corresponding set of outputs. (A tree joining an input to a set of outputs is a directed tree whose root is the input and whose leaves are the outputs.) Let d(n) denote the minimum possible number of edges in a generalized n-connector; that d(n)<= 10n log2 n + O(n) for n a power of 2 is implicit in the work of Ofman [1]. Thompson [5] has recently shown that d(n)<= 12n log3 n + O(n) for n a power of 3. The object of this note is to show that d(n)=c(n)+O(n), * Received by the editors May 13, 1977. t Mathematical Sciences Department, IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598.