Multitree

File:Butterfly multitree.svg

The butterfly network, a multitree used in distributed computation, showing the subtree reachable from one of its nodes

See also: MultiTree

In combinatorics and order-theoretic mathematics, a multitree may describe either of two equivalent structures: a directed acyclic graph in which the set of nodes reachable from any node form a tree, or a partially ordered set that does not have four items a, b, c, and d forming a diamond suborder with a ≤ b ≤ d and a ≤ c ≤ d but with b and c incomparable to each other (also called a diamond-free poset).

Equivalence between directed acyclic graph and poset definitions

If G is a directed acyclic graph ("DAG") in which the nodes reachable from each vertex form a tree (or equivalently, if G is a directed graph in which there is at most one directed path between any two nodes, in either direction) then the reachability relation in G forms a diamond-free partial order. Conversely, if P is a diamond-free partial order, its transitive reduction forms a DAG in which the successors of any node form a tree.

Diamond-free families

A diamond-free family of sets is a family F of sets whose inclusion ordering forms a diamond-free poset. If D(n) denotes the largest possible diamond-free family of subsets of an n-element set, then it is known that

$2\le \lim_{n\to\infty} D(n) \Big/ \binom{n}{\lfloor n/2\rfloor}\le 2\frac{3}{11}$

and it is conjectured that the limit is 2.

Applications

Multitrees may be used to represent multiple overlapping taxonomies over the same ground set. If a family tree may contain multiple marriages from one family to another, but does not contain marriages between any two blood relatives, then it forms a multitree. In the context of computational complexity theory, multitrees have also been called strongly unambiguous graphs or mangroves; they can be used to model nondeterministic algorithms in which there is at most one computational path connecting any two states.

Related structures

A polytree, a directed acyclic graph formed by assigning an orientation to each edge of an undirected tree, may be viewed as a special case of a multitree.

The set of all nodes connected to any node u in a multitree forms an arborescence.

The word "multitree" has also been used to refer to a series-parallel partial order, or to other structures formed by combining multiple trees.