Antimatroid, The

thoughts on computer science, electronics, mathematics

One tree, many possibilities

Introduction

Given the finite set $S$ and the surjective function $\sigma$, Let $T_{n}\left(S, \sigma\right)$ be a complete $|S|$-ary tree of depth n such that $T_{0}\left(S, \sigma\right) = \emptyset$, $T_n(S, \sigma) = T_{n - 1}(S, \sigma) \{ \left( T_{n - 1}(S, \sigma) \oplus s \right) | s \in \sigma(S) \}$. For example, Let $A = \{0, 1\}$ and $\\f = S \to S$, $T_{2}\left(A, \\f\right) = \emptyset \{ \left(\emptyset, 0\right) \{ \left(\emptyset, 0, 0\right), \left(\emptyset, 0, 1\right) \}, \left(\emptyset, 1\right) \{ \left(\emptyset, 1, 0\right), \left(\emptyset, 1, 1\right) \} \}$. Pictured bellow is $T_{3}\left(A, \\f \right)$ with omitting $\emptyset$ for brevity.

It should be evident from the example that the set of nodes at depth $n$ is the $n$-ary Cartesian product of $S, S^{n} = S \times \ldots \times S$.

Now, if we change $\\f$ to now be $\\f : S \to \{ s | s \in S \wedge \nu(s) \neq \nu\left(parent\left(s\right)\right)\}$, where $\nu : S \leftrightarrow \mathbb{N}_{0}$, the set of all leaf nodes at depth $|S|$ is the set of permutations of $S, S!$. Extending the example above, $T_{2} \left( A, \\f \right) = \{ \{ \left( \emptyset, 0, 1 \right) \}, \{ \left( \emptyset, 1, 0 \right) \} \}$.

Without changing the definition of $\\f$, it is possible to produce all possible $k$-permutations from $T, P(S, k)$ by selecting all nodes at depth $k$.

At this point, if we make a small change to $\\f : S \to \{ s | s \in S \wedge \nu\left(s\right) > \nu\left( parent\left(s\right)\right)\}$, every node in $T$ now represents the power set of $S, 2^{S}$. Again, using the example from above, $T_{2} \left( A, \\f \right) = \emptyset \{ \left( \emptyset, 0 \right), \left( \emptyset, 1 \right) \{ \left( \emptyset, 1, 1 \right) \} \}$.

Finally, without any further changes to the definition of $\\f$, it is possible to produce all possible $n$-combinations from $T, C(S, n)$, by selecting all nodes at depth $n$.

$T$ is a very elegant way of representing and enumerating some of the most rudimentary combinatorial operations of introductory mathematics. In this series of post we’ll explore each operation and look at implementations, time complexities and possible applications.

Written by lewellen

2008-06-29 at 8:25 pm

Posted in Combinatorics