# order.optimal: Optimal Leaf Ordering of Binary Trees. In cba: Clustering for Business Analytics

## Description

Find an optimal linear leaf ordering of a binary merge tree as produced by a hierarchical cluster algorithm.

## Usage

 `1` ```order.optimal(dist, merge) ```

## Arguments

 `dist` an object of class `dist`. `merge` a binary merge tree (see `hclust`).

## Details

A binary tree has 2^(n-1) internal nodes (subtrees) and the same number of leaf orderings. That is, at each internal node the left and right subtree (or leaves) can be swapped, or, in terms of a dendrogram, be flipped.

An objective measure of a leaf ordering is the sum of the distances along the path connecting the leaves in the given order. An ordering with a minimal path length is defined to be an optimal ordering.

This function provides an interface to the optimal leaf ordering algorithm (see references) for tree representations that are used by hierarchical cluster algorithms such as `hclust`.

Note that non-finite distance values are not allowed.

## Value

A list with the following components:

 `merge` a matrix containing the merge tree corresponding with the optimal leaf order. `order` a vector containing the optimal leaf order. `length` the length of the ordering.

## Note

The time complexity of the algorithm is O(n^3).

Christian Buchta

## References

Z. Bar-Joseph, E. D. Demaine, D. K. Gifford, and T. Jaakkola. (2001). Fast Optimal Leaf Ordering for Hierarchical Clustering. Bioinformatics, Vol. 17 Suppl. 1, pp. 22-29.

`hclust` for hierarchical clustering and `order.length` for computing the objective value of a leaf ordering.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```d <- dist(matrix(runif(30), ncol=2)) hc <- hclust(d) co <- order.optimal(d, hc\$merge) ### compare dendrograms ho <- hc ho\$merge <- co\$merge ho\$order <- co\$order op <- par(mfrow=c(2,2), pty="s") plot(hc, main="hclust") plot(ho, main="optimal") # compare images implot(d[[hc\$order]]) implot(d[[co\$order]]) par(op) ### compare lengths order.length(d, hc\$order) order.length(d, co\$order) cat("compare: ",co\$length,"\n") ```