cba (version 0.2-5)

order.optimal: Optimal Leaf Ordering of Binary Trees.

Description

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

Usage

order.optimal(dist, merge)

Arguments

dist

an object of class dist.

merge

a binary merge tree (see hclust).

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.

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.

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.

See Also

hclust for hierarchical clustering and order.length for computing the objective value of a leaf ordering.

Examples

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$order
op <- par(mfrow=c(2,2), pty="s")
plot(hc, main="hclust")
plot(ho, main="optimal")
# compare images
image(d[[hc$order]])
image(d[[co$order]])
par(op)
### compare lengths
order.length(d, hc$order)
order.length(d, co$order)
cat("compare: ",co$length,"")