GromovHausdorfftype distances of labelled metric spaces
Description
The function gromovdist
calculates the matched Gromov\ell^p distances of two metrics on a finite space X:
D_p(ρ_1,ρ_2)=\inf\{\(d(φ_1(x),φ_2(x)))_{x \in X}\_p\}.
There, the infimum is taken over all isometric embeddings φ_1 of (X,ρ_1), φ_2 of (X,ρ_2) into a common metric space (Y,d). Only 1≤ p≤ ∞ is considered.
At the basis is the reformulation of the metric as value of a convex program, see Liebscher (2015).
Methods for various classes are provided:

dist,dissimilarity
which are distance matrices. 
matrix
for matrices containing the individual distances of the elements of X. 
igraph
for connected graphs. The metric on the nodes or just the leaves (nodes of degree 1) of the graph is the length of the shortest path. 
phylo
for phylogenetic trees. Again the metric is induced by the graph.
Usage
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  gromovdist(d1,d2=NULL,type="l1",p=NULL,...)
## S3 method for class 'list'
gromovdist(d1,d2=NULL,type="l1",p=NULL,...)
## S3 method for class 'multiPhylo'
gromovdist(d1,d2=NULL,type="l1",p=NULL,...)
## S3 method for class 'phylo'
gromovdist(d1,d2=NULL,type="l1",p=NULL,...)
## S3 method for class 'dist'
gromovdist(d1,d2=NULL,type="l1",p=NULL,...)
## S3 method for class 'dissimilarity'
gromovdist(d1,d2=NULL,type="l1",p=NULL,...)
## S3 method for class 'matrix'
gromovdist(d1,d2=NULL,type="l1",p=NULL,...)
## S3 method for class 'igraph'
gromovdist(d1,d2=NULL,type="l1",p=NULL,leavesonly=TRUE,...)

Arguments
type 
type of metric to use 
d1,d2 
distance object(s). 
p 
if 
leavesonly 
compute the distances between the leaves of the graph/tree only? 
... 
further parameters 
Details
type="l1"
yields p=1
type="l2"
yields p=2
type="linfinity"
yields p=∞
type="lp"
is for (not so efficient) computation using constrOptim for arbitrary 1≤ p<∞
If d1
is a list, the distance matrix between all elements of the list is computed. It is represented as an object of class dissimiliarity
, see dissimilarity.object.
The distance is only computed for that part of the objects where the labels are present in both objects. If there are no labels the elements are numbered consecutively.
Value
The distance (one numeric) or a distance matrix for the list
and multiPhylo
methods
Author(s)
Volkmar Liebscher
References
V.Liebscher, Gromov meets Phylogenetics  new Animals for the Zoo of Metrics on Tree Space. preprint 2015 arXiv:1504.05795
See Also
dist, dissimilarity.object,phangornpackage, igraphpackage,constrOptim.
Examples
1 2 3 4 5 6 7 8  library("ape")
tr1<rtree(n=10)
tr2<rtree(n=10)
gromovdist(tr1,tr2,"l1")
gromovdist(tr1,tr2,"l2")
#thesame, but slower
gromovdist(d1=tr1,d2=tr2,type="lp",p=2)
gromovdist(tr1,tr2,"linf")

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