Integer partitions of size K - r

Question

Given a vector v of F non-negative integers, I want to create, one by one, all possible sets of K vectors with size F whose sum is v. I call C the matrix of these K vectors; the row sum of C gives v.
For instance, the vector (1,2) of size F=2, if we set K=2, can be decomposed in:
# all sets of K vectors such that their sum is (1,2)
C_1 = 1,0 C_2 = 1,0 C_3 = 1,0 C_4 = 0,1 C_5 = 0,1 C_6 = 0,1
2,0 1,1 0,2 2,0 1,1 0,2
The goal is to apply some function to each possible C. Currently, I use this code, where I pre-compute all possible C and then go through them.
library(partitions)
K <- 3
F <- 5
v <- 1:F
partitions <- list()
for(f in 1:F){
partitions[[f]] <- compositions(n=v[f],m=K)
}
# Each v[f] has multiple partitions. Now we create an index to consider
# all possible combinations of partitions for the whole vector v.
npartitions <- sapply(partitions, ncol)
indices <- lapply(npartitions, function(x) 1:x)
grid <- as.matrix(do.call(expand.grid, indices)) # breaks if too big
for(n in 1:nrow(grid)){
selected <- c(grid[n,])
C <- t(sapply(1:F, function(f) partitions[[f]][,selected[f]]))
# Do something with C
#...
print(C)
}
However, when the dimensions are too big, of F, K are large, then the number of combinations explodes and expand.grid can't deal with that.
I know that, for a given position v[f], I can create a partition at a time
partition <- firstcomposition(n=v[f],m=K)
nextcomposition(partition, v[f],m=K)
But how can I use this to generate all possible C as in the above code?

Answer 1

npartitions <- ......
indices <- lapply(npartitions, function(x) 1:x)
grid <- as.matrix(do.call(expand.grid, indices))
You can avoid the generation of grid and successively generate its rows thanks to a Cantor expansion.
Here is the function returning the Cantor expansion of the integer n:
aryExpansion <- function(n, sizes){
l <- c(1, cumprod(sizes))
nmax <- tail(l,1)-1
if(n > nmax){
stop(sprintf("n cannot exceed %d", nmax))
}
epsilon <- numeric(length(sizes))
while(n>0){
k <- which.min(l<=n)
e <- floor(n/l[k-1])
epsilon[k-1] <- e
n <- n-e*l[k-1]
}
return(epsilon)
}
For example:
expand.grid(1:2, 1:3)
## Var1 Var2
## 1 1 1
## 2 2 1
## 3 1 2
## 4 2 2
## 5 1 3
## 6 2 3
aryExpansion(0, sizes = c(2,3)) + 1
## [1] 1 1
aryExpansion(1, sizes = c(2,3)) + 1
## [1] 2 1
aryExpansion(2, sizes = c(2,3)) + 1
## [1] 1 2
aryExpansion(3, sizes = c(2,3)) + 1
## [1] 2 2
aryExpansion(4, sizes = c(2,3)) + 1
## [1] 1 3
aryExpansion(5, sizes = c(2,3)) + 1
## [1] 2 3
So, instead of generating grid:
npartitions <- ......
indices <- lapply(npartitions, function(x) 1:x)
grid <- as.matrix(do.call(expand.grid, indices))
for(n in 1:nrow(grid)){
selected <- grid[n,]
......
}
you can do:
npartitions <- ......
for(n in seq_len(prod(npartitions))){
selected <- 1 + aryExpansion(n-1, sizes = npartitions)
......
}