CODEWITHSUNDEEP
ralgorithmcombinatorics
alberto
alberto

Reputation: 2645

Integer partitions of size K

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?

Upvotes: 2

Views: 270

Answers (1)

St&#233;phane Laurent
St&#233;phane Laurent

Reputation: 84529

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)
  ......
}

Upvotes: 1

Related Questions