Multiplying Combinations of a list of lists in R

Question

Given a list of two lists, I am trying to obtain, without using for loops, a list of all element-wise products of the first list with the second. For example:

> a <- list(c(1,2), c(2,3), c(4,5))
> b <- list(c(1,3), c(3,4), c(6,2))
> c <- list(a, b)

The function should return a list with 9 entries, each of size two. For example,

> answer
[[1]]
[1] 1 6

[[2]]
[1] 3 8

[[3]]
[1] 6 4

[[4]]
[1] 2 9

[[5]]
[1] 6 12

etc...

Any suggestions would be much appreciated!

Answer 1

A fast (but memory-intensive) way would be to use the mechanism of mapply in combination with argument recycling, something like this:

mapply(`*`,a,rep(b,each=length(a)))

Gives :

> mapply(`*`,a,rep(b,each=length(a)))
     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[1,]    1    2    4    3    6   12    6   12   24
[2,]    6    9   15    8   12   20    4    6   10

Or replace a with c[[1]] and b with c[[2]] to obtain the same. To get a list, set the argument SIMPLIFY = FALSE.

Answer 2

Have no idea if this is fast or memory intensive just that it works, Joris Meys's answer is more eloquent:

x <- expand.grid(1:length(a), 1:length(b))
x <- x[order(x$Var1), ]    #gives the order you asked for
FUN <- function(i)  diag(outer(a[[x[i, 1]]], b[[x[i, 2]]], "*"))
sapply(1:nrow(x), FUN)     #I like this out put
lapply(1:nrow(x), FUN)     #This one matches what you asked for

EDIT: Now that Brian introduced benchmarking (which I love (LINK)) I have to respond. I actually have a faster answer using what I call expand.grid2 that's a lighter weight version of the original that I stole from HERE. I was going to throw it up before but when I saw how fast Joris's is I figured why bother, both short and sweet but also fast. But now that Diggs has dug I figured I'd throw up here the expand.grid2 for educational purposes.

expand.grid2 <-function(seq1,seq2) {
    cbind(Var1 = rep.int(seq1, length(seq2)), 
    Var2 = rep.int(seq2, rep.int(length(seq1),length(seq2))))
}

x <- expand.grid2(1:length(a), 1:length(b))
x <- x[order(x[,'Var1']), ]    #gives the order you asked for
FUN <- function(i)  diag(outer(a[[x[i, 1]]], b[[x[i, 2]]], "*"))
lapply(1:nrow(x), FUN)

Here's the results (same labeling as Bryan's except TylerEG2 is using the expand.grid2):

Unit: microseconds
            expr      min       lq   median       uq      max
1   DiggsL(a, b) 5102.296 5307.816 5471.578 5887.516 70965.58
2   DiggsM(a, b)  384.912  428.769  443.466  461.428 36213.89
3    Joris(a, b)   91.446  105.210  123.172  130.171 16833.47
4 TylerEG2(a, b)  392.377  425.503  438.100  453.263 32208.94
5   TylerL(a, b) 1752.398 1808.852 1847.577 1975.880 49214.10
6   TylerM(a, b) 1827.515 1888.867 1925.959 2090.421 75766.01
7 Wojciech(a, b) 1719.740 1771.760 1807.686 1924.325 81666.12

And if I take the ordering step out I can squeak out even more but it still isn't close to Joris's answer.

Answer 3

Pulling ideas from the other answers together, I'll throw another one-liner in for fun:

do.call(mapply, c(FUN=`*`, as.list(expand.grid(b, a))))

which gives

     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[1,]    1    3    6    2    6   12    4   12   24
[2,]    6    8    4    9   12    6   15   20   10

If you really need it in the format you gave, then you can use the plyr library to transform it into that:

library("plyr")
as.list(unname(alply(do.call(mapply, c(FUN=`*`, as.list(expand.grid(b, a)))), 2)))

which gives

[[1]]
[1] 1 6

[[2]]
[1] 3 8

[[3]]
[1] 6 4

[[4]]
[1] 2 9

[[5]]
[1]  6 12

[[6]]
[1] 12  6

[[7]]
[1]  4 15

[[8]]
[1] 12 20

[[9]]
[1] 24 10

---

Just for fun, benchmarking:

Joris <- function(a, b) {
    mapply(`*`,a,rep(b,each=length(a)))
}

TylerM <- function(a, b) {
    x <- expand.grid(1:length(a), 1:length(b))
    x <- x[order(x$Var1), ]    #gives the order you asked for
    FUN <- function(i)  diag(outer(a[[x[i, 1]]], b[[x[i, 2]]], "*"))
    sapply(1:nrow(x), FUN)
}

TylerL <- function(a, b) {
    x <- expand.grid(1:length(a), 1:length(b))
    x <- x[order(x$Var1), ]    #gives the order you asked for
    FUN <- function(i)  diag(outer(a[[x[i, 1]]], b[[x[i, 2]]], "*"))
    lapply(1:nrow(x), FUN)
}

Wojciech <- function(a, b) {
    # Matrix with indicies for elements to multiply
    G <- expand.grid(1:3,1:3)

    # Coversion of G to list
    L <- lapply(1:nrow(G),function(x,d=G) d[x,])

    lapply(L,function(i,x=a,y=b) x[[i[[2]]]]*y[[i[[1]]]])
}

DiggsM <- function(a, b) {
    do.call(mapply, c(FUN=`*`, as.list(expand.grid(b, a))))
}

DiggsL <- function(a, b) {
    as.list(unname(alply(t(do.call(mapply, c(FUN=`*`, as.list(expand.grid(b, a))))), 1)))
}

and the benchmarks

> library("rbenchmark")
> benchmark(Joris(b,a),
+           TylerM(a,b),
+           TylerL(a,b),
+           Wojciech(a,b),
+           DiggsM(a,b),
+           DiggsL(a,b),
+           order = "relative", 
+           replications = 1000,
+           columns = c("test", "elapsed", "relative"))
            test elapsed relative
1    Joris(b, a)    0.08    1.000
5   DiggsM(a, b)    0.26    3.250
4 Wojciech(a, b)    1.34   16.750
3   TylerL(a, b)    1.36   17.000
2   TylerM(a, b)    1.40   17.500
6   DiggsL(a, b)    3.49   43.625

and to show they are equivalent:

> identical(Joris(b,a), TylerM(a,b))
[1] TRUE
> identical(Joris(b,a), DiggsM(a,b))
[1] TRUE
> identical(TylerL(a,b), Wojciech(a,b))
[1] TRUE
> identical(TylerL(a,b), DiggsL(a,b))
[1] TRUE