Product between all combinations of a vector's elements

Question

Suppose I have a vector c(1, 2, 3, 4) with no duplicated values. I need a vector c(1 * 2, 1 * 3, 1 * 4, 2 * 3, 2 * 4, 3 * 4), so the multiplication

Answer 1

Here is the "outer + upper triangular part option"

m <- outer(1:4, 1:4)
as.numeric(m[upper.tri(m)])
#[1]  2  3  6  4  8 12

Another method is to index the elements of the upper/lower triangular part of a matrix directly and then calculate the pairwise product for those elements (adapted from this post)

upperouter <- function(x) {
    N <- length(x)
    i <- sequence(1:N)
    j <- rep(1:N, 1:N)
    (1:N)[i[i != j]] * (1:N)[j[j != i]]
}
upperouter(1:4)
#[1]  2  3  6  4  8 12

---

Benchmark analysis

It's interesting to compare the different methods in a microbenchmark analysis for a larger vector (e.g. 1:100):

upperouter <- function(x) {
    N <- length(x)
    i <- sequence(1:N)
    j <- rep(1:N, 1:N)
    (1:N)[i[i != j]] * (1:N)[j[j != i]]
}

finv <- function (n) {
  k <- 1:(n * (n - 1) / 2)
  j <- floor(((2 * n + 1) - sqrt((2 * n - 1) ^ 2 - 8 * (k - 1))) / 2)
  i <- j + k - (2 * n - j) * (j - 1) / 2
  cbind(i, j)
  }


N <- 100
library(microbenchmark)
res <- microbenchmark(
    combn  = combn(1:N, 2, prod),
    outer = {
        m <- outer(1:N, 1:N)
        as.numeric(m[upper.tri(m)])
    },
    upperouter = {
        upperouter(1:N)
    },
    finv = {
        vec <- 1:N
        ind <- finv(length(vec))
        vec[ind[, 2]] * vec[ind[, 1]]
    },
    sapply = {
        m <- sapply(1:N, "*", 1:N)
        as.numeric(m[upper.tri(m)])
    })
res
#Unit: microseconds
#       expr      min        lq      mean    median        uq       max neval
#      combn 6584.938 6896.0545 7584.8084 7035.9575 7886.5720 12020.626   100
#      outer  106.791  113.6535  157.3774  138.9205  160.5985   950.706   100
# upperouter  201.943  210.1515  277.0989  227.6370  259.1975  2806.962   100
#       finv  308.447  324.1960  442.3220  332.7250  375.3490  4128.325   100
#     sapply  232.805  249.9080  298.3674  283.8580  315.9145   556.463   100

library(ggplot2)
autoplot(res)