Creating a Random Matrix without One Number

In this article, we will explore how to generate a random matrix of size n-1 x n such that the i-th column contains all numbers from 1 to n without containing i. We’ll dive into various approaches and their implementations.

Problem Statement

Given a matrix of size n-1 x n, we want to ensure that each column follows a specific pattern: the first column should contain all numbers from 2 to n, the second column should contain 1, 3, 4,…, the third column should contain 1, 2, 4,… and so on. We are looking for an efficient way to generate such matrices.

Approach using combn

One possible approach is to use R’s combn function, which generates a matrix of size m x n containing all combinations of length n from the vector 1:1:…:m without repetition. However, this method does not guarantee that each column will contain numbers in the desired sequence.

Here is an example implementation using combn:

n <- 5
out <- combn(n:1, n - 1)
print(out)

Output:

     [,1] [,2] [,3] [,4] [,5]
[1,]    5    5    5    5    4
[2,]    4    4    4    3    3
[3,]    3    3    2    2    2
[4,]    2    1    1    1    1

As we can see, the generated matrix does not follow the desired pattern. We need to shuffle each column if needed.

Shuffling columns

To achieve the desired sequence, we can use the sample function in R to randomly select numbers from a set for each column.

Here is an example implementation:

set.seed(1)
out <- apply(combn(n:1, n - 1), 2, sample)
print(out)

Output:

     [,1] [,2] [,3] [,4] [,5]
[1,]    5    3    1    2    4
[2,]    4    2    5    1    3
[3,]    3    4    2    5    1
[4,]    2    5    4    3    2

Efficiency Comparison

If you are looking for efficiency, we can compare the performance of different approaches using R’s microbenchmark package.

Here is an example implementation:

library(microbenchmark)

n <- 1000
set.seed(1)
benchmark <- microbenchmark(
  akrun = vapply(seq_len(n), function(i) sample(setdiff(seq_len(n), i)), numeric(n-1)),
  markus = apply(combn(n:1, n - 1), 2, sample),
  A_Stam = create_matrix(n),
  times = 30L
)
print(benchmark)

Output:

#Unit: milliseconds
#   expr        min         lq       mean     median         uq        max neval
#  akrun   64.32350   66.99177   73.61685   71.15608   78.79612 104.99161    30
# markus   51.65092   53.01034   59.80802   58.48310   64.76143   78.35348    30
# A_Stam 1331.52882 1379.70371 1470.31044 1407.89861 1548.28011 1896.22913    30

As we can see, the akrun approach is the most efficient, followed by the markus approach.

Conclusion

In this article, we explored how to generate a random matrix of size n-1 x n such that each column follows a specific pattern. We discussed three approaches: using combn, shuffling columns, and efficiency comparison using microbenchmark. The most efficient approach is the akrun method, which uses vapply to create the desired sequence.

Note that this implementation assumes that the numbers in the matrix are integers between 1 and n. If you need to generate matrices with other types of numbers or with different patterns, you will need to modify the code accordingly.

Last modified on 2023-11-28