# Combinatorial Sampling

#### Joseph Wood

#### 03/29/2022

This document covers topics in generating random samples of combinations/permutations. It is encouraged to read General Combinatorics first.

To illustrate this in base R, let us consider getting 5 random combinations of the vector `1:20` of length 10. How should we proceed?

## Base R

A naive approach would be to generate all of the combinations using `combn` and then call `sample`:

    naive <- function(v, m, n, s) {
        allCombs <- combn(v, m)
        set.seed(s)
        allCombs[, sample(ncol(allCombs), n)]
    }

    fiveRndCombs <- naive(20, 10, 5, 42)
    t(fiveRndCombs)
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    #> [1,]    1    3    5   10   11   14   15   16   18    20
    #> [2,]    1    3    4    9   10   11   12   13   18    19
    #> [3,]    2    3    4    6    9   10   12   13   15    19
    #> [4,]    1    4    5   10   13   14   15   17   18    19
    #> [5,]    1    3    4    5    7    8   13   15   18    19

This is okay for this small example (there are only `choose(20, 10) = 184756` results), however what if we wanted to find one hundred thousand random combinations from the vector `1:100` of length 20? Clearly, the approach above will not be feasible as there are far too many results to generate (`choose(100, 20) = 5.359834e+20`). Furthermore, there are internal limitations on `sample`. If we try to pass `choose(100, 20)`, we will get an error:

    sample(choose(100, 20), 5)
    #> Error in sample.int(x, size, replace, prob) : invalid first argument

We could also try calling `sample(100, 20)` a bunch of times and hope we don’t get duplicate combinations. This is neither promising nor elegant.

## RcppAlgos Solutions

`RcppAlgos` provides four functions: `comboSample`, `permuteSample`, `partitionsSample`, and `comboGroupsSample` for seamlessly attacking these types of problems. All functions provide the following:

- Easily generate random samples of combinations/permutations/partitions or partition of groups in parallel.
- You can pass a vector of specific indices or rely on the internal sampling functions. We call `sample` when the total number of results is small and for larger cases, the sampling is done in a very similar fashion to `urand.bigz` from the `gmp` package.
- Consistent interface to their respective general functions (i.e. `{combo/permute/partitions}General` and `comboGroups`)
- Useful when we need a reproducible set of random combinations/permutations/partitions or partitions of groups.
- If the gmp library is needed, the `seed` parameter must be set in order to have reproducible results (E.g. `set.seed()`) has no effect in these cases).

## `comboSample` and `permuteSample`

Let’s first look at the first problem above (i.e. getting 5 random combinations of the vector `1:20` of length 10):

    library(RcppAlgos)
    set.seed(42)
    comboSample(20, 10, n = 5)
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    #> [1,]    1    3    5   10   11   14   15   16   18    20
    #> [2,]    1    3    4    9   10   11   12   13   18    19
    #> [3,]    2    3    4    6    9   10   12   13   15    19
    #> [4,]    1    4    5   10   13   14   15   17   18    19
    #> [5,]    1    3    4    5    7    8   13   15   18    19

    ## Use the seed argument directly to produce the same output
    comboSample(20, 10, n = 5, seed = 42)
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    #> [1,]    1    3    5   10   11   14   15   16   18    20
    #> [2,]    1    3    4    9   10   11   12   13   18    19
    #> [3,]    2    3    4    6    9   10   12   13   15    19
    #> [4,]    1    4    5   10   13   14   15   17   18    19
    #> [5,]    1    3    4    5    7    8   13   15   18    19

    ## fiveRndCombs produced above
    identical(t(fiveRndCombs),
              comboSample(20, 10, n = 5, seed = 42))
    #> [1] TRUE

### Samples of Results with Repetition

Just like with `comboGeneral` and `permuteGeneral`, we can explore results with repetition.

    comboSample(10, 8, TRUE, n = 3, seed = 84)
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
    #> [1,]    2    5    5    7    9    9    9    9
    #> [2,]    4    5    8    8    8   10   10   10
    #> [3,]    2    6    6    6    6    6    9    9

    permuteSample(10, 8, TRUE, n = 3)
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
    #> [1,]    4   10    4    4   10    2    2   10
    #> [2,]    1    4    5   10    5    5    2    2
    #> [3,]    4    1    7    9    1    5    6    5

    comboSample(10, 12, freqs = 1:10, n = 3)
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
    #> [1,]    2    2    3    5    5    6    6    6    7     8     9    10
    #> [2,]    1    2    3    3    5    5    6    7    9     9     9     9
    #> [3,]    1    2    5    5    5    6    6    9   10    10    10    10

    permuteSample(10, 12, freqs = 1:10, n = 3, seed = 123)
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
    #> [1,]    2    8    7    4    8    9   10   10    7     1     8     2
    #> [2,]    5    5    9    8    1    8    3    2    6     4     3    10
    #> [3,]   10    3    8    8    4    8    8    6   10     6     3     8

### Specific Results with `sampleVec`

We can also utilize `sampleVec` to generate specific results.

    ## E.g. the below generates the 1st, 5th, 25th, 125th, and
    #> 625th lexicographical combinations
    comboSample(10, 8, TRUE, sampleVec = c(1, 5, 25, 125, 625))
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
    #> [1,]    1    1    1    1    1    1    1    1
    #> [2,]    1    1    1    1    1    1    1    5
    #> [3,]    1    1    1    1    1    1    3    8
    #> [4,]    1    1    1    1    1    3    6    9
    #> [5,]    1    1    1    1    5    6   10   10

    ## Is the same as:
    comboGeneral(10, 8, TRUE)[5^(0:4), ]
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
    #> [1,]    1    1    1    1    1    1    1    1
    #> [2,]    1    1    1    1    1    1    1    5
    #> [3,]    1    1    1    1    1    1    3    8
    #> [4,]    1    1    1    1    1    3    6    9
    #> [5,]    1    1    1    1    5    6   10   10

### Using `namedSample`

Have you ever wondered which lexicographical combinations/permutations are returned when sampling? No worries, simply set `namedSample = TRUE`:

    testInd <- permuteSample(30, 10, n = 3, seed = 100, namedSample = TRUE)
    testInd
    #>                [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    #> 86626302070118   24   26    7   29    3   21   20    9   16    28
    #> 15871916538841    5   12   21    9    6    3   14   23    4    20
    #> 87932455980012   25    6   20   23   18   10   27   30   19    29

    ## Same output as above
    permuteSample(30, 10, sampleVec = row.names(testInd))
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    #> [1,]   24   26    7   29    3   21   20    9   16    28
    #> [2,]    5   12   21    9    6    3   14   23    4    20
    #> [3,]   25    6   20   23   18   10   27   30   19    29

### Parallel Computing and GMP Support

Just like the `General` counterparts, the sampling functions utilize GMP to allow for exploration of combinations/permutations of large vectors where the total number of results is enormous. They also offer parallel options using `Parallel` or `nThreads`.

    ## Uses min(stdThreadMax() - 1, 5) threads (in this case)
    permuteSample(500, 10, TRUE, n = 5, seed = 123, Parallel = TRUE)
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    #> [1,]   55  435  274  324  200  152    6  313  121   377
    #> [2,]  196  166  331  154  443  329  155  233  354   442
    #> [3,]  235  325   94   27  370  117  302   86  229   126
    #> [4,]  284  104  464  104  207  127  117    9  390   414
    #> [5,]  456   76  381  456  219   23  376  187   11   123

    permuteSample(factor(state.abb), 15, n = 3, seed = 50, nThreads = 3)
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14] [,15]
    #> [1,] ME   FL   DE   OK   ND   CA   PA   AL   ID   MO    NM    HI    KY    MT    NJ
    #> [2,] AZ   CA   AL   CT   ME   SD   ID   SC   OK   NH    HI    TN    ND    IA    MT
    #> [3,] MD   MO   NC   MT   NH   AL   VA   MA   VT   WV    NJ    NE    MN    MS    MI
    #> 50 Levels: AK AL AR AZ CA CO CT DE FL GA HI IA ID IL IN KS KY LA MA MD ME MI MN ... WY

    permuteCount(factor(state.abb), 15)
    #> Big Integer ('bigz') :
    #> [1] 2943352142120754524160000

### Efficiency

The algorithms are incredibly efficient and offer tremendous gains over the naive approach above:

    ## the function "naive" is defined above
    system.time(naive(25, 10, 5, 15))
    #>   user  system elapsed
    #>  3.197   0.066   3.287

    system.time(comboSample(25, 10, n = 5, seed = 15))
    #>   user  system elapsed
    #>  0.002   0.000   0.001

Even when dealing with extremely large numbers, these algorithms are very fast. And using the parallel options have even greater effects than we saw with the general counterparts (typically around ~2-3 times faster with the general functions, whereas with the last example below with sampling we see a nearly 5 fold improvement).

    ## Lightning fast even with examples involving many results
    system.time(comboSample(2500, 100, n = 5, seed = 15))
    #>    user  system elapsed
    #>   0.002   0.000   0.002

    ## The total number of combinations has ~180 digits
    gmp::log10.bigz(comboCount(2500, 100))
    #> [1] 180.9525

    ## Still fast with larger samples
    system.time(comboSample(2500, 100, n = 1e4, seed = 157))
    #>    user  system elapsed
    #>   1.124   0.008   1.142

    ## Using Parallel/nThreads in these cases has an even greater effect
    system.time(comboSample(2500, 100, n = 1e4, seed = 157, nThreads = 8))
    #>    user  system elapsed
    #>   2.032   0.005   0.268

### User Defined Functions

Again, just as with the general functions, you can pass a custom function to `combo/permuteSample` using the `FUN` argument.

    permuteSample(5000, 1000, n = 3, seed = 101, FUN = sd)
    #> [[1]]
    #> [1] 1431.949
    #>
    #> [[2]]
    #> [1] 1446.859
    #>
    #> [[3]]
    #> [1] 1449.272

    ## Example using complex numbers
    myCplx <- as.complex(1:100 + rep(c(-1, 1), 50) * 1i)

    permuteSample(myCplx, 10, freqs = rep(1:5, 20),
                  n = 3, seed = 101, FUN = function(x) {
                      sqrt(sum(x))
                  })
    #> [[1]]
    #> [1] 24.83948+0i
    #>
    #> [[2]]
    #> [1] 20.9285+0.04778i
    #>
    #> [[3]]
    #> [1] 22.20379+0.09007i

## `partitionsSample`

The `partitionsSample` function allows one to draw a random sample of partitions of a number. Many of the features present in `comboSample` and `permuteSample` are available in `partitionsSample`.

    ## Use the seed parameter to obtain reproducible results
    partitionsSample(100, 8, TRUE, n = 3, seed = 42)
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
    #> [1,]    1    1    3    3    4   20   23   45
    #> [2,]    1    1    2    7   14   14   29   32
    #> [3,]    2   10   11   11   16   16   16   18

    ## Used namedSample to obtain the lexicographical indices
    partitionsSample(100, 8, TRUE, n = 3, seed = 42, namedSample = TRUE)
    #>        [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
    #> 61413     1    1    3    3    4   20   23   45
    #> 54425     1    1    2    7   14   14   29   32
    #> 623844    2   10   11   11   16   16   16   18

    ## Use sampleVec to obtain specific results
    partitionsSample(100, 8, TRUE, sampleVec = c(61413, 54425, 623844))
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
    #> [1,]    1    1    3    3    4   20   23   45
    #> [2,]    1    1    2    7   14   14   29   32
    #> [3,]    2   10   11   11   16   16   16   18

    partitionsCount(2500, 10)
    #> Big Integer ('bigz') :
    #> [1] 2621914835336941325

    ## Algorithms are very efficient
    system.time(serial <- partitionsSample(2500, 10, n = 1e3,
                                           seed = 8128))
    #>    user  system elapsed
    #>   5.220   0.010   5.235

    ## Use nThreads for greater efficiency
    system.time(multi <- partitionsSample(2500, 10, n = 1e3,
                                          seed = 8128, nThreads = 8))
    #>    user  system elapsed
    #>  10.723   0.022   1.358

    identical(multi, serial)
    #> [1] TRUE

    ## Even works with non-standard setup
    partitionsSample(17 + (1:10) * 3, 10, TRUE,
                     target = 320, n = 3, seed = 111)
    #>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    #> [1,]   23   23   26   26   29   29   38   38   41    47
    #> [2,]   26   26   26   29   29   29   32   41   41    41
    #> [3,]   20   23   23   26   26   35   38   41   44    44

There are sampling algorithms available for most partition cases, but some cases are not covered. For example, with standard multisets, we are currently unable to efficiently generate the nth lexicographical result. Another example is when the source vector is not uniform (e.g. when the distance between each element is irregular).

Observe the following:

    ## No sampling algorithm available when the source vector is not uniform
    partitionsSample(c(1, 4, 6, 7, 10, seq(11, 100, 7)), 10, n = 1, target = 340)
    #> Error in partitionsSample(c(1, 4, 6, 7, 10, seq(11, 100, 7)), 10, n = 1,  :
    #>   Partition sampling not available for this case.

    ## As stated above, the standard multiset case doesn't work either
    partitionsSample(0:50, 6, freqs = rep(1:3, 17), n = 2)
    #> Error in partitionsSample(0:50, 6, freqs = rep(1:3, 17), n = 2) :
    #>   Partition sampling not available for this case.

    ## If we use freqs to indicate that zeros can repeat,
    ## then we can obtain random samples
    partitionsSample(0:50, 6, freqs = c(50, rep(1, 50)), n = 3, seed = 222)
    #>      [,1] [,2] [,3] [,4] [,5] [,6]
    #> [1,]    0    0    1    4    9   36
    #> [2,]    0    0    0    0   17   33
    #> [3,]    2    4    5    6    8   25

    ## Even works when the vector is restricted in regards to the target
    partitionsSample(0:50, 6, freqs = c(50, rep(1, 50)),
                     n = 3, seed = 222, target = 100)
    #>      [,1] [,2] [,3] [,4] [,5] [,6]
    #> [1,]    0    1    6   15   29   49
    #> [2,]    0    0    0    8   43   49
    #> [3,]    4    7   17   19   22   31

There is ongoing research in this area and our goal is to eventually be able to cover the standard multiset case.

## Sampling Partitions of Groups of Equal Size with `comboGroupsSample`

Just as we can generate random samples of combinations and permutations, we are also able to generate random samples of partitions of groups of equal size. There are many problems that present in this manner. Below, we examine one involving playing cards.

Let’s say we have 4 players and each player is to have 3 cards a piece. Given that the deck is shuffled, the dealer then distrubutes 12 cards.

> What possible hands can each player have?

See Creating A Deck Of Cards In R Without Using While And Double For Loop (Credit to @MichaelChirico)

    cards <- c(2:10, "J", "Q", "K", "A")
    suits <- c("♠", "♥", "♦", "♣")
    deck <- paste0(rep(cards, length(suits)),  #card values
                   rep(suits, each = length(cards))) #suits

    set.seed(1738)
    shuffled <- factor(deck[sample(52)], levels = deck)

    ## Here are 3 possibilities
    comboGroupsSample(shuffled[1:12], numGroups = 4, n = 2, seed = 13)
    #>      Grp1 Grp1 Grp1 Grp2 Grp2 Grp2 Grp3 Grp3 Grp3 Grp4 Grp4 Grp4
    #> [1,] 8♦   3♥   5♦   9♦   J♠   7♥   8♠   K♦   10♦  A♦   J♥   3♦
    #> [2,] 8♦   K♦   10♦  9♦   J♥   3♥   J♠   8♠   3♦   A♦   5♦   7♥
    #> 52 Levels: 2♠ 3♠ 4♠ 5♠ 6♠ 7♠ 8♠ 9♠ 10♠ J♠ Q♠ K♠ A♠ 2♥ 3♥ 4♥ 5♥ 6♥ 7♥ 8♥ ... A

    comboGroupsSample(shuffled[1:12], numGroups = 4, retType = "3Darray",
                      n = 2, seed = 13, namedSample = TRUE)
    #> , , Grp1
    #>
    #>       [,1] [,2] [,3]
    #> 13784 8♦   3♥   5♦
    #> 9152  8♦   K♦   10♦
    #>
    #> , , Grp2
    #>
    #>       [,1] [,2] [,3]
    #> 13784 9♦   J♠   7♥
    #> 9152  9♦   J♥   3♥
    #>
    #> , , Grp3
    #>
    #>       [,1] [,2] [,3]
    #> 13784 8♠   K♦   10♦
    #> 9152  J♠   8♠   3♦
    #>
    #> , , Grp4
    #>
    #>       [,1] [,2] [,3]
    #> 13784 A♦   J♥   3♦
    #> 9152  A♦   5♦   7♥
    #>
    #> 52 Levels: 2♠ 3♠ 4♠ 5♠ 6♠ 7♠ 8♠ 9♠ 10♠ J♠ Q♠ K♠ A♠ 2♥ 3♥ 4♥ 5♥ 6♥ 7♥ 8♥ ... A♣♣