# High Performance Benchmarks

#### Joseph Wood

#### 11/30/2023

This document serves as an overview for measuring the performance of `RcppAlgos` against other tools for generating combinations, permutations, and partitions. This stackoverflow post: How to generate permutations or combinations of object in R? has some benchmarks. You will note that the examples in that post are relatively small. The benchmarks below will focus on larger examples where performance really matters and for this reason we only consider the packages arrangements, partitions, and RcppAlgos.

## Setup Information

For the benchmarks below, we used a `2022 Macbook Air Apple M2 24 GB` machine.

    library(RcppAlgos)
    library(partitions)
    library(arrangements)
    #>
    #> Attaching package: 'arrangements'
    #> The following object is masked from 'package:partitions':
    #>
    #>     compositions
    library(microbenchmark)

    options(digits = 4)
    options(width = 90)

    pertinent_output <- capture.output(sessionInfo())
    cat(paste(pertinent_output[1:3], collapse = "\n"))
    #> R version 4.3.1 (2023-06-16)
    #> Platform: aarch64-apple-darwin20 (64-bit)
    #> Running under: macOS Ventura 13.4.1

    pkgs <- c("RcppAlgos", "arrangements", "partitions", "microbenchmark")
    sapply(pkgs, packageVersion, simplify = FALSE)
    #> $RcppAlgos
    #> [1] '2.8.3'
    #>
    #> $arrangements
    #> [1] '1.1.9'
    #>
    #> $partitions
    #> [1] '1.10.7'
    #>
    #> $microbenchmark
    #> [1] '1.4.10'

    numThreads <- min(as.integer(RcppAlgos::stdThreadMax() / 2), 6)
    numThreads
    #> [1] 4

## Combinations

### Combinations - Distinct

    set.seed(13)
    v1 <- sort(sample(100, 30))
    m <- 21
    t1 <- comboGeneral(v1, m, Parallel = T)
    t2 <- combinations(v1, m)
    stopifnot(identical(t1, t2))
    dim(t1)
    #> [1] 14307150       21
    rm(t1, t2)
    invisible(gc())
    microbenchmark(cbRcppAlgosPar = comboGeneral(v1, m, nThreads = numThreads),
                   cbRcppAlgosSer = comboGeneral(v1, m),
                   cbArrangements = combinations(v1, m),
                   times = 15, unit = "relative")
    #> Warning in microbenchmark(cbRcppAlgosPar = comboGeneral(v1, m, nThreads = numThreads), :
    #> less accurate nanosecond times to avoid potential integer overflows
    #> Unit: relative
    #>            expr   min    lq  mean median    uq   max neval cld
    #>  cbRcppAlgosPar 1.000 1.000 1.000  1.000 1.000 1.000    15  a
    #>  cbRcppAlgosSer 3.495 3.040 3.040  2.996 2.997 2.905    15   b
    #>  cbArrangements 3.520 3.068 3.065  3.028 3.019 2.934    15   b

### Combinations - Repetition

    v2 <- v1[1:10]
    m <- 20
    t1 <- comboGeneral(v2, m, repetition = TRUE, nThreads = numThreads)
    t2 <- combinations(v2, m, replace = TRUE)
    stopifnot(identical(t1, t2))
    dim(t1)
    #> [1] 10015005       20
    rm(t1, t2)
    invisible(gc())
    microbenchmark(cbRcppAlgosPar = comboGeneral(v2, m, TRUE, nThreads = numThreads),
                   cbRcppAlgosSer = comboGeneral(v2, m, TRUE),
                   cbArrangements = combinations(v2, m, replace = TRUE),
                   times = 15, unit = "relative")
    #> Unit: relative
    #>            expr   min    lq  mean median    uq   max neval cld
    #>  cbRcppAlgosPar 1.000 1.000 1.000  1.000 1.000 1.000    15  a
    #>  cbRcppAlgosSer 2.840 2.944 2.983  2.934 2.929 3.727    15   b
    #>  cbArrangements 2.952 2.927 2.905  2.919 2.908 2.761    15   b

### Combinations - Multisets

    myFreqs <- c(2, 4, 4, 5, 3, 2, 2, 2, 3, 4, 1, 4, 2, 5)
    v3 <- as.integer(c(1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610))
    t1 <- comboGeneral(v3, 20, freqs = myFreqs, nThreads = numThreads)
    t2 <- combinations(freq = myFreqs, k = 20, x = v3)
    stopifnot(identical(t1, t2))
    dim(t1)
    #> [1] 14594082       20
    rm(t1, t2)
    invisible(gc())
    microbenchmark(cbRcppAlgosPar = comboGeneral(v3, 20, freqs = myFreqs, nThreads = numThreads),
                   cbRcppAlgosSer = comboGeneral(v3, 20, freqs = myFreqs),
                   cbArrangements = combinations(freq = myFreqs, k = 20, x = v3),
                   times = 10, unit = "relative")
    #> Unit: relative
    #>            expr   min    lq  mean median    uq   max neval cld
    #>  cbRcppAlgosPar 1.000 1.000 1.000  1.000 1.000 1.000    10 a
    #>  cbRcppAlgosSer 3.086 3.078 3.036  3.051 2.999 2.958    10  b
    #>  cbArrangements 5.661 5.782 5.683  5.735 5.618 5.522    10   c

## Permutations

### Permutations - Distinct

    v4 <- as.integer(c(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59))
    t1 <- permuteGeneral(v4, 6, nThreads = numThreads)
    t2 <- permutations(v4, 6)
    stopifnot(identical(t1, t2))
    dim(t1)
    #> [1] 8910720       6
    rm(t1, t2)
    invisible(gc())
    microbenchmark(cbRcppAlgosPar = permuteGeneral(v4, 6, nThreads = numThreads),
                   cbRcppAlgosSer = permuteGeneral(v4, 6),
                   cbArrangements = permutations(v4, 6),
                   times = 15, unit = "relative")
    #> Unit: relative
    #>            expr   min    lq  mean median    uq   max neval cld
    #>  cbRcppAlgosPar 1.000 1.000 1.000  1.000 1.000 1.000    15 a
    #>  cbRcppAlgosSer 1.489 1.498 1.392  1.463 1.179 1.322    15  b
    #>  cbArrangements 2.521 2.521 2.311  2.469 2.127 1.984    15   c

    ## Indexing permutation example with the partitions package
    t1 <- permuteGeneral(11, nThreads = 4)
    t2 <- permutations(11)
    t3 <- perms(11)

    dim(t1)
    #> [1] 39916800       11

    stopifnot(identical(t1, t2), identical(t1, t(as.matrix(t3))))
    rm(t1, t2, t3)
    invisible(gc())

    microbenchmark(cbRcppAlgosPar = permuteGeneral(11, nThreads = 4),
                   cbRcppAlgosSer = permuteGeneral(11),
                   cbArrangements = permutations(11),
                   cbPartitions   = perms(11),
                   times = 5, unit = "relative")
    #> Unit: relative
    #>            expr   min    lq  mean median    uq   max neval  cld
    #>  cbRcppAlgosPar 1.000 1.000 1.000  1.000 1.000 1.000     5 a
    #>  cbRcppAlgosSer 2.456 2.466 2.175  2.421 1.793 1.975     5  b
    #>  cbArrangements 4.003 4.008 3.651  4.088 3.169 3.316     5   c
    #>    cbPartitions 7.857 7.936 7.062  8.019 5.915 6.338     5    d

### Permutations - Repetition

    v5 <- v3[1:5]
    t1 <- permuteGeneral(v5, 10, repetition = TRUE, nThreads = numThreads)
    t2 <- permutations(v5, 10, replace = TRUE)
    stopifnot(identical(t1, t2))
    dim(t1)
    #> [1] 9765625      10
    rm(t1, t2)
    invisible(gc())
    microbenchmark(cbRcppAlgosPar = permuteGeneral(v5, 10, TRUE, nThreads = numThreads),
                   cbRcppAlgosSer = permuteGeneral(v5, 10, TRUE),
                   cbArrangements = permutations(x = v5, k = 10, replace = TRUE),
                   times = 10, unit = "relative")
    #> Unit: relative
    #>            expr   min    lq  mean median    uq   max neval cld
    #>  cbRcppAlgosPar 1.000 1.000 1.000  1.000 1.000 1.000    10 a
    #>  cbRcppAlgosSer 2.958 2.842 2.946  2.825 2.781 3.835    10  b
    #>  cbArrangements 3.591 3.478 3.590  3.447 3.391 4.590    10   c

### Permutations - Multisets

    v6 <- sort(runif(12))
    t1 <- permuteGeneral(v6, 7, freqs = rep(1:3, 4), nThreads = numThreads)
    t2 <- permutations(freq = rep(1:3, 4), k = 7, x = v6)
    stopifnot(identical(t1, t2))
    dim(t1)
    #> [1] 19520760        7
    rm(t1, t2)
    invisible(gc())
    microbenchmark(cbRcppAlgosPar = permuteGeneral(v6, 7, freqs = rep(1:3, 4), nThreads = numThreads),
                   cbRcppAlgosSer = permuteGeneral(v6, 7, freqs = rep(1:3, 4)),
                   cbArrangements = permutations(freq = rep(1:3, 4), k = 7, x = v6),
                   times = 10, unit = "relative")
    #> Unit: relative
    #>            expr   min    lq  mean median    uq   max neval cld
    #>  cbRcppAlgosPar 1.000 1.000 1.000  1.000 1.000 1.000    10 a
    #>  cbRcppAlgosSer 3.214 3.312 3.065  3.285 3.251 1.954    10  b
    #>  cbArrangements 3.635 3.636 3.374  3.609 3.572 2.131    10   c

## Partitions

### Partitions - Distinct

#### All Distinct Partitions

    t1 <- comboGeneral(0:140, freqs=c(140, rep(1, 140)),
                       constraintFun = "sum", comparisonFun = "==",
                       limitConstraints = 140)
    t2 <- partitions(140, distinct = TRUE)
    t3 <- diffparts(140)

    # Each package has different output formats... we only examine dimensions
    # and that each result is a partition of 140
    stopifnot(identical(dim(t1), dim(t2)), identical(dim(t1), dim(t(t3))),
                        all(rowSums(t1) == 140), all(rowSums(t2) == 140),
                        all(colSums(t3) == 140))
    dim(t1)
    #> [1] 9617150      16
    rm(t1, t2, t3)
    invisible(gc())
    microbenchmark(cbRcppAlgosPar = partitionsGeneral(0:140, freqs=c(140, rep(1, 140)), nThreads = numThreads),
                   cbRcppAlgosSer = partitionsGeneral(0:140, freqs=c(140, rep(1, 140))),
                   cbArrangements = partitions(140, distinct = TRUE),
                   cbPartitions   = diffparts(140),
                   times = 10, unit = "relative")
    #> Unit: relative
    #>            expr    min     lq   mean median     uq    max neval  cld
    #>  cbRcppAlgosPar  1.000  1.000  1.000  1.000  1.000  1.000    10 a
    #>  cbRcppAlgosSer  3.320  3.252  3.036  3.510  2.670  2.539    10  b
    #>  cbArrangements  2.638  2.592  2.361  2.566  2.152  1.974    10   c
    #>    cbPartitions 17.403 17.723 16.021 18.527 14.143 13.119    10    d

#### Restricted Distinct Partitions

    t1 <- comboGeneral(160, 10,
                       constraintFun = "sum", comparisonFun = "==",
                       limitConstraints = 160)
    t2 <- partitions(160, 10, distinct = TRUE)
    stopifnot(identical(t1, t2))
    dim(t1)
    #> [1] 8942920      10
    rm(t1, t2)
    invisible(gc())
    microbenchmark(cbRcppAlgosPar = partitionsGeneral(160, 10, nThreads = numThreads),
                   cbRcppAlgosSer = partitionsGeneral(160, 10),
                   cbArrangements = partitions(160, 10, distinct = TRUE),
                   times = 10, unit = "relative")
    #> Unit: relative
    #>            expr   min    lq  mean median    uq   max neval cld
    #>  cbRcppAlgosPar 1.000 1.000 1.000  1.000 1.000 1.000    10 a
    #>  cbRcppAlgosSer 3.439 3.433 3.007  3.332 2.235 2.564    10  b
    #>  cbArrangements 4.509 4.519 3.834  4.379 2.919 2.928    10   c

### Partitions - Repetition

#### All Partitions

    t1 <- comboGeneral(0:65, repetition = TRUE, constraintFun = "sum",
                       comparisonFun = "==", limitConstraints = 65)
    t2 <- partitions(65)
    t3 <- parts(65)

    # Each package has different output formats... we only examine dimensions
    # and that each result is a partition of 65
    stopifnot(identical(dim(t1), dim(t2)), identical(dim(t1), dim(t(t3))),
              all(rowSums(t1) == 65), all(rowSums(t2) == 65),
              all(colSums(t3) == 65))
    dim(t1)
    #> [1] 2012558      65
    rm(t1, t2, t3)
    invisible(gc())
    microbenchmark(cbRcppAlgosPar = partitionsGeneral(0:65, repetition = TRUE,
                                                      nThreads = numThreads),
                   cbRcppAlgosSer = partitionsGeneral(0:65, repetition = TRUE),
                   cbArrangements = partitions(65),
                   cbPartitions   = parts(65),
                   times = 20, unit = "relative")
    #> Unit: relative
    #>            expr   min    lq  mean median     uq    max neval cld
    #>  cbRcppAlgosPar 1.000 1.000 1.000  1.000  1.000 1.0000    20 a
    #>  cbRcppAlgosSer 2.902 2.644 2.376  2.624  2.571 0.8792    20  b
    #>  cbArrangements 2.310 2.030 1.916  1.999  1.974 1.2587    20  b
    #>    cbPartitions 8.951 8.916 9.342 11.188 10.938 3.7147    20   c

#### Restricted Partitions

    t1 <- comboGeneral(100, 15, TRUE, constraintFun = "sum",
                       comparisonFun = "==", limitConstraints = 100)
    t2 <- partitions(100, 15)
    stopifnot(identical(t1, t2))
    dim(t1)
    #> [1] 9921212      15
    rm(t1, t2)

    # This takes a really long time... not because of restrictedparts,
    # but because apply is not that fast. This transformation is
    # needed for proper comparisons. As a result, we will compare
    # a smaller example
    # t3 <- t(apply(as.matrix(restrictedparts(100, 15, include.zero = F)), 2, sort))
    t3 <- t(apply(as.matrix(restrictedparts(50, 15, include.zero = F)), 2, sort))
    stopifnot(identical(partitions(50, 15), t3))
    rm(t3)
    invisible(gc())
    microbenchmark(cbRcppAlgosPar = partitionsGeneral(100, 15, TRUE,
                                                      nThreads = numThreads),
                   cbRcppAlgosSer = partitionsGeneral(100, 15, TRUE),
                   cbArrangements = partitions(100, 15),
                   cbPartitions   = restrictedparts(100, 15,
                                                    include.zero = FALSE),
                   times = 10, unit = "relative")
    #> Unit: relative
    #>            expr    min     lq   mean median     uq   max neval  cld
    #>  cbRcppAlgosPar  1.000  1.000  1.000   1.00  1.000 1.000    10 a
    #>  cbRcppAlgosSer  3.439  3.445  2.586   2.74  2.802 1.290    10  b
    #>  cbArrangements  4.210  4.210  3.126   3.35  3.266 1.534    10   c
    #>    cbPartitions 15.238 15.460 11.480  12.26 11.754 6.212    10    d

### Partitions - Multisets

Currenlty, `RcppAlgos` is the only package capable of efficiently generating partitions of multisets. Therefore, we will only time `RcppAlgos` and use this as a reference for future improvements.

    t1 <- comboGeneral(120, 10, freqs=rep(1:8, 15),
                       constraintFun = "sum", comparisonFun = "==",
                       limitConstraints = 120)
    dim(t1)
    #> [1] 7340225      10
    stopifnot(all(rowSums(t1) == 120))
    microbenchmark(cbRcppAlgos = partitionsGeneral(120, 10, freqs=rep(1:8, 15)),
                   times = 10)
    #> Unit: milliseconds
    #>         expr   min    lq  mean median    uq   max neval
    #>  cbRcppAlgos 273.9 275.5 279.1  276.4 283.6 286.8    10

## Compositions

### Compositions - Repetition

#### All Compositions (Small case)

    t1 <- compositionsGeneral(0:15, repetition = TRUE)
    t2 <- arrangements::compositions(15)
    t3 <- partitions::compositions(15)

    # Each package has different output formats... we only examine dimensions
    # and that each result is a partition of 15
    stopifnot(identical(dim(t1), dim(t2)), identical(dim(t1), dim(t(t3))),
              all(rowSums(t1) == 15), all(rowSums(t2) == 15),
              all(colSums(t3) == 15))
    dim(t1)
    #> [1] 16384    15
    rm(t1, t2, t3)
    invisible(gc())
    microbenchmark(cbRcppAlgosSer = compositionsGeneral(0:15, repetition = TRUE),
                   cbArrangements = arrangements::compositions(15),
                   cbPartitions   = partitions::compositions(15),
                   times = 20, unit = "relative")
    #> Unit: relative
    #>            expr     min      lq    mean  median     uq     max neval cld
    #>  cbRcppAlgosSer   1.000   1.000   1.000   1.000   1.00   1.000    20  a
    #>  cbArrangements   1.201   1.178   1.189   1.175   1.19   1.195    20  a
    #>    cbPartitions 129.521 148.587 169.709 173.088 190.08 207.932    20   b

For the next two examples, we will exclude the `partitions` package for efficiency reasons.

#### All Compositions (Larger case)

    t1 <- compositionsGeneral(0:23, repetition = TRUE)
    t2 <- arrangements::compositions(23)

    # Each package has different output formats... we only examine dimensions
    # and that each result is a partition of 23
    stopifnot(identical(dim(t1), dim(t2)), all(rowSums(t1) == 23),
              all(rowSums(t2) == 23))
    dim(t1)
    #> [1] 4194304      23
    rm(t1, t2)
    invisible(gc())
    microbenchmark(cbRcppAlgosPar = compositionsGeneral(0:23, repetition = TRUE,
                                                        nThreads = numThreads),
                   cbRcppAlgosSer = compositionsGeneral(0:23, repetition = TRUE),
                   cbArrangements = arrangements::compositions(23),
                   times = 20, unit = "relative")
    #> Unit: relative
    #>            expr   min    lq  mean median    uq   max neval cld
    #>  cbRcppAlgosPar 1.000 1.000 1.000  1.000 1.000 1.000    20 a
    #>  cbRcppAlgosSer 3.388 3.397 3.419  3.445 3.404 3.439    20  b
    #>  cbArrangements 3.741 3.786 3.822  3.835 3.829 3.869    20   c

#### Restricted Compositions

    t1 <- compositionsGeneral(30, 10, repetition = TRUE)
    t2 <- arrangements::compositions(30, 10)

    stopifnot(identical(t1, t2), all(rowSums(t1) == 30))
    dim(t1)
    #> [1] 10015005       10
    rm(t1, t2)
    invisible(gc())
    microbenchmark(cbRcppAlgosPar = compositionsGeneral(30, 10, repetition = TRUE,
                                                        nThreads = numThreads),
                   cbRcppAlgosSer = compositionsGeneral(30, 10, repetition = TRUE),
                   cbArrangements = arrangements::compositions(30, 10),
                   times = 20, unit = "relative")
    #> Unit: relative
    #>            expr   min    lq  mean median    uq   max neval cld
    #>  cbRcppAlgosPar 1.000 1.000 1.000  1.000 1.000 1.000    20  a
    #>  cbRcppAlgosSer 3.057 3.088 3.167  3.083 3.032 4.699    20   b
    #>  cbArrangements 3.165 3.131 3.154  3.143 3.196 3.085    20   b

## Iterators

We will show one example from each category to demonstrate the efficiency of the iterators in `RcppAlgos`. The results are similar for the rest of the cases not shown.

### Combinations

    pkg_arrangements <- function(n, total) {
        a <- icombinations(n, as.integer(n / 2))
        for (i in 1:total) a$getnext()
    }

    pkg_RcppAlgos <- function(n, total) {
        a <- comboIter(n, as.integer(n / 2))
        for (i in 1:total) a@nextIter()
    }

    total <- comboCount(18, 9)
    total
    #> [1] 48620

    microbenchmark(cbRcppAlgos    = pkg_RcppAlgos(18, total),
                   cbArrangements = pkg_arrangements(18, total),
                   times = 15, unit = "relative")
    #> Unit: relative
    #>            expr   min    lq  mean median   uq   max neval cld
    #>     cbRcppAlgos  1.00  1.00  1.00   1.00  1.0  1.00    15  a
    #>  cbArrangements 19.29 19.19 18.54  19.08 18.8 14.04    15   b

### Permutations

    pkg_arrangements <- function(n, total) {
        a <- ipermutations(n)
        for (i in 1:total) a$getnext()
    }

    pkg_RcppAlgos <- function(n, total) {
        a <- permuteIter(n)
        for (i in 1:total) a@nextIter()
    }

    total <- permuteCount(8)
    total
    #> [1] 40320

    microbenchmark(cbRcppAlgos    = pkg_RcppAlgos(8, total),
                   cbArrangements = pkg_arrangements(8, total),
                   times = 15, unit = "relative")
    #> Unit: relative
    #>            expr   min    lq  mean median    uq   max neval cld
    #>     cbRcppAlgos  1.00  1.00  1.00   1.00  1.00  1.00    15  a
    #>  cbArrangements 19.14 18.88 18.24  17.93 18.21 17.09    15   b

### Partitions

    pkg_partitions <- function(n, total) {
        a <- firstpart(n)
        for (i in 1:(total - 1)) a <- nextpart(a)
    }

    pkg_arrangements <- function(n, total) {
        a <- ipartitions(n)
        for (i in 1:total) a$getnext()
    }

    pkg_RcppAlgos <- function(n, total) {
        a <- partitionsIter(0:n, repetition = TRUE)
        for (i in 1:total) a@nextIter()
    }

    total <- partitionsCount(0:40, repetition = TRUE)
    total
    #> [1] 37338

    microbenchmark(cbRcppAlgos    = pkg_RcppAlgos(40, total),
                   cbArrangements = pkg_arrangements(40, total),
                   cbPartitions   = pkg_partitions(40, total),
                   times = 15, unit = "relative")
    #> Unit: relative
    #>            expr   min    lq  mean median    uq   max neval cld
    #>     cbRcppAlgos  1.00  1.00  1.00   1.00  1.00  1.00    15 a
    #>  cbArrangements 15.50 15.53 15.28  15.53 15.68 13.35    15  b
    #>    cbPartitions 25.28 25.30 25.00  25.50 24.79 23.28    15   c

### Compositions

    pkg_partitions <- function(n, total) {
        a <- firstcomposition(n)
        for (i in 1:(total - 1)) a <- nextcomposition(a, FALSE)
    }

    pkg_arrangements <- function(n, total) {
        a <- icompositions(n)
        for (i in 1:total) a$getnext()
    }

    pkg_RcppAlgos <- function(n, total) {
        a <- compositionsIter(0:n, repetition = TRUE)
        for (i in 1:total) a@nextIter()
    }

    total <- compositionsCount(0:15, repetition = TRUE)
    total
    #> [1] 16384

    microbenchmark(cbRcppAlgos    = pkg_RcppAlgos(15, total),
                   cbArrangements = pkg_arrangements(15, total),
                   cbPartitions   = pkg_partitions(15, total),
                   times = 15, unit = "relative")
    #> Unit: relative
    #>            expr   min    lq  mean median    uq   max neval cld
    #>     cbRcppAlgos  1.00  1.00  1.00   1.00  1.00  1.00    15 a
    #>  cbArrangements 13.87 13.69 13.47  13.53 13.23 13.00    15  b
    #>    cbPartitions 45.04 44.26 43.93  43.14 43.83 47.16    15   c