RcppAlgos
A collection of high performance functions and iterators implemented in C++ for solving problems in combinatorics and computational mathematics.
Featured Functions
{combo|permute}General: Generate all combinations/permutations of a vector (including multisets) meeting specific criteria.{partitions|compositions}General: Efficient algorithms for partitioning numbers under various constraints{expandGrid|comboGrid}: Generate traditional Cartesian product as well as the product where order does not matter.{combo|permute|partitions|compositions|expandGrid|comboGroups}Sample: Generate reproducible random samples{combo|permute|partitions|compositions|expandGrid|comboGroups}Iter: Flexible iterators allow for bidirectional iteration as well as random access.primeSieve: Fast prime number generatorprimeCount: Prime counting function using Legendre's formula
The primeSieve function and the primeCount function are both based off of the excellent work by Kim Walisch. The respective repos can be found here: kimwalisch/primesieve; kimwalisch/primecount
Additionally, many of the sieving functions make use of the fast integer division library libdivide by ridiculousfish.
Benchmarks
Installation
install.packages("RcppAlgos")
## install the development version
devtools::install_github("jwood000/RcppAlgos")Usage
Combinatorics Basics
## Find all 3-tuples combinations of 1:4
comboGeneral(4, 3)
#> [,1] [,2] [,3]
#> [1,] 1 2 3
#> [2,] 1 2 4
#> [3,] 1 3 4
#> [4,] 2 3 4
## Alternatively, iterate over combinations
a = comboIter(4, 3)
a@nextIter()
#> [1] 1 2 3
a@back()
#> [1] 2 3 4
a[[2]]
#> [1] 1 2 4
## Pass any atomic type vector
permuteGeneral(letters, 3, upper = 4)
#> [,1] [,2] [,3]
#> [1,] "a" "b" "c"
#> [2,] "a" "b" "d"
#> [3,] "a" "b" "e"
#> [4,] "a" "b" "f"
## Generate a reproducible sample
comboSample(10, 8, TRUE, n = 5, seed = 84)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1,] 3 3 3 6 6 10 10 10
#> [2,] 1 3 3 4 4 7 9 10
#> [3,] 3 7 7 7 9 10 10 10
#> [4,] 3 3 3 9 10 10 10 10
#> [5,] 1 2 2 3 3 4 4 7Integer Partitions and Constraints
## Flexible partitioning algorithms
partitionsGeneral(0:5, 3, freqs = rep(1:2, 3), target = 6)
#> [,1] [,2] [,3]
#> [1,] 0 1 5
#> [2,] 0 2 4
#> [3,] 0 3 3
#> [4,] 1 1 4
#> [5,] 1 2 3
## And compositions
compositionsGeneral(0:3, repetition = TRUE)
#> [,1] [,2] [,3]
#> [1,] 0 0 3
#> [2,] 0 1 2
#> [3,] 0 2 1
#> [4,] 1 1 1
## Get combinations such that the product is between
## 3600 and 4000 (including 3600 but not 4000)
comboGeneral(5, 7, TRUE, constraintFun = "prod",
comparisonFun = c(">=","<"),
limitConstraints = c(3600, 4000),
keepResults = TRUE)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1,] 1 2 3 5 5 5 5 3750
#> [2,] 1 3 3 4 4 5 5 3600
#> [3,] 1 3 4 4 4 4 5 3840
#> [4,] 2 2 3 3 4 5 5 3600
#> [5,] 2 2 3 4 4 4 5 3840
#> [6,] 3 3 3 3 3 3 5 3645
#> [7,] 3 3 3 3 3 4 4 3888
## We can even iterate over constrained cases. These are
## great when we don't know how many results there are upfront.
## Save on memory and still at the speed of C++!!
p = permuteIter(5, 7, TRUE, constraintFun = "prod",
comparisonFun = c(">=","<"),
limitConstraints = c(3600, 4000),
keepResults = TRUE)
## Get the next n results
t = p@nextNIter(1048)
## N.B. keepResults = TRUE adds the 8th column
tail(t)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1043,] 5 4 4 3 4 1 4 3840
#> [1044,] 5 4 4 3 4 4 1 3840
#> [1045,] 5 4 4 4 1 3 4 3840
#> [1046,] 5 4 4 4 1 4 3 3840
#> [1047,] 5 4 4 4 3 1 4 3840
#> [1048,] 5 4 4 4 3 4 1 3840
## Continue iterating from where we left off
p@nextIter()
#> [1] 5 4 4 4 4 1 3 3840
p@nextIter()
#> [1] 5 4 4 4 4 3 1 3840
p@nextIter()
#> [1] 2 2 3 3 4 5 5 3600
## N.B. totalResults and totalRemaining are NA because there is no
## closed form solution for determining this.
p@summary()
#> $description
#> [1] "Permutations with repetition of 5 choose 7 where the prod is between 3600 and 4000"
#>
#> $currentIndex
#> [1] 1051
#>
#> $totalResults
#> [1] NA
#>
#> $totalRemaining
#> [1] NACartesian Products
## Base R expand.grid returns a data.frame by default
## and varies the first column the fastest
bR = expand.grid(rep(list(1:3), 3))
head(bR, n = 3)
#> Var1 Var2 Var3
#> 1 1 1 1
#> 2 2 1 1
#> 3 3 1 1
tail(bR, n = 3)
#> Var1 Var2 Var3
#> 25 1 3 3
#> 26 2 3 3
#> 27 3 3 3
## RcppAlgos::expandGrid returns a matrix if the input is of
## the same class, otherwise it returns a data.frame. Also
## varies the first column the slowest.
algos = expandGrid(rep(list(1:3), 3))
head(algos, n = 3)
#> Var1 Var2 Var3
#> [1,] 1 1 1
#> [2,] 1 1 2
#> [3,] 1 1 3
tail(algos, n = 3)
#> Var1 Var2 Var3
#> [25,] 3 3 1
#> [26,] 3 3 2
#> [27,] 3 3 3
## N.B. Since we are passing more than one type, a data.frame is returned
expandGrid(
c(rep(list(letters[1:3]), 3), list(1:3)),
upper = 3
)
#> Var1 Var2 Var3 Var4
#> 1 a a a 1
#> 2 a a a 2
#> 3 a a a 3
## With RcppAlgos::comboGrid order doesn't matter, so c(1, 1, 2),
## c(1, 2, 1), and c(2, 1, 1) are the same.
comboGrid(rep(list(1:3), 3))
#> Var1 Var2 Var3
#> [1,] 1 1 1
#> [2,] 1 1 2
#> [3,] 1 1 3
#> [4,] 1 2 2
#> [5,] 1 2 3
#> [6,] 1 3 3
#> [7,] 2 2 2
#> [8,] 2 2 3
#> [9,] 2 3 3
#> [10,] 3 3 3
## If you don't want any repeats, set repetition = FALSE
comboGrid(rep(list(1:3), 3), repetition = FALSE)
#> Var1 Var2 Var3
#> [1,] 1 2 3Partitions of Groups
Efficiently partition a vector into groups with comboGroups. For example, the code below finds all possible pairings of groups of size 3 vs groups of size 2 (See this stackoverflow post: Find all possible team pairing schemes).
players <- c("Ross", "Bobby", "Max", "Casper", "Jake")
comboGroups(players, grpSizes = c(2, 3))
#> Grp1 Grp1 Grp2 Grp2 Grp2
#> [1,] "Ross" "Bobby" "Max" "Casper" "Jake"
#> [2,] "Ross" "Max" "Bobby" "Casper" "Jake"
#> [3,] "Ross" "Casper" "Bobby" "Max" "Jake"
#> [4,] "Ross" "Jake" "Bobby" "Max" "Casper"
#> [5,] "Bobby" "Max" "Ross" "Casper" "Jake"
#> [6,] "Bobby" "Casper" "Ross" "Max" "Jake"
#> [7,] "Bobby" "Jake" "Ross" "Max" "Casper"
#> [8,] "Max" "Casper" "Ross" "Bobby" "Jake"
#> [9,] "Max" "Jake" "Ross" "Bobby" "Casper"
#> [10,] "Casper" "Jake" "Ross" "Bobby" "Max"Computational Mathematics
## Generate prime numbers
primeSieve(50)
#> [1] 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
## Many of the functions can produce results in
## parallel for even greater performance
p = primeSieve(1e15, 1e15 + 1e8, nThreads = 4)
head(p)
#> [1] 1000000000000037 1000000000000091 1000000000000159
#> [4] 1000000000000187 1000000000000223 1000000000000241
tail(p)
#> [1] 1000000099999847 1000000099999867 1000000099999907
#> [4] 1000000099999919 1000000099999931 1000000099999963
## Count prime numbers less than n
primeCount(1e10)
#> [1] 455052511
## Get the prime factorization
set.seed(24028)
primeFactorize(sample(1e15, 3), namedList = TRUE)
#> $`701030825091514`
#> [1] 2 149 2352452433193
#>
#> $`83054168594779`
#> [1] 3098071 26808349
#>
#> $`397803024735610`
#> [1] 2 5 13 13 235386405169Further Reading
- Function Documentation
- Computational Mathematics Overview
- Combination and Permutation Basics
- Combinatorial Sampling
- Constraints, Partitions, and Compositions
- Attacking Problems Related to the Subset Sum Problem
- Combinatorial Iterators in RcppAlgos
- Cartesian Products and Partitions of Groups
Why RcppAlgos but no Rcpp?
Previous versions of RcppAlgos relied on Rcpp to ease the burden of exposing C++ to R. While the current version of RcppAlgos does not utilize Rcpp, it would not be possible without the myriad of excellent contributions to Rcpp.
Contact
If you would like to report a bug, have a question, or have suggestions for possible improvements, please file an issue.