High Performance Tools for Combinatorics and Computational Mathematics

Provides optimized functions and flexible combinatorial iterators implemented in C++ with 'Rcpp' for solving problems in combinatorics and computational mathematics. Utilizes parallel programming via 'RcppThread' for maximal performance. Also makes use of the RMatrix class from the 'RcppParallel' library. There are combination/permutation functions with constraint parameters that allow for generation of all results of a vector meeting specific criteria (e.g. generating integer partitions/compositions or finding all combinations such that the sum is between two bounds). Capable of generating specific combinations/permutations (e.g. retrieve only the nth lexicographical result) which sets up nicely for parallelization as well as random sampling. Gmp support permits exploration where the total number of results is large (e.g. comboSample(10000, 500, n = 4)). Additionally, there are several high performance number theoretic functions that are useful for problems common in computational mathematics. Some of these functions make use of the fast integer division library 'libdivide'. The primeSieve function is based on the segmented sieve of Eratosthenes implementation by Kim Walisch. It is also efficient for large numbers by using the cache friendly improvements originally developed by Tomás Oliveira. Finally, there is a prime counting function that implements Legendre's formula based on the work of Kim Walisch.

Overview

A collection of high performance functions implemented in C++ with Rcpp for solving problems in combinatorics and computational mathematics. Utilizes the library RcppThread where multithreading is needed. We also make use of the RMatrix.h header file from RcppParallel for thread safe accessors for Rcpp matrices. Featured functions:

• primeSieve - Generates all primes less than a billion in under 0.5 seconds.

system.time(b <- primeSieve(10^9, nThreads = 8))
 user  system elapsed 
2.339   0.047   0.418

• primeCount - Counts the number of primes below a 1e12 in ~130 milliseconds and 1e15 in under 50 seconds.

system.time(primeCount(1e12, nThreads = 8))
   user  system elapsed 
  0.804   0.005   0.127

• comboGeneral/permuteGeneral - Generate all combinations/permutations of a vector (including multisets) meeting specific criteria.
  • Produce results in parallel using the Parallel argument. You can also apply each of the five compiled functions given by the argument constraintFun in parallel as well. E.g. Obtaining the row sums of all combinations:
    • comboGeneral(20, 10, constraintFun = "sum", Parallel = TRUE)
  • Alternatively, the arguments lower and upper make it possible to generate combinations/permutations in chunks allowing for parallelization via the package parallel. This is convenient when you want to apply a custom function to the output in parallel as well (see this stackoverflow post for a use case).
  • GMP support allows for exploration of combinations/permutations of vectors with many elements.
• comboSample/permuteSample - Easily generate random samples of combinations/permutations 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.

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.

Installation

install.packages("RcppAlgos")
 
## Or install the development version
devtools::install_github("jwood000/RcppAlgos")

Usage

Common Combinatorial Functions

Easily executed with a very simple interface. Output is in lexicographical order.

## Find all 3-tuples combinations without 
## repetition of the numbers c(1, 2, 3, 4).
comboGeneral(4, 3)
     [,1] [,2] [,3]
[1,]   1    2    3
[2,]   1    2    4
[3,]   1    3    4
[4,]   2    3    4
 
 
## Find all 3-tuples permutations without
## repetition of the numbers c(1, 2, 3, 4).
permuteGeneral(4, 3)
      [,1] [,2] [,3]
 [1,]    1    2    3
 [2,]    1    2    4
 [3,]    1    3    2
 [4,]    1    3    4
  .      .    .    .
  .      .    .    .
[21,]    4    2    1
[22,]    4    2    3
[23,]    4    3    1
[24,]    4    3    2
 
## For combinations/permutations with repetition, simply
## set the repetition argument to TRUE
comboGeneral(4, 3, repetition = TRUE)
      [,1] [,2] [,3]
 [1,]    1    1    1
 [2,]    1    1    2
 [3,]    1    1    3
  .      .    .    .
  .      .    .    .
  
## They are very efficient
system.time(comboGeneral(25, 13))
   user  system elapsed 
  0.116   0.062   0.178
 
system.time(comboGeneral(25, 13, nThreads = 8))
   user  system elapsed 
  0.192   0.228   0.058
 
nrow(comboGeneral(25,13))
[1] 5200300

Using Constraints

Oftentimes, one needs to find combinations/permutations that meet certain requirements.

## Generate some random data
set.seed(101)
s <- sample(500, 20)
## Find all 5-tuples permutations without repetition
## of s (defined above) such that the sum is equal to 1176.
p <- permuteGeneral(v = s,
                    m = 5, 
                    constraintFun = "sum", 
                    comparisonFun = "==", 
                    limitConstraints = 1176,
                    keepResults = TRUE)
 
tail(p)
        [,1] [,2] [,3] [,4] [,5] [,6]
[3955,]  354  287  149  187  199 1176
[3956,]  354  287  149  199  187 1176
[3957,]  354  287  187  149  199 1176
[3958,]  354  287  187  199  149 1176
[3959,]  354  287  199  149  187 1176
[3960,]  354  287  199  187  149 1176
 
 
## 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

Working with Multisets

Sometimes, the standard combination/permutation functions don't quite get us to our desired goals. For example, one may need all permutations of a vector with some of the elements repeated a specific amount of times (i.e. a multiset). Consider the following vector a <- c(1,1,1,1,2,2,2,7,7,7,7,7) and one would like to find permutations of a of length 6. Using traditional methods, we would need to generate all permutations, then eliminate duplicate values. Even considering that permuteGeneral is very efficient, this approach is clunky and not as fast as it could be. Observe:

getPermsWithSpecificRepetition <- function(z, n) {
    b <- permuteGeneral(z, n)
    myDupes <- duplicated(apply(b, 1, paste, collapse=""))
    b[!myDupes, ]
}
 
a <- c(1,1,1,1,2,2,2,7,7,7,7,7)
 
system.time(test <- getPermsWithSpecificRepetition(a, 6))
   user  system elapsed 
  4.168   0.046   4.230

Enter freqs

Situations like this call for the use of the freqs argument. Simply, enter the number of times each unique element is repeated and Voila!

system.time(test2 <- permuteGeneral(unique(a), 6, freqs = rle(a)$lengths))
   user  system elapsed 
      0       0       0
      
identical(test, test2)
[1] TRUE

Here are some more general examples with multisets:

## Generate all permutations of a vector with specific
## length of repetition for each element (i.e. multiset)
permuteGeneral(3, freqs = c(1,2,2))
      [,1] [,2] [,3] [,4] [,5]
 [1,]    1    2    2    3    3
 [2,]    1    2    3    2    3
 [3,]    1    2    3    3    2
 [4,]    1    3    2    2    3
 [5,]    1    3    2    3    2
  .      .    .    .    .    .
  .      .    .    .    .    .
[26,]    3    2    3    1    2
[27,]    3    2    3    2    1
[28,]    3    3    1    2    2
[29,]    3    3    2    1    2
[30,]    3    3    2    2    1
 
## or combinations of a certain length
comboGeneral(3, 2, freqs = c(1,2,2), constraintFun = "prod")
     [,1] [,2] [,3]
[1,]    1    2    2
[2,]    1    3    3
[3,]    2    2    4
[4,]    2    3    6
[5,]    3    3    9

All Combinatorial Functions Work with Factors

facPerms <- permuteGeneral(factor(c("low", "med", "high"),
                                   levels = c("low", "med", "high"),
                                   ordered = TRUE),
                                   freqs = c(1,4,2))
     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] low  med  med  med  med  high high
[2,] low  med  med  med  high med  high
[3,] low  med  med  med  high high med 
[4,] low  med  med  high med  med  high
[5,] low  med  med  high med  high med 
Levels: low < med < high

Parallel Computing

Using the parameter Parallel, we can easily generate combinations/permutations with great efficiency.

## RcppAlgos uses the number of cores available minus one
RcppAlgos::stdThreadMax()
[1] 8
 
identical(comboGeneral(20, 10, freqs = rep(1:4, 5)),
          comboGeneral(20, 10, freqs = rep(1:4, 5), Parallel = TRUE))
[1] TRUE
 
## Using 7 cores
library(microbenchmark)
microbenchmark(serial = comboGeneral(20, 10, freqs = rep(1:4, 5)),
             parallel = comboGeneral(20, 10, freqs = rep(1:4, 5), Parallel = TRUE))
Unit: milliseconds
     expr       min        lq      mean    median        uq      max neval
   serial 237.80421 244.06242 253.76806 249.45499 263.65091 284.7350   100
 parallel  82.11218  85.88929  88.78034  87.72153  90.01558 126.3167   100

And applying any of the constraint functions in parallel is highly efficient as well. Consider obtaining the row sums of all combinations:

## base R using combn and FUN
combnSum <- combn(20, 10, sum)
algosSum <- comboGeneral(20, 10, constraintFun = "sum")
 
identical(as.integer(combnSum), algosSum[,11])
[1] TRUE
 
## Using parallel
paralSum <- comboGeneral(20, 10, constraintFun = "sum", Parallel = TRUE)
identical(paralSum, algosSum)
[1] TRUE
 
microbenchmark(serial = comboGeneral(20, 10, constraintFun = "sum"),
             parallel = comboGeneral(20, 10, constraintFun = "sum", Parallel = TRUE),
             combnSum = combn(20, 10, sum))
Unit: milliseconds
     expr        min         lq       mean     median         uq        max neval
   serial   3.426027   3.736279   4.285025   3.908142   4.312603  10.247652   100
 parallel   1.233491   1.371532   1.742455   1.450557   1.970033   6.102456   100
 combnSum 206.819174 222.254082 227.762687 224.331548 227.135461 293.065110   100

Faster than rowSums and rowMeans

In fact, finding row sums or row means is even faster than simply applying the highly efficient rowSums/rowMeans after the combinations have already been generated:

## Pre-generate combinations
combs <- comboGeneral(25, 10)
 
## Testing rowSums alone against generating combinations as well as summing
microbenchmark(serial = comboGeneral(25, 10, constraintFun = "sum"),
             parallel = comboGeneral(25, 10, constraintFun = "sum", Parallel = TRUE),
              rowsums = rowSums(combs))
Unit: milliseconds
     expr       min        lq      mean    median        uq       max neval
   serial 113.74535 124.91470 127.85210 127.15044 130.47335 172.37043   100
 parallel  41.23766  49.78379  51.46558  51.07019  52.58048  88.31258   100
  rowsums  95.92139  98.21449 104.66987 101.32041 105.46913 152.69522   100
 
all.equal(rowSums(combs), 
          comboGeneral(25, 10, 
                       constraintFun = "sum",
                       Parallel = TRUE)[,11])
[1] TRUE
 
## Testing rowMeans alone against generating combinations as well as obtain row means
microbenchmark(serial = comboGeneral(25, 10, constraintFun = "mean"),
             parallel = comboGeneral(25, 10, constraintFun = "mean", Parallel = TRUE),
             rowmeans = rowMeans(combs))
Unit: milliseconds
     expr       min        lq      mean    median        uq      max neval
   serial 173.55130 183.94174 203.09916 202.17554 217.77389 274.7437   100
 parallel  58.11018  61.17694  79.59311  79.21929  97.96688 127.3256   100
 rowmeans 102.56019 103.72126 113.12717 106.89584 117.95641 165.1638   100
 
all.equal(rowMeans(combs), 
          comboGeneral(25, 10, 
                       constraintFun = "mean",
                       Parallel = TRUE)[,11])
[1] TRUE

In both cases above, RcppAlgos is doing double the work nearly twice as fast!!!

Using arguments lower and upper

There are arguments lower and upper that can be utilized to generate chunks of combinations/permutations without having to generate all of them followed by subsetting. As the output is in lexicographical order, these arguments specify where to start and stop generating. For example, comboGeneral(5, 3) outputs 10 combinations of the vector 1:5 chosen 3 at a time. We can set lower to 5 in order to start generation from the 5^th lexicographical combination. Similarly, we can set upper to 4 in order only generate the first 4 combinations. We can also use them together to produce only a certain chunk of combinations. For example, setting lower to 4 and upper to 6 only produces the 4^th, 5^th, and 6^th lexicographical combinations. Observe:

comboGeneral(5, 3, lower = 4, upper = 6)
     [,1] [,2] [,3]
[1,]    1    3    4
[2,]    1    3    5
[3,]    1    4    5
 
## is equivalent to the following:
comboGeneral(5, 3)[4:6, ]
     [,1] [,2] [,3]
[1,]    1    3    4
[2,]    1    3    5
[3,]    1    4    5

In addition to being useful by avoiding the unnecessary overhead of generating all combination/permutations followed by subsetting just to see a few specific results, lower and upper can be utilized to generate large number of combinations/permutations in parallel. Observe:

## Over 3 billion results
comboCount(35, 15)
[1] 3247943160
 
## 10086780 evenly divides 3247943160
 
system.time(lapply(seq(1, 3247943160, 10086780), function(x) {
        temp <- comboGeneral(35, 15, lower = x, upper = x + 10086779)
        ## do something
        x
}))
   user  system elapsed 
 99.495  49.993 149.467
 
## Enter parallel
 
library(parallel)
system.time(mclapply(seq(1, 3247943160, 10086780), function(x) {
     temp <- comboGeneral(35, 15, lower = x, upper = x + 10086779)
     ## do something
     x
}, mc.cores = 6))
   user  system elapsed 
125.361  85.916  35.641

These arguments are also useful when one needs to explore combinations/permutations of really large vectors:

set.seed(222)
myVec <- rnorm(1000)
 
## HUGE number of combinations
comboCount(myVec, 50, repetition = TRUE)
Big Integer ('bigz') :
[1] 109740941767310814894854141592555528130828577427079559745647393417766593803205094888320
 
## Let's look at one hundred thousand combinations in the range (1e15 + 1, 1e15 + 1e5)
system.time(b <- comboGeneral(myVec, 50, TRUE, 
                              lower = 1e15 + 1,
                              upper = 1e15 + 1e5))
   user  system elapsed 
  0.008   0.001   0.010
  
b[1:5, 45:50]
          [,1]      [,2]      [,3]     [,4]      [,5]       [,6]
[1,] 0.5454861 0.4787456 0.7797122 2.004614 -1.257629 -0.7740501
[2,] 0.5454861 0.4787456 0.7797122 2.004614 -1.257629  0.1224679
[3,] 0.5454861 0.4787456 0.7797122 2.004614 -1.257629 -0.2033493
[4,] 0.5454861 0.4787456 0.7797122 2.004614 -1.257629  1.5511027
[5,] 0.5454861 0.4787456 0.7797122 2.004614 -1.257629  1.0792094

Sampling

We can also produce random samples of combinations/permutations with comboSample and permuteSample. This is really useful when we need a reproducible set of random combinations/permutations. Many of the traditional ways of doing this involved relying on heavy use of sample and hoping that we don't generate duplicate results. Both functions have a similar interface to their respective General functions. Observe:

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    7
 
## We can also use 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

Just like the General counterparts (i.e. combo/permuteGeneral), we can easily explore combinations/permutations of large vectors where the total number of results is enormous in parallel (using Parallel or nThreads).

## Uses stdThreadMax() - 1 = 7 thread (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 = 4)
     [,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

User Defined Functions

You can also pass user defined functions by utilizing the argument FUN. This feature's main purpose is for convenience, however it is somewhat more efficient than generating all combinations/permutations and then using a function from the apply family (N.B. the argument Parallel has no effect when FUN is employed).

funCustomComb = function(n, r) {
    combs = comboGeneral(n, r)
    lapply(1:nrow(combs), function(x) cumprod(combs[x,]))
}
 
identical(funCustomComb(15, 8), comboGeneral(15, 8, FUN = cumprod))
[1] TRUE
 
library(microbenchmark)
microbenchmark(f1 = funCustomComb(15, 8),
                f2 = comboGeneral(15, 8, FUN = cumprod), unit = "relative")
unit: relative
 expr      min       lq     mean   median       uq      max neval
   f1 6.946481 6.891553 6.334866 6.821221 6.934111 2.686777   100
   f2 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000   100
   
comboGeneral(15, 8, FUN = cumprod, upper = 3)
[[1]]
[1]     1     2     6    24   120   720  5040 40320
 
[[2]]
[1]     1     2     6    24   120   720  5040 45360
 
[[3]]
[1]     1     2     6    24   120   720  5040 50400
 
## Works with the sampling functions as well
permuteSample(5000, 1000, n = 3, seed = 101, FUN = sd)
[[1]]
[1] 1431.949
 
[[2]]
[1] 1446.859
 
[[3]]
[1] 1449.272

Mathematical Computation

RcppAlgos comes equipped with several functions for quickly generating essential components for problems common in computational mathematics. All functions below can be executed in parallel by using the argument nThreads.

The following sieving functions (primeFactorizeSieve, divisorsSieve, numDivisorSieve, & eulerPhiSieve) are very useful and flexible. Generate components up to a number or between two bounds.

## get the number of divisors for every number from 1 to n
numDivisorSieve(20)
 [1] 1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6
 
## If you want the complete factorization from 1 to n, use divisorsList
system.time(allFacs <- divisorsSieve(10^5, namedList = TRUE))
   user  system elapsed 
  0.045   0.004   0.049
 
allFacs[c(4339, 15613, 22080)]
$`4339`
[1]    1 4339
 
$`15613`
[1]     1    13  1201 15613
 
$`22080`
 [1]     1     2     3     4     5     6     8    10    12    15
[11]    16    20    23    24    30    32    40    46    48    60
[21]    64    69    80    92    96   115   120   138   160   184
[31]   192   230   240   276   320   345   368   460   480   552
[41]   690   736   920   960  1104  1380  1472  1840  2208  2760
[51]  3680  4416  5520  7360 11040 22080
 
 
## Between two bounds
primeFactorizeSieve(10^12, 10^12 + 5)
[[1]]
 [1] 2 2 2 2 2 2 2 2 2 2 2 2 5 5 5 5 5 5 5 5 5 5 5 5
 
[[2]]
[1]       73      137 99990001
 
[[3]]
[1]            2            3 166666666667
 
[[4]]
[1]      61   14221 1152763
 
[[5]]
[1]      2      2     17    149    197 501001
 
[[6]]
[1]           3           5 66666666667
 
 
## Creating a named object
eulerPhiSieve(20, namedVector = TRUE)
 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 
 1  1  2  2  4  2  6  4  6  4 10  4 12  6  8  8 16  6 18  8
 
 
system.time(a <- eulerPhiSieve(1e12, 1e12 + 1e7))
   user  system elapsed 
  1.092   0.044   1.143
 
## Using nThreads for greater efficiency
system.time(b <- eulerPhiSieve(1e12, 1e12 + 1e7, nThreads = 8))
   user  system elapsed 
  3.784   0.014   0.518
  
identical(a, b)
[1] TRUE

Vectorized Functions

There are three very fast vectorized functions for general factoring (e.g. all divisors of number), primality testing, as well as prime factoring (divisorsRcpp, isPrimeRcpp, primeFactorize)

## get result for individual numbers
primeFactorize(123456789)
[1]    3    3 3607 3803
 
 
## or for an entire vector
set.seed(100)
myVec <- sample(-100000000:100000000, 5)
divisorsRcpp(myVec, namedList = TRUE)
$`-38446778`
[1] -38446778 -19223389        -2        -1         1
[6]         2  19223389  38446778
 
$`-48465500`
 [1] -48465500 -24232750 -12116375  -9693100  -4846550
 [6]  -2423275  -1938620   -969310   -484655   -387724
[11]   -193862    -96931      -500      -250      -125
[16]      -100       -50       -25       -20       -10
[21]        -5        -4        -2        -1         1
[26]         2         4         5        10        20
[31]        25        50       100       125       250
[36]       500     96931    193862    387724    484655
[41]    969310   1938620   2423275   4846550   9693100
[46]  12116375  24232750  48465500
 
$`10464487`
[1]        1       11      317     3001     3487    33011
[7]   951317 10464487
 
$`-88723370`
 [1] -88723370 -44361685 -17744674  -8872337       -10
 [6]        -5        -2        -1         1         2
[11]         5        10   8872337  17744674  44361685
[16]  88723370
 
$`-6290143`
[1] -6290143       -1        1  6290143
 
 
## Creating a named object
isPrimeRcpp(995:1000, namedVector = TRUE)
  995   996   997   998   999  1000 
FALSE FALSE  TRUE FALSE FALSE FALSE
 
 
system.time(a <- primeFactorize(1e12:(1e12 + 3e4)))
   user  system elapsed 
  1.202   0.002   1.207
 
## Using nThreads for greater efficiency  
system.time(b <- primeFactorize(1e12:(1e12 + 3e4), nThreads = 8))
   user  system elapsed 
  1.674   0.003   0.220
  
identical(a, b)
[1] TRUE

primeSieve & primeCount

Both of these functions are based on the excellent algorithms developed by Kim Walisch. First primeSieve:

## Quickly generate large primes over small interval
options(scipen = 50)
system.time(myPs <- primeSieve(10^13+10^3, 10^13))
   user  system elapsed 
  0.035   0.002   0.036
  
myPs
 [1] 10000000000037 10000000000051 10000000000099 10000000000129
 [5] 10000000000183 10000000000259 10000000000267 10000000000273
 [9] 10000000000279 10000000000283 10000000000313 10000000000343
[13] 10000000000391 10000000000411 10000000000433 10000000000453
[17] 10000000000591 10000000000609 10000000000643 10000000000649
[21] 10000000000657 10000000000687 10000000000691 10000000000717
[25] 10000000000729 10000000000751 10000000000759 10000000000777
[29] 10000000000853 10000000000883 10000000000943 10000000000957
[33] 10000000000987 10000000000993
 
## Object created is small
object.size(myPs)
312 bytes
 
## primes under a billion!!!
system.time(a <- primeSieve(10^9))
   user  system elapsed 
  1.263   0.095   1.368
 
## Using nThreads  
system.time(b <- primeSieve(10^9, nThreads = 8))
   user  system elapsed 
  2.339   0.047   0.418

Larger primes

Since version 2.3.0, we are implementing the cache-friendly improvements for larger primes originally developed by Tomás Oliveira.

## Version <= 2.2.0.. i.e. older versions
system.time(old <- RcppAlgos2.2::primeSieve(1e15, 1e15 + 1e9))
   user  system elapsed 
  7.615   0.140   7.792
 
## v2.3.0 is over 3x faster!  
system.time(a <- primeSieve(1e15, 1e15 + 1e9))
   user  system elapsed 
  2.530   0.197   2.744 
  
## And using nThreads we are ~8x faster
system.time(b <- primeSieve(1e15, 1e15 + 1e9, nThreads = 8))
   user  system elapsed 
  5.087   0.778   0.997 
  
identical(a, b)
[1] TRUE
 
identical(a, old)
[1] TRUE

primeCount

The library by Kim Walisch relies on OpenMP for parallel computation with Legendre's Formula. Currently, the default compiler on macOS is clang, which does not support OpenMP. James Balamuta (a.k.a. TheCoatlessProfessor... well at least we think so) has written a great article on this topic, which you can find here: https://thecoatlessprofessor.com/programming/openmp-in-r-on-os-x/. One of the goals of RcppAlgos is to be accessible by all users. With this in mind, we set out to count primes in parallel without OpenMP.

At first glance, this seems trivial as we have a function in Primes.cpp called phiWorker that counts the primes up to x. If you look in phi.cpp in the primecount library by Kim Walisch, we see that OpenMP does its magic on a for loop that makes repeated calls to phi (which is what phiWorker is based on). All we need to do is break this loop into n intervals where n is the number of threads. Simple, right?

We can certainly do this, but what you will find is that n - 1 threads will complete very quickly and the n^th thread will be left with a heavy computation. In order to alleviate this unbalanced load, we take advantage of thread pooling provided by RcppThread which allows us to reuse threads efficiently as well as breaking up the loop mentioned above into smaller intervals. The idea is to completely calculate phi up to a limit m using all n threads and then gradually increase m. The advantage here is that we are benefiting greatly from the caching done by the work of the previous n threads.

With this is mind, here are some results:

## Enumerate the number of primes below trillion
system.time(underOneTrillion <- primeCount(10^12))
   user  system elapsed 
  0.481   0.000   0.483
  
underOneTrillion
[1] 37607912018
 
 
## Enumerate the number of primes below a billion in 2 milliseconds
library(microbenchmark)
microbenchmark(primeCount(10^9))
Unit: milliseconds
             expr       min      lq    mean   median       uq      max neval
 primeCount(10^9) 1.958806 1.960901 2.05824 1.994066 2.099495 3.610174   100
 
 
primeCount(10^9)
[1] 50847534
 
system.time(underOneHundredTrillion <- primeCount(1e14, nThreads = 8))
   user  system elapsed 
 49.894   0.102   6.774
 
underOneHundredTrillion
[1] 3204941750802
 
## From Kim Walisch's primecount library:
## Josephs-MBP:primecount-4 josephwood$ ./primecount 1e14 --legendre --time
## 3204941750802
## Seconds: 4.441

Contact

I welcome any and all feedback. If you would like to report a bug, have a question, or have suggestions for possible improvements, please contact me here: [email protected]