Combinatorial Iterators in RcppAlgos

Joseph Wood

03/29/2022

This document covers working with combinatorial iterators in RcppAlgos. Combinatorial iterators in RcppAlgos are memory efficient like traditional iterator objects. They allow traversal of combinations/permutations one by one without the necessity for storing all results in memory.

Unlike traditional combinatorial iterators, the iterators in RcppAlgos offers random access via the [[ operator. This means, we can access the nth lexicographical order result on demand without having to first iterate over the previous n - 1 results.

Iterating over Combinations and Permutations

In order to iterate, we must initialize an iterator via comboIter or permuteIter. The interface is very similar to comboGeneral and permuteGeneral.

library(RcppAlgos)

## Initialize the iterator
a = comboIter(5, 3)

## Get the first combination
a$nextIter()
#> [1] 1 2 3

## And the next
a$nextIter()
#> [1] 1 2 4

## Set the current iterator to a variable
iter = a$currIter()
i = 1

## Iterate until there are no more
while (!is.null(iter)) {
    cat(i, " ", iter, "\n")
    iter = a$nextIter()
    i = i + 1
}
#> 1   1 2 4
#> 2   1 2 5
#> 3   1 3 4
#> 4   1 3 5
#> 5   1 4 5
#> 6   2 3 4
#> 7   2 3 5
#> 8   2 4 5
#> 9   3 4 5
#> No more results. To see the last result, use the prevIter method(s)

## See the output of comboGeneral for comparison
comboGeneral(5, 3, lower = 2)
#>       [,1] [,2] [,3]
#>  [1,]    1    2    4
#>  [2,]    1    2    5
#>  [3,]    1    3    4
#>  [4,]    1    3    5
#>  [5,]    1    4    5
#>  [6,]    2    3    4
#>  [7,]    2    3    5
#>  [8,]    2    4    5
#>  [9,]    3    4    5

## Call the summary method to see information about our iterator
a$summary()
#> $description
#> [1] "Combinations of 5 choose 3"
#>
#> $currentIndex
#> [1] 11
#>
#> $totalResults
#> [1] 10
#>
#> $totalRemaining
#> [1] -1

Bidirectional Iterators

Some of the combinatorial iterators in RcppAlgos are bidirectional iterators. This means that not only can we iterate in a forward manner (i.e. lexicographically), but we can also iterate backwards (i.e. Reverse Lexicographical Order) via the prevIter method(s).

## Using the same iterable from the previous section
a$currIter()
#> No more results. To see the last result, use the prevIter method(s)
#>
#> NULL

## As the comment says, we call the prevIter method to see the last result
a$prevIter()
#> [1] 3 4 5

## Get the previous result
a$prevIter()
#> [1] 2 4 5

## As in the previous example, we set the current iterator to a variable
iter = a$currIter()

## Defined above
print(i)
#> [1] 10

## Iterate until we are at the very beginning. Note that the
## output is exactly the same as above, but in reverse order
while (!is.null(iter)) {
    i = i - 1
    cat(i, " ", iter, "\n")
    iter = a$prevIter()
}
#> 9   2 4 5
#> 8   2 3 5
#> 7   2 3 4
#> 6   1 4 5
#> 5   1 3 5
#> 4   1 3 4
#> 3   1 2 5
#> 2   1 2 4
#> 1   1 2 3
#> Iterator Initialized. To see the first result, use the nextIter method(s)

## Call the summary method to see information about our iterator
a$summary()
#> $description
#> [1] "Combinations of 5 choose 3"
#>
#> $currentIndex
#> [1] 0
#>
#> $totalResults
#> [1] 10
#>
#> $totalRemaining
#> [1] 10

Retrieving More than One Result at a Time

There are four methods which allow for obtaining more than one result at a time: nextNIter, prevNIter, nextRemaining, and prevRemaining.

## Reset the iterator
a$startOver()

## Get the next 4 combinations
a$nextNIter(4)
#>      [,1] [,2] [,3]
#> [1,]    1    2    3
#> [2,]    1    2    4
#> [3,]    1    2    5
#> [4,]    1    3    4

## Get the summary. Note that the index has been updated
a$summary()
#> $description
#> [1] "Combinations of 5 choose 3"
#>
#> $currentIndex
#> [1] 4
#>
#> $totalResults
#> [1] 10
#>
#> $totalRemaining
#> [1] 6

## View the current combination
a$currIter()
#> [1] 1 3 4

## Get the remaining combinations with nextRemaining
a$nextRemaining()
#>      [,1] [,2] [,3]
#> [1,]    1    3    5
#> [2,]    1    4    5
#> [3,]    2    3    4
#> [4,]    2    3    5
#> [5,]    2    4    5
#> [6,]    3    4    5

a$summary()
#> $description
#> [1] "Combinations of 5 choose 3"
#>
#> $currentIndex
#> [1] 11
#>
#> $totalResults
#> [1] 10
#>
#> $totalRemaining
#> [1] -1

Now, we look at the opposite direction.

## Get the previous 4 combinations
a$prevNIter(4)
#>      [,1] [,2] [,3]
#> [1,]    3    4    5
#> [2,]    2    4    5
#> [3,]    2    3    5
#> [4,]    2    3    4

## Get the summary. Note that the index has been updated
a$summary()
#> $description
#> [1] "Combinations of 5 choose 3"
#>
#> $currentIndex
#> [1] 7
#>
#> $totalResults
#> [1] 10
#>
#> $totalRemaining
#> [1] 3

## View the current combination
a$currIter()
#> [1] 2 3 4

## Get the remaining previous combinations with prevRemaining
a$prevRemaining()
#>      [,1] [,2] [,3]
#> [1,]    1    4    5
#> [2,]    1    3    5
#> [3,]    1    3    4
#> [4,]    1    2    5
#> [5,]    1    2    4
#> [6,]    1    2    3

a$summary()
#> $description
#> [1] "Combinations of 5 choose 3"
#>
#> $currentIndex
#> [1] 0
#>
#> $totalResults
#> [1] 10
#>
#> $totalRemaining
#> [1] 10

Random Access Iterator

As with the bidirectional iterators, with some of the combinatorial iterators in RcppAlgos, we can jump to the nth result without the need for iterating over the first n - 1 results.

## Reset the iterator
a$startOver()

## How many total combinations do we have?
a$summary()$totalResults
#> [1] 10

## Let's get the 3rd combination
a[[3]]
#> [1] 1 2 5

## See the summary. Note that the index has been updated
a$summary()
#> $description
#> [1] "Combinations of 5 choose 3"
#>
#> $currentIndex
#> [1] 3
#>
#> $totalResults
#> [1] 10
#>
#> $totalRemaining
#> [1] 7

## Let's see the 9th combination
a[[9]]
#> [1] 2 4 5

## What about the first and last combination?
a$front()
#> [1] 1 2 3

a$back()
#> [1] 3 4 5

## Again the index has been updated
a$summary()
#> $description
#> [1] "Combinations of 5 choose 3"
#>
#> $currentIndex
#> [1] 10
#>
#> $totalResults
#> [1] 10
#>
#> $totalRemaining
#> [1] 0

a$currIter()
#> [1] 3 4 5

We can also easily return a random sample of combinations with the [[ operator by passing a vector of indices. In these cases, it should be noted that the current index will not be updated.

## Set the current index to the second combination
a[[2]]
#> [1] 1 2 4

set.seed(121)
samp = sample(a$summary()$totalResults, 4)

samp
#> [1]  4  7 10  1

a[[samp]]
#>      [,1] [,2] [,3]
#> [1,]    1    3    4
#> [2,]    2    3    4
#> [3,]    3    4    5
#> [4,]    1    2    3

## Note that the current index remains unchanged
a$summary()
#> $description
#> [1] "Combinations of 5 choose 3"
#>
#> $currentIndex
#> [1] 2
#>
#> $totalResults
#> [1] 10
#>
#> $totalRemaining
#> [1] 8

User Defined Functions

Just as with comboGeneral and permuteGeneral, we can pass a user defined function to comboIter and permuteIter.

## Initialize the iterator
b = permuteIter(LETTERS[1:4], 3, FUN = function(p) paste(p, collapse = ""),
                FUN.VALUE = "a")
b$nextIter()
#> [1] "ABC"

b$nextNIter(5)
#> [1] "ABD" "ACB" "ACD" "ADB" "ADC"

b$back()
#> [1] "DCB"

b$prevIter()
#> [1] "DCA"

b$prevNIter(5)
#> [1] "DBC" "DBA" "DAC" "DAB" "CDB"

b$nextRemaining()
#> [1] "DAB" "DAC" "DBA" "DBC" "DCA" "DCB"

## Random access
b[[5]]
#> [1] "ADB"

b$prevRemaining()
#> [1] "ACD" "ACB" "ABD" "ABC"

## View the source vector
b$sourceVector()
#> [1] "A" "B" "C" "D"

New in Verison 2.5.0

As of version 2.5.0, we no longer rely on Rcpp as a dependency, which means that we do not utilize Rcpp modules for exposing C++ classes. This is now carried out using external pointers (See External pointers and weak references) along with S4 Classes. We use the slots of S4 classes for exposing each method so access is carried out with the “at sign”, @. We have also added the ability to access each method with the “dollar sign”, $, for backwards compatibility.

Access Efficiency in 2.5.0

Our tests show that accessing methods is much more efficient in 2.5.0 compared to prior versions. In the below tests, we measure excecution time of calling nextIter multiple times in different versions. We will use the function test_nextIter for our testing. If one needs to reproduce, simply download the 2.4.3 tar here: https://cran.r-project.org/src/contrib/Archive/RcppAlgos/, change RcppAlgos to RcppAlgos243 in a few place (e.g. DESCRIPTION, NAMESPACE, etc.), and rebuild.

test_nextIter <- function(n, m, get_val = FALSE, v = 243) {
    a <- if (v == 243) {
        RcppAlgos243::comboIter(n, m)
    } else {
        comboIter(n, m)
    }

    total <- comboCount(n, m)

    if (get_val) {
        mat <- matrix(0L, nrow = total, ncol = m)
        for (i in 1:total) mat[i, ] <- a$nextIter()
        return(mat)
    } else {
        if (v == 243) {
            for (i in 1:total) a$nextIter()
        } else {
            for (i in 1:total) a@nextIter()
        }

        invisible(NULL)
    }
}

Version 2.4.3 Using Rcpp

library(microbenchmark)
## Using R version 4.1.3
comboCount(15, 8)
#> [1] 6435

microbenchmark(test_nextIter(15, 8))
#> Unit: milliseconds
#>                  expr      min       lq     mean   median       uq      max neval
#>  test_nextIter(15, 8) 46.31869 51.43212 57.28727 51.93104 52.92773 516.2674   100

identical(test_nextIter(15, 8, get_val = TRUE), comboGeneral(15, 8))
#> [1] TRUE

comboCount(25, 10)
#> [1] 3268760

system.time(test_nextIter(25, 10))
#>    user  system elapsed
#>  23.537   0.070  23.614

Rprof("Version243.out", memory.profiling = TRUE)
test_nextIter(25, 10)
Rprof(NULL)
summaryRprof("Version243.out", memory = "both")
#> $by.self
#>                  self.time self.pct total.time total.pct mem.total
#> "as.environment"      6.98    32.37       6.98     32.37    4045.5
#> "$"                   6.28    29.13      16.40     76.07    9433.6
#> "test_nextIter"       2.36    10.95      21.56    100.00   12438.3
#> ".External"           2.02     9.37       2.02      9.37    1172.9
#> "exists"              1.50     6.96       1.50      6.96     823.7
#> "get"                 1.50     6.96       1.50      6.96     856.2
#> "a$nextIter"          0.78     3.62       2.80     12.99    1620.6
#> "is.symbol"           0.14     0.65       0.14      0.65      72.5
#>
#> $by.total
#>                  total.time total.pct mem.total self.time self.pct
#> "test_nextIter"       21.56    100.00   12438.3      2.36    10.95
#> "$"                   16.40     76.07    9433.6      6.28    29.13
#> "as.environment"       6.98     32.37    4045.5      6.98    32.37
#> "a$nextIter"           2.80     12.99    1620.6      0.78     3.62
#> ".External"            2.02      9.37    1172.9      2.02     9.37
#> "exists"               1.50      6.96     823.7      1.50     6.96
#> "get"                  1.50      6.96     856.2      1.50     6.96
#> "is.symbol"            0.14      0.65      72.5      0.14     0.65
#>
#> $sample.interval
#> [1] 0.02
#>
#> $sampling.time
#> [1] 21.56

Version 2.5.0 (No Rcpp)

microbenchmark(test_nextIter(15, 8, v = 250))
#> Unit: milliseconds
#>                           expr      min       lq     mean   median       uq      max neval
#>  test_nextIter(15, 8, v = 251) 3.872621 4.105784 4.464846 4.376151 4.647602 9.961372   100

system.time(test_nextIter(25, 10, v = 250))
#>    user  system elapsed
#>   2.093   0.011   2.110

identical(test_nextIter(15, 8, get_val = TRUE, v = 250), comboGeneral(15, 8))
#> [1] TRUE

Rprof("Version250.out", memory.profiling = TRUE)
test_nextIter(25, 10, v = 250)
Rprof(NULL)
summaryRprof("Version250.out", memory = "both")
#> $by.self
#>                 self.time self.pct total.time total.pct mem.total
#> "<Anonymous>"        1.10    55.56       1.66     83.84     701.2
#> ".Call"              0.56    28.28       0.56     28.28     255.6
#> "test_nextIter"      0.32    16.16       1.98    100.00     831.8
#>
#> $by.total
#>                 total.time total.pct mem.total self.time self.pct
#> "test_nextIter"       1.98    100.00     831.8      0.32    16.16
#> "<Anonymous>"         1.66     83.84     701.2      1.10    55.56
#> ".Call"               0.56     28.28     255.6      0.56    28.28
#>
#> $sample.interval
#> [1] 0.02
#>
#> $sampling.time
#> [1] 1.98

Conclusions

It appears that memory is the issue in previous versions. Indeed, if we look at Memory statistics from Rprof, and view both files with memory = "stats" we see that the C funciton, duplicate, appears to be the main culprit.

### Verison 2.4.3
summaryRprof("Version243.out", memory = "stats")
#> index: "test_nextIter"
#>      vsize.small  max.vsize.small      vsize.large  max.vsize.large            nodes        max.nodes     duplications tot.duplications          samples
#>           594132           981272                0                0         11705787         19332264             9200          1085637              118
#> --------------------------------------------------------------------------------------------------------------------------------------------------------
#> index: "test_nextIter":"$"
#>      vsize.small  max.vsize.small      vsize.large  max.vsize.large            nodes        max.nodes     duplications tot.duplications          samples
#>           602919         16615120            19309         15833656         11636867        128195480             9069          7436251              820
#> --------------------------------------------------------------------------------------------------------------------------------------------------------
#> index: "test_nextIter":"a$nextIter"
#>      vsize.small  max.vsize.small      vsize.large  max.vsize.large            nodes        max.nodes     duplications tot.duplications          samples
#>           586247           980856                0                0         11551771         19329128             9169          1283636              140

## Version 2.5.0
summaryRprof("Version250.out", memory = "stats")
#> index: "test_nextIter"
#>      vsize.small  max.vsize.small      vsize.large  max.vsize.large            nodes        max.nodes     duplications tot.duplications          samples
#>          1901132          3243968                0                0          6653888         11353832                0                0               16
#> --------------------------------------------------------------------------------------------------------------------------------------------------------
#> index: "test_nextIter":"<Anonymous>"
#>      vsize.small  max.vsize.small      vsize.large  max.vsize.large            nodes        max.nodes     duplications tot.duplications          samples
#>          2412388         36833808           196363         16298112          9270795        197594096                0                7               83

With verison 2.5.0 there are only 7 tot.duplications whereas with version 2.4.3 there are millions of tot.duplications. In fact, there are a total of 1085637 + 7436251 + 1283636 = 9,805,524 duplications with version 2.4.3. This together with comboCount(25, 10) = 3,268,760 implies that the C funciton, duplicate, is called about 3 times per iteration with older versions (i.e. 9805524 / 3268760 ~= 2.999769).

Iterating over Partitions of a Number

For most partition cases, we have all of the capabilities of the standard comboIter and permuteIter except for bidirectionality (i.e. the prevIter methods). For cases involving standard multisets we also don’t have random access methods.

## Similar illustration of comboIter(5, 3) at the top
p = partitionsIter(16, 4)
p@nextIter()
#> [1]  1  2  3 10

p@nextIter()
#> [1] 1 2 4 9

iter = p@currIter()
i = 1

while (!is.null(iter)) {
    cat(i, " ", iter, "\n")
    iter = p@nextIter()
    i = i + 1
}
#> 1   1 2 4 9
#> 2   1 2 5 8
#> 3   1 2 6 7
#> 4   1 3 4 8
#> 5   1 3 5 7
#> 6   1 4 5 6
#> 7   2 3 4 7
#> 8   2 3 5 6
#> No more results.

partitionsGeneral(16, 4, lower = 2)
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    2    4    9
#> [2,]    1    2    5    8
#> [3,]    1    2    6    7
#> [4,]    1    3    4    8
#> [5,]    1    3    5    7
#> [6,]    1    4    5    6
#> [7,]    2    3    4    7
#> [8,]    2    3    5    6

p@summary()
#> $description
#> [1] "Partitions of 16 into 4 parts"
#>
#> $currentIndex
#> [1] 10
#>
#> $totalResults
#> [1] 9
#>
#> $totalRemaining
#> [1] -1

## Using random access
p[[7]]
#> [1] 1 4 5 6

## No previous iterators
p@prevIter()
#> Error: no slot of name "prevIter" for this object of class "Partitions"

Iterating over Constrained Combinations/Permutations

Now, the combinatorial iterators have all of the features of their “general” analogs (I.e. {combo|permute|partitions}General), which includes constrained results.

For general constrained cases, these iterators offer huge advantages over their “general” counterparts. Previously, one had to guess how many results there would be using the upper parameter as executing the function with no constraints meant the user could be waiting for a while or consume a large amount of resources.

Another drawback is that it difficult to start generating from a particular point. With the “general” functions, if the lower parameter is used, we have to make a decision in order to disambiguate the use. Without constraints, using lower is easy to understand. It simply means to start generating results starting at a particular lexicographical result, which we can do efficiently (i.e. no need to generate the first lower - 1 results). With constraints, it could mean one of two things:

  1. Start checking from a particular lexicographical result without considering the constraint (as we do normally).
  2. Start generating results from a particular result with regards to the final constrained output.

In RcppAlgos we have always used the first interpretation. A big downside for the second point is that we don’t have any fast algorithms for enumerating the total number of results, which reduces determining the nth result to a brute force approach.

With iterators, we can generate n results with nextNIter(n) or calling nextIter() n times (or some combination of the two). Then, if we want to continue iterating, we pick up where we left off fetching the (n + 1)th result and beyond (if there are any results left). This allows us to keep memory low without sacrificing our current state.

set.seed(55)
s = runif(10, -5, 5)

print(s)
#> [1]  0.478135161 -2.818403214 -4.650360052  2.915492940
#> [5]  0.602420762 -4.257748260 -3.684770642 -2.058761222
#> [9]  0.007612633 -4.116755421

## Using comboGeneral to retrieve all results
comboGeneral(s, 5, constraintFun = "mean",
             comparisonFun = "<", limitConstraints = -3)
#>            [,1]      [,2]      [,3]      [,4]         [,5]
#>  [1,] -4.650360 -4.257748 -4.116755 -3.684771 -2.818403214
#>  [2,] -4.650360 -4.257748 -4.116755 -3.684771 -2.058761222
#>  [3,] -4.650360 -4.257748 -4.116755 -3.684771  0.007612633
#>  [4,] -4.650360 -4.257748 -4.116755 -3.684771  0.478135161
#>  [5,] -4.650360 -4.257748 -4.116755 -3.684771  0.602420762
#>  [6,] -4.650360 -4.257748 -4.116755 -2.818403 -2.058761222
#>  [7,] -4.650360 -4.257748 -4.116755 -2.818403  0.007612633
#>  [8,] -4.650360 -4.257748 -4.116755 -2.818403  0.478135161
#>  [9,] -4.650360 -4.257748 -4.116755 -2.818403  0.602420762
#> [10,] -4.650360 -4.257748 -4.116755 -2.058761  0.007612633
#> [11,] -4.650360 -4.257748 -3.684771 -2.818403 -2.058761222
#> [12,] -4.650360 -4.257748 -3.684771 -2.818403  0.007612633
#> [13,] -4.650360 -4.116755 -3.684771 -2.818403 -2.058761222
#> [14,] -4.650360 -4.116755 -3.684771 -2.818403  0.007612633
#> [15,] -4.257748 -4.116755 -3.684771 -2.818403 -2.058761222


## Using comboIter
a = comboIter(s, 5, constraintFun = "mean",
              comparisonFun = "<", limitConstraints = -3)

## See the first result
a@nextIter()
#> [1] -4.650360 -4.257748 -4.116755 -3.684771 -2.818403

## Get the next three
a@nextNIter(3)
#>          [,1]      [,2]      [,3]      [,4]         [,5]
#> [1,] -4.65036 -4.257748 -4.116755 -3.684771 -2.058761222
#> [2,] -4.65036 -4.257748 -4.116755 -3.684771  0.007612633
#> [3,] -4.65036 -4.257748 -4.116755 -3.684771  0.478135161

## See the summary... Note the totalResults and totalRemaining
## fields are NA as we are not able to calculate this upfront.
a@summary()
#> $description
#> [1] "Combinations of 10 choose 5 where the mean is < -3"
#>
#> $currentIndex
#> [1] 4
#>
#> $totalResults
#> [1] NA
#>
#> $totalRemaining
#> [1] NA


a@nextNIter(3)
#>          [,1]      [,2]      [,3]      [,4]         [,5]
#> [1,] -4.65036 -4.257748 -4.116755 -3.684771  0.602420762
#> [2,] -4.65036 -4.257748 -4.116755 -2.818403 -2.058761222
#> [3,] -4.65036 -4.257748 -4.116755 -2.818403  0.007612633

## Get the rest
a@nextRemaining()
#>           [,1]      [,2]      [,3]      [,4]         [,5]
#> [1,] -4.650360 -4.257748 -4.116755 -2.818403  0.478135161
#> [2,] -4.650360 -4.257748 -4.116755 -2.818403  0.602420762
#> [3,] -4.650360 -4.257748 -4.116755 -2.058761  0.007612633
#> [4,] -4.650360 -4.257748 -3.684771 -2.818403 -2.058761222
#> [5,] -4.650360 -4.257748 -3.684771 -2.818403  0.007612633
#> [6,] -4.650360 -4.116755 -3.684771 -2.818403 -2.058761222
#> [7,] -4.650360 -4.116755 -3.684771 -2.818403  0.007612633
#> [8,] -4.257748 -4.116755 -3.684771 -2.818403 -2.058761222

They are very efficient as well. Consider the example below where we use comboGeneral to generate all results without capping the output. Again, we are in a situation where we don’t know a priori how many results we will obtain.

set.seed(77)
s = runif(50, 20, 100)

## Over one trillion results to sift through
comboCount(s, 15)
#> [1] 2.25083e+12

system.time({
    print(
        nrow(
            comboGeneral(s, 15,
                         constraintFun = "mean",
                         comparisonFun = ">",
                         limitConstraints = 83)
        )
    )
})
#> [1] 38935252
#>    user  system elapsed
#>   6.140   4.446  11.208

## Over 4 GBs of results
(38935252 * 15 * 8) / 2^30
#> [1] 4.351353

Just over 11 seconds isn’t bad, however 4 GBs could put a strain on your computer.

Let’s use iterators instead and only generate ten thousand at a time to keep memory low. We should mention here that the iterators are “smart” in that there is no fear in requesting more results than what is actually left. For example, if in the problem above, we had iterated to the 38th million result and requested 10 million more, we would only obtain 935,252 results.

system.time({
    a = comboIter(s, 15,
                  constraintFun = "mean",
                  comparisonFun = ">",
                  limitConstraints = 83)
    while (!is.null(a@nextNIter(1e4))) {}
    print(a@summary())
})
#> No more results.
#>
#> $description
#> [1] "Combinations of 50 choose 15 where the mean is > 83"
#>
#> $currentIndex
#> [1] 38935252
#>
#> $totalResults
#> [1] NA
#>
#> $totalRemaining
#> [1] NA
#>
#>    user  system elapsed
#>   2.852   1.051   3.907


## Only 11 MBs per iteration
(1e4 * 15 * 8) / 2^20
#> [1] 1.144409

Wow! Using the iterator approach is almost 3 times faster (11.208 / 3.907 ~= 2.869)! Our gains came strictly from memory efficiency (From over 4 GBs to just over 1 MB) as the underlying algorithm is exactly the same.