The bit package provides the following S3 functionality:

• alternatives to the logical data type which need less memory with methods that are often faster
• the bit type for boolean data (without NA, factor 32 smaller)
• the bitwhich type for very skewed boolean data (without NA, potentially even smaller)
• further alternatives for representing Boolean selections
• the which type for positive selections maintaining the original vector length
• the ri range index
• very fast methods for integer, particularly
• methods for unsorted integers leveraging bit vectors rather than hash tables
• methods for sorting integers leveraging $$O(N)$$ synthetic sorting rather than $$O(N \log N)$$ divide and conquer
• methods for sorted integers leveraging merging rather than hash tables
• some foundations for package ff, particularly
• bit and bitwhich vectors and ri range indices for filtering and subscripting ff objects
• rlepack compressing sets of sorted integers for hi hybrid indexing
• methods for chunking
• helper methods for avoiding unwanted data copies

## Boolean data types

R's logical vectors cost 32 bit per element in order to code three states FALSE, TRUE and NA. By contrast bit vectors cost only 1 bit per element and support only the two Boolean states FALSE and TRUE. Internally bit are stored as integer. bitwhich vectors manage very skewed Boolean data as a set of sorted unique integers denoting either included or excluded positions, whatever costs less memory. Internally bitwhich are stored as integer or as logical for the extreme cases all included (TRUE), all excluded (FALSE) or zero length (logical()). Function bitwhich_representation allows to distinguish these five cases without the copying-cost of unclass(bitwhich). All three Boolean types logical, bit, and bitwhich are unified in a super-class booltype and have is.booltype(x)==TRUE. Classes which and ri can be somewhat considered as a fourth and fifth even more special Boolean types (for skewed data with mostly FALSE that represents the sorted positions of the few TRUE, and as a consecutive series of the latter), and hence have booltype 4 and 5 but coerce differently to integer and double and hence are not is.booltype.

### Available classes

The booltypes bit and bitwhich behave very much like logical with the following exceptions

• length is limited to .Machine\$integer.max
• currently only vector methods are supported, not matrix or array
• subscripting from them does return logical rather than returning their own class (like ff boolean vectors do)
• values assigned to them are internally coerced to logical before being processed, hence x[] <- x is a potentially expensive operation
• they do not support names and character subscripts
• only scalar logical subscripts are allowed, i.e. FALSE, TRUE
• assigned or coerced NA are silently converted to FALSE
• increasing the length of bit has semantics differing from logical: new elements are consistently initialized with FALSE instead of NA
• increasing the length of bitwhich has semantics differing from logical and bit: new elements are initialized with the more frequent value of the skewed distribution (FALSE winning in the equally frequent case).
• aggregation methods min, max and range and summary have special meaning, for example min corresponds to which.max.

Note the following features

• aggregation functions support a range argument for chunked processing
• sorted positive subscripts marked as class which are processed faster than 'just positive subscripts', and unlike the result of the function which, class which retains the length of the Boolean vector as attribute maxindex, but its length is the number of positive positions
• ri range indices are allowed as subscripts which supports chunked looping over in-memory bit or bitwhich vectors not unlike chunked looping over on-disk ff objects.
• binary Boolean operations will promote differing data type to the data type implying fewer assumptions, i.e. logical wins over bit, bit wins over bitwhich and which wins over bitwhich.

Note the following warnings

• currently bit and bitwhich may answer is.logical and is.integer according to their internal representation. Do not rely on this, it may be subject to change. Do use booltype, is.booltype, is.bit, is.bitwhich and bitwhich_representation for reasoning about the data type.

### Available methods

Basic methods

is as length length<- [ [<- [[ [[<- rev rep c print

Boolean operations

is.na ! | & == != xor

Aggregation methods

anyNA any all sum min max range summary

### Creating and manipulating

bit and bitwhich vectors are created like logical, for example zero length vectors

logical()
#> logical(0)
bit()
#> bit length=0 occupying only 0 int32
bitwhich()
#> bitwhich: 0/0 occupying only 0 int32 in  representation


or vectors of a certain length initialized to FALSE

logical(3)
#> [1] FALSE FALSE FALSE
bit(3)
#> bit length=3 occupying only 1 int32
#>     1     2     3
#> FALSE FALSE FALSE
bitwhich(3)
#> bitwhich: 0/3 occupying only 1 int32 in FALSE representation
#>     1     2     3
#> FALSE FALSE FALSE


bitwhich can be created initialized to all elements TRUE with

bitwhich(3, TRUE)
#> bitwhich: 3/3 occupying only 1 int32 in TRUE representation
#>    1    2    3
#> TRUE TRUE TRUE


or can be created initialized to a few included or excluded elements

bitwhich(3, 2)
#> bitwhich: 1/3 occupying only 1 int32 in 1 representation
#>     1     2     3
#> FALSE  TRUE FALSE
bitwhich(3, -2)
#> bitwhich: 2/3 occupying only 1 int32 in -1 representation
#>     1     2     3
#>  TRUE FALSE  TRUE


Note that logical behaves somewhat inconsistent, when creating it, the default is FALSE, when increasing the length, the default is NA:

l <- logical(3)
length(l) <- 6
l
#> [1] FALSE FALSE FALSE    NA    NA    NA


Note that the default in bit is always FALSE, for creating and increasing the length.

b <- bit(3)
length(b) <- 6
b
#> bit length=6 occupying only 1 int32
#>     1     2     3     4     5     6
#> FALSE FALSE FALSE FALSE FALSE FALSE


Increasing the length of bitwhich initializes new elements to the majority of the old elements, hence a bitwhich with a few exclusions has majority TRUE and will have new elements initialized to TRUE (if both, TRUE and FALSE have equal frequency the default is FALSE).

w <- bitwhich(3,2)
length(w) <- 6
w
#> bitwhich: 1/6 occupying only 1 int32 in 1 representation
#>     1     2     3     4     5     6
#> FALSE  TRUE FALSE FALSE FALSE FALSE
w <- bitwhich(3,-2)
length(w) <- 6
w
#> bitwhich: 5/6 occupying only 1 int32 in -1 representation
#>     1     2     3     4     5     6
#>  TRUE FALSE  TRUE  TRUE  TRUE  TRUE


Vector subscripting non-existing elements returns NA

l <- logical(3); l[6]
#> [1] NA
b <- bit(3); b[6]
#> [1] NA
#> attr(,"vmode")
#> [1] "boolean"
w <- bitwhich(3); w[6]
#> [1] NA
#> attr(,"vmode")
#> [1] "boolean"


while assigned NA turn into FALSE and assigning to a non-existing element does increase vector length

l[6] <- NA; l
#> [1] FALSE FALSE FALSE    NA    NA    NA
b[6] <- NA; b
#> bit length=6 occupying only 1 int32
#>     1     2     3     4     5     6
#> FALSE FALSE FALSE FALSE FALSE FALSE
w[6] <- NA; w
#> bitwhich: 0/6 occupying only 1 int32 in FALSE representation
#>     1     2     3     4     5     6
#> FALSE FALSE FALSE FALSE FALSE FALSE


As usual list subscripting is only allowed for existing elements

l[[6]]
#> [1] NA
b[[6]]
#> [1] FALSE
#> attr(,"vmode")
#> [1] "boolean"
w[[6]]
#> [1] FALSE
#> attr(,"vmode")
#> [1] "boolean"


while assignments to non-existing elements do increase length.

l[[9]] <- TRUE
b[[9]] <- TRUE
w[[9]] <- TRUE
#> Warning in [[<-.bitwhich(*tmp*, 9, value = TRUE): increasing length of
#> bitwhich, which has non-standard semantics
l
#> [1] FALSE FALSE FALSE    NA    NA    NA    NA    NA  TRUE
b
#> bit length=9 occupying only 1 int32
#>     1     2     3     4     5     6     7     8     9
#> FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE
w
#> bitwhich: 1/6 occupying only 1 int32 in 1 representation
#>     1     2     3     4     5     6  <NA>  <NA>  <NA>
#> FALSE FALSE FALSE FALSE FALSE FALSE    NA    NA  TRUE


### Coercion

There are coercion functions between classes logical, bit, bitwhich, which, integer and double. However, only the first three of those represent Boolean vectors, whereas which represents subscript positions, and integer and double are ambiguous in that they can represent both, Booleans or positions.

Remember first that coercing logical to integer (or double) gives 0 and 1 and not integer subscript positions:

l <- c(FALSE, TRUE, FALSE)
i <- as.integer(l)
as.logical(i)
#> [1] FALSE  TRUE FALSE


To obtain integer subscript positions use which or better as.which because the latter S3 class remembers the original vector length and hence we can go coerce back to logical

l <- c(FALSE, TRUE, FALSE)
w <- as.which(l)
w
#> [1] 2
#> attr(,"maxindex")
#> [1] 3
#> attr(,"class")
#> [1] "booltype" "which"
as.logical(w)
#> [1] FALSE  TRUE FALSE


coercing back to logical fails using just which

l <- c(FALSE, TRUE, FALSE)
w <- which(l)
w
#> [1] 2
as.logical(w)  # does not coerce back
#> [1] TRUE


Furthermore from class which we can deduce that the positions are sorted which can be leveraged for performance. Note that as.which.integer is a core method for converting integer positions to class which and it enforces sorting

i <- c(7,3)
w <- as.which(i, maxindex=12)
w
#> [1] 3 7
#> attr(,"maxindex")
#> [1] 12
#> attr(,"class")
#> [1] "booltype" "which"


and as.integer gives us back those positions (now sorted)

as.integer(w)
#> [1] 3 7


You see that the as.integer generic is ambiguous as the integer data type in giving positions on some, and zeroes and ones on other inputs. The following set of Boolean types can be coerced without loss of information

logical bit bitwhich which


The same is true for this set

logical bit bitwhich integer double


Furthermore positions are retained in this set

which integer double


although the length of the Boolean vector is lost when coercing from which to integer or double, therefore coercing to them is not reversible. Let's first create all six types and compare their sizes:

r <- ri(1, 2^16, 2^20) # sample(2^20, replace=TRUE, prob=c(.125,875))
all.as <- list(
double = as.double
, integer= as.integer
, logical = as.logical
, bit = as.bit
, bitwhich = as.bitwhich
, which = as.which
, ri = function(x)x
)
all.types <- lapply(all.as, function(f)f(r))
sapply(all.types, object.size)
#>   double  integer  logical      bit bitwhich    which       ri
#>  8388656  4194352  4194352   132584   262832   262656      360


Now let's create all combinations of coercion:

all.comb <- vector('list', length(all.types)^2)
all.id <- rep(NA, length(all.types)^2)
dim(all.comb)      <- dim(all.id)      <-    c(from=length(all.types),  to=length(all.types))
dimnames(all.comb) <- dimnames(all.id) <- list(from= names(all.types) , to= names(all.types))
for (i in seq_along(all.types))
for (j in seq_along(all.as)){
# coerce all types to all types (FROM -> TO)
all.comb[[i,j]] <- all.as[[j]](all.types[[i]])
# and test whether coercing back to the FROM type gives the orginal object
all.id[i,j] <- identical(all.as[[i]](all.comb[[i,j]]),  all.types[[i]])
}
all.id
#>           to
#> from       double integer logical   bit bitwhich which   ri
#>   double     TRUE    TRUE    TRUE  TRUE     TRUE FALSE TRUE
#>   integer    TRUE    TRUE    TRUE  TRUE     TRUE FALSE TRUE
#>   logical    TRUE    TRUE    TRUE  TRUE     TRUE  TRUE TRUE
#>   bit        TRUE    TRUE    TRUE  TRUE     TRUE  TRUE TRUE
#>   bitwhich   TRUE    TRUE    TRUE  TRUE     TRUE  TRUE TRUE
#>   which     FALSE   FALSE    TRUE  TRUE     TRUE  TRUE TRUE
#>   ri        FALSE   FALSE   FALSE FALSE    FALSE FALSE TRUE


Do understand the FALSE above!

The functions booltype and is.booltype diagnose the Boolean type as follows

data.frame(booltype=sapply(all.types, booltype), is.boolean=sapply(all.types, is.booltype), row.names=names(all.types))
#>          booltype is.boolean
#> double     nobool      FALSE
#> integer    nobool      FALSE
#> logical   logical       TRUE
#> bit           bit       TRUE
#> bitwhich bitwhich       TRUE
#> which       which       TRUE
#> ri             ri       TRUE


Class which and ri are currently not is.boolean (no subscript, assignment and Boolean operators), but since it is even more specialized than bitwhich (assuming skew towards TRUE), we have ranked it as the most specialized booltype.

### Boolean operations

The usual Boolean operators

! | & == != xor is.na


are implemented and the binary operators work for all combinations of logical, bit and bitwhich.

Technically this is achieved in S3 by giving bit and bitwhich another class booltype. Note that this is not inheritance where booltype implements common methods and bit and bitwhich overrule this with more specific methods. Instead the method dispatch to booltype dominates bit and bitwhich and coordinates more specific methods, this was the only way to realize binary logical operators that combine bit and bitwhich in S3, because in this case R dispatches to neither bit nor bitwhich: if both arguments are custom classes R does a non-helpful dispatch to integer or logical.

Anyhow, if a binary Boolean operator meets two different types, argument and result type is promoted to less assumptions, hence bitwhich is promoted to bit dropping the assumption of strong skew and bit is promoted to logical dropping the assumption that no NA are present.

Such promotion comes at the price of increased memory requirements, for example the following multiplies the memory requirement by factor 32

x <- bit(1e6)
y <- x | c(FALSE, TRUE)
object.size(y) / object.size(x)
#> 31.6 bytes


Better than lazily relying on automatic propagation is

x <- bit(1e6)
y <- x | as.bit(c(FALSE, TRUE))
object.size(y) / object.size(x)
#> 1 bytes


### Manipulation methods

Concatenation follows the same promotion rules as Boolean operators. Note that c dispatches on the first argument only, hence when concatenating multiple Boolean types the first must not be logical, otherwise we get corrupt results:

l <- logical(6)
b <- bit(6)
c(l,b)
#> [1] 0 0 0 0 0 0 0


because c.logical treats the six bits as a single value. The following expressions work

c(b,l)
#>  [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
c(l, as.logical(b))
#>  [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE


and of course the most efficient is

c(as.bit(l), b)
#> bit length=12 occupying only 1 int32
#>     1     2     3     4     5     6     7     8     9    10    11    12
#> FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE


If you want your code to process any is.booltype, you can use c.booltype directly

c.booltype(l, b)
#>  [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE


Both, bit and bitwhich also have replication (rep) and reverse (rev) methods that work as expected:

b <- as.bit(c(FALSE,TRUE))
rev(b)
#> bit length=2 occupying only 1 int32
#>     1     2
#>  TRUE FALSE
rep(b, 3)
#> bit length=6 occupying only 1 int32
#>     1     2     3     4     5     6
#> FALSE  TRUE FALSE  TRUE FALSE  TRUE
rep(b, length.out=6)
#> bit length=6 occupying only 1 int32
#>     1     2     3     4     5     6
#> FALSE  TRUE FALSE  TRUE FALSE  TRUE


### Aggregation methods for booltype

The usual logical aggregation functions length, all, any and anyNA work as expected. Note the exception that length(which) does not give the length of the Boolean vector but the length of the vector of positive integers (like the result of function which). sum gives the number of TRUE for all Boolean types. For the booltype > 1 min gives the first position of a TRUE (i.e. which.max), max gives the last position of a TRUE, range gives both, the range at which we find TRUE, and finally summary gives the the counts of FALSE and TRUE as well as min and max. For example

l <- c(NA,NA,FALSE,TRUE,TRUE)
b <- as.bit(l)
length(b)
#> [1] 5
anyNA(b)
#> [1] FALSE
any(b)
#> [1] TRUE
all(b)
#> [1] FALSE
sum(b)
#> [1] 2
min(b)
#> [1] 4
max(b)
#> [1] 5
range(b)
#> [1] 4 5
summary(b)
#> FALSE  TRUE  Min.  Max.
#>     3     2     4     5


These special interpretations of min, max, range and summary can be enforced for type logical, integer, and double by using the booltype methods directly as in

# minimum after coercion to integer
min(c(FALSE,TRUE))
#> [1] 0
# minimum  position of first TRUE
min.booltype(c(FALSE,TRUE))
#> [1] 2


Except for length and anyNA the aggregation functions support an optional argument range which restricts evaluation the specified range of the Boolean vector. This is useful in the context of chunked processing. For example analyzing the first 30\% of a million Booleans

b <- as.bit(sample(c(FALSE, TRUE), 1e6, TRUE))
summary(b,range=c(1,3e5))
#>  FALSE   TRUE   Min.   Max.
#> 150256 149744      4 300000


and analyzing all such chunks

sapply(chunk(b, by=3e5, method="seq"), function(i)summary(b, range=i))
#>       1:300000 300001:600000 600001:900000 900001:1000000
#> FALSE   150256        149862        150197          50265
#> TRUE    149744        150138        149803          49735
#> Min.         4        300003        600002         900002
#> Max.    300000        599998        899996         999999


or better balanced

sapply(chunk(b, by=3e5), function(i)summary(b, range=i))
#>       1:250000 250001:500000 500001:750000 750001:1000000
#> FALSE   125135        124897        125194         125354
#> TRUE    124865        125103        124806         124646
#> Min.         4        250001        500005         750001
#> Max.    250000        499998        750000         999999


The real use-case for chunking is ff objects, where instead of processing huge objects at once

x <- ff(vmode="single", length=length(b))   # create a huge ff vector
x[as.hi(b)] <- runif(sum(b))      # replace some numbers at filtered positions
summary(x[])
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
#>  0.0000  0.0000  0.0000  0.2498  0.4996  1.0000


we can process the ff vector in chunks

sapply(chunk(x, by=3e5), function(i)summary(x[i]))
#>          1:250000 250001:500000 500001:750000 750001:1000000
#> Min.    0.0000000  0.0000000000     0.0000000      0.0000000
#> 1st Qu. 0.0000000  0.0000000000     0.0000000      0.0000000
#> Median  0.0000000  0.0007963628     0.0000000      0.0000000
#> Mean    0.2502720  0.2499065427     0.2496453      0.2494940
#> 3rd Qu. 0.4999921  0.5001179874     0.4988354      0.4992141
#> Max.    0.9999930  0.9999995828     0.9999888      0.9999956


and even can process a bit-filtered ff vector in chunks

sapply(chunk(x, by=3e5), function(i)summary(x[as.hi(b, range=i)]))
#>             1:250000 250001:500000 500001:750000 750001:1000000
#> Min.    2.003042e-06  1.240987e-06  8.684583e-08   3.883149e-06
#> 1st Qu. 2.508616e-01  2.494814e-01  2.500706e-01   2.505730e-01
#> Median  5.007221e-01  4.997281e-01  4.996362e-01   5.005408e-01
#> Mean    5.010852e-01  4.994016e-01  5.000667e-01   5.004052e-01
#> 3rd Qu. 7.519348e-01  7.493132e-01  7.503459e-01   7.490826e-01
#> Max.    9.999930e-01  9.999996e-01  9.999888e-01   9.999956e-01


## Fast methods for integer set operations

R implements set methods using hash tables, namely match, %in%, unique, duplicated, union, intersect, setdiff, setequal. Hashing is a powerful method, but it is costly in terms of random access and memory consumption. For integers package 'bit' implements faster set methods using bit vectors (which can be an order of magnitude faster and saves up to factor 64 on temporary memory) and merging (which is about two orders of magnitude faster, needs no temporary memory but requires sorted input). While many set methods return unique sets, the merge_* methods can optionally preserve ties, the range_* methods below allow to specify one of the sets as a range.

R hashing bit vectors merging range merging rlepack
match merge_match
%in% bit_in merge_in merge_rangein
!(%in%) !bit_in merge_notin merge_rangenotin
duplicated bit_duplicated
unique bit_unique merge_unique unique(rlepack)
union bit_union merge_union
intersect bit_intersect merge_intersect merge_rangesect
setdiff bit_setdiff merge_setdiff merge_rangediff
(a:b)[-i] bit_rangediff merge_rangediff merge_rangediff
bit_symdiff merge_symdiff
setequal bit_setequal merge_setequal
anyDuplicated bit_anyDuplicated anyDuplicated(rlepack)
sum(duplicated) bit_sumDuplicated

Furthermore there are very fast methods for sorting integers (unique or keeping ties), reversals that simultaneously change the sign to preserve ascending order and methods for finding min or max in sorted vectors or within ranges of sorted vectors.

R bit vectors merging range merging
sort(unique) bit_sort_unique
sort bit_sort
rev rev(bit) reverse_vector
-rev copy_vector(revx=TRUE)
min min(bit) merge_first merge_firstin
max max(bit) merge_last merge_lastin
min(!bit) merge_firstnotin
max(!bit) merge_lastnotin

Set operations using hash tables incur costs for populating and querying the hash table: this is random access cost to a relative large table. In order to avoid hash collisions hash tables need more elements than those being hashed, typically 2*N for N elements. That is hashing N integer elements using a int32 hash function costs random access to 64*N bits of memory. If the N elements are within a range of N values, it is much (by factor 64) cheaper to register them in a bit-vector of N bits. The bit_* functions first determine the range of the values (and count of NA), and then use an appropriately sized bit vector. Like the original R functions the bit_* functions keep the original order of values (although some implementations could be faster by delivering an undefined order). If N is small relative to the range of the values the bit_* fall back to the standard R functions using hash tables. Where the bit_* functions return Boolean vectors they do so by default as bit vectors, but but you can give a different coercion function as argument retFUN.

Bit vectors cannot communicate positions, hence cannot replace match, however they can replace the %in% operator:

set.seed(1); n <- 9
x <- sample(n, replace=TRUE); x
#> [1] 9 4 7 1 2 7 2 3 1
y <- sample(n, replace=TRUE); y
#> [1] 5 5 6 7 9 5 5 9 9
x %in% y
#> [1]  TRUE FALSE  TRUE FALSE FALSE  TRUE FALSE FALSE FALSE
bit_in(x,y)
#> bit length=9 occupying only 1 int32
#>     1     2     3     4     5     6     7     8     9
#>  TRUE FALSE  TRUE FALSE FALSE  TRUE FALSE FALSE FALSE
bit_in(x,y, retFUN=as.logical)
#> [1]  TRUE FALSE  TRUE FALSE FALSE  TRUE FALSE FALSE FALSE


The bit_in function combines a bit vector optimization with reverse look-up, i.e. if the range of x is smaller than the range of table, we build the bit vector on x instead of the table.

The bit_duplicated function can handle NA in three different ways

x <- c(NA,NA,1L,1L,2L,3L)
duplicated(x)
#> [1] FALSE  TRUE FALSE  TRUE FALSE FALSE
bit_duplicated(x, retFUN=as.logical)
#> [1] FALSE  TRUE FALSE  TRUE FALSE FALSE
bit_duplicated(x, na.rm=NA, retFUN=as.logical)
#> [1] FALSE  TRUE FALSE  TRUE FALSE FALSE

duplicated(x, incomparables = NA)
#> [1] FALSE FALSE FALSE  TRUE FALSE FALSE
bit_duplicated(x, na.rm=FALSE, retFUN=as.logical)
#> [1] FALSE FALSE FALSE  TRUE FALSE FALSE

bit_duplicated(x, na.rm=TRUE, retFUN=as.logical)
#> [1]  TRUE  TRUE FALSE  TRUE FALSE FALSE


The bit_unique function can also handle NA in three different ways

x <- c(NA,NA,1L,1L,2L,3L)
unique(x)
#> [1] NA  1  2  3
bit_unique(x)
#> [1] NA  1  2  3

unique(x, incomparables = NA)
#> [1] NA NA  1  2  3
bit_unique(x, na.rm=FALSE)
#> [1] NA NA  1  2  3

bit_unique(x, na.rm=TRUE)
#> [1] 1 2 3


The bit_union function build a bit vector spanning the united range of both input sets and filters all unites duplicates:

x <- c(NA,NA,1L,1L,3L)
y <- c(NA,NA,2L,2L,3L)
union(x,y)
#> [1] NA  1  3  2
bit_union(x,y)
#> [1] NA  1  3  2


The bit_intersect function builds a bit vector spanning only the intersected range of both input sets and filters all elements outside and the duplicates inside:

x <- c(0L,NA,NA,1L,1L,3L)
y <- c(NA,NA,2L,2L,3L,4L)
intersect(x,y)
#> [1] NA  3
bit_intersect(x,y)
#> [1] NA  3


The bit_setdiff function builds a bit vector spanning the range of the first input set, marks elements of the second set within this range as tabooed, and then outputs the remaining elements of the first set unless they are duplicates:

x <- c(0L,NA,NA,1L,1L,3L)
y <- c(NA,NA,2L,2L,3L,4L)
setdiff(x,y)
#> [1] 0 1
bit_setdiff(x,y)
#> [1] 0 1


The bit_symdiff function implements symmetric set difference. It builds two bit vectors spanning the full range and then outputs those elements of both sets that are marked at exactly one of the bit vectors.

x <- c(0L,NA,NA,1L,1L,3L)
y <- c(NA,NA,2L,2L,3L,4L)
union(setdiff(x,y),setdiff(y,x))
#> [1] 0 1 2 4
bit_symdiff(x,y)
#> [1] 0 1 2 4


The bit_setequal function terminates early if the ranges of the two sets (or the presence of NA) differ. Otherwise it builds two bit vectors spanning the identical range; finally it checks the two vectors for being equal with early termination if two unequal integers are found.

x <- c(0L,NA,NA,1L,1L,3L)
y <- c(NA,NA,2L,2L,3L,4L)
setequal(y,x)
#> [1] FALSE
bit_setequal(x,y)
#> [1] FALSE


The bit_rangediff function works like bit_setdiff with two differences: the first set is specified as a range of integers, and it has two arguments revx and revy which allow to reverse order and sign of the two sets before the set-diff operation is done. The order of the range is significant, e.g. c(1L,7L) is different from c(7L,1L), while the order of the second set has no influence:

bit_rangediff(c(1L,7L), (3:5))
#> [1] 1 2 6 7
bit_rangediff(c(7L,1L), (3:5))
#> [1] 7 6 2 1
bit_rangediff(c(1L,7L), -(3:5), revy=TRUE)
#> [1] 1 2 6 7
bit_rangediff(c(1L,7L), -(3:5), revx=TRUE)
#> [1] -7 -6 -2 -1


If the range and the second set don't overlap, for example due to different signs, the full range is returned:

bit_rangediff(c(1L,7L), (1:7))
#> integer(0)
bit_rangediff(c(1L,7L), -(1:7))
#> [1] 1 2 3 4 5 6 7
bit_rangediff(c(1L,7L), (1:7), revy=TRUE)
#> [1] 1 2 3 4 5 6 7


Note that bit_rangediff provides faster negative subscripting from a range of integers than the usual phrase (1:n)[-i]:

(1:9)[-7]
#> [1] 1 2 3 4 5 6 8 9
bit_rangediff(c(1L,9L), -7L, revy=TRUE)
#> [1] 1 2 3 4 5 6 8 9


Functions bit_anyDuplicated is a faster version of anyDuplicated

x <- c(NA,NA,1L,1L,2L,3L)
any(duplicated(x))  # full hash work, returns FALSE or TRUE
#> [1] TRUE
anyDuplicated(x)   # early termination of hash work, returns 0 or position of first duplicate
#> [1] 2
any(bit_duplicated(x))  # full bit work, returns FALSE or TRUE
#> [1] TRUE
bit_anyDuplicated(x)   # early termination of bit work, returns 0 or position of first duplicate
#> [1] 2


For the meaning of the na.rm parameter see bit_duplicated. Function bit_sumDuplicated is a faster version of sum(bit_duplicated)

x <- c(NA,NA,1L,1L,2L,3L)
sum(duplicated(x))  # full hash work, returns FALSE or TRUE
#> [1] 2
sum(bit_duplicated(x))  # full bit work, returns FALSE or TRUE
#> [1] 2
bit_sumDuplicated(x)   # early termination of bit work, returns 0 or position of first duplicated
#> [1] 2


### Methods using bit vectors for sorting integers

A bit vector cannot replace a hash table for all possible kind of tasks (when it comes to counting values or to their positions), but on the other hand a bit vector allows something impossible with a hash table: sorting keys (without payload). Sorting a large subset of unique integers in [1,N] using a bit vector has been described in “Programming pearls – cracking the oyster” by Jon Bentley (where 'uniqueness' is not an input condition but an output feature). This is easily generalized to sorting a large subset of integers in a range [min,max]. A first scan over the data determines the range of the keys, arranges NAs according to the usual na.last= argument and checks for presortedness (see range_sortna), then the range is projected to a bit vector of size max-min+1, all keys are registered causing random access and a final scan over the bit vector sequentially writes out the sorted keys. This is a synthetical sort because unlike a comparison sort the elements are not moved towards their ordered positions, instead they are synthesized from the bit-representation.

For N consecutive permuted integers this is by an order of magnitude faster than quicksort. For a density $$d=N/(max-min) \ge 1$$ this sort beats quicksort, but for $$d << 1$$ the bit vector becomes too large relative to N and hence quicksort is faster. bit_sort_unique implements a hybrid algorithm automatically choosing the faster of both, hence for integers the following gives identical results

x <- sample(9, 9, TRUE)
unique(sort(x))
#> [1] 1 2 3 4 5 6 9
sort(unique(x))
#> [1] 1 2 3 4 5 6 9
bit_sort_unique(x)
#> [1] 1 2 3 4 5 6 9


What if duplicates shall be kept?

• A: sort all non-duplicated occurrences using a bit vector
• B: sort the rest
• merge A and B

The crucial question is how to sort the rest? Recursively using a bit-vector can again be faster than quicksort, however it is non-trivial to determine an optimal recursion-depth before falling back to quicksort. Hence a safe bet is using the bit vector only once and sort the rest via quicksort, let's call that bitsort. Again, using bitsort is fast only at medium density in the data range. For low density quicksort is faster. For high density (duplicates) another synthetic sort is faster: counting sort. bit_sort implements a sandwhich sort algorithm which given density uses bitsort between two complementing algorithms:

low density    medium density    high density
quicksort   << bitsort        << countsort

x <- sample(9, 9, TRUE)
sort(x)
#> [1] 4 4 6 7 7 8 9 9 9
bit_sort(x)
#> [1] 4 4 6 7 7 8 9 9 9


Both, bit_sort_unique and bit_sort can sort decreasing, however, currently this requires an extra pass over the data in bit_sort and in the quicksort fallback of bit_sort_unique. So far, bit_sort does not leverage radix sort for very large N.

### Methods for sets of sorted integers

Efficient handling of sorted sets is backbone of class bitwhich. The merge_* and merge_range* integer functions expect sorted input (non-decreasing sets or increasing ranges) and return sorted sets (some return Boolean vectors or scalars). Exploiting the sortedness makes them even faster than the bit_ functions. Many of them have revx or revy arguments, which reverse the scanning direction of an input vector and the interpreted sign of its elements, hence we can change signs of input vectors in an order-preserving way and without any extra pass over the data. By default these functions return unique sets which have each element not more than once. However, the binary merge_* functions have a method="exact" which in both sets treats consecutive occurrences of the same value as different. With method="exact" for example merge_setdiff behaves as if counting the values in the first set and subtraction the respective counts of the second set (and capping the lower end at zero). Assuming positive integers and equal tabulate(, nbins=) with method="exact" the following is identical:

merging then counting counting then combining
tabulate(merge_union(x,y)) pmax(tabulate(x) ,tabulate(y))
tabulate(merge_intersect(x,y)) pmin(tabulate(x) ,tabulate(y))
tabulate(merge_setdiff(x,y)) pmax(tabulate(x) -tabulate(y), 0)
tabulate(merge_symdiff(x,y)) abs(tabulate(x) -tabulate(y))
tabulate(merge_setequal(x,y)) all(tabulate(x)==tabulate(y))

Note further that method="exact" delivers unique output if the input is unique, and in this case works faster than method="unique".

Note further, that the merge_* and merge_range* functions have no special treatment for NA. If vectors with NA are sorted ith NA in the first positions (na.last=FALSE) and arguments revx= or revy= have not been used, then NAs are treated like ordinary integers. NA sorted elsewhere or using revx= or revy= can cause unexpected results (note for example that revx= switches the sign on all integers but NAs).

The unary merge_unique function transform a sorted set into a unique sorted set:

x = sample(12)
bit_sort(x)
#>  [1]  1  2  3  4  5  6  7  8  9 10 11 12
merge_unique(bit_sort(x))
#>  [1]  1  2  3  4  5  6  7  8  9 10 11 12
bit_sort_unique(x)
#>  [1]  1  2  3  4  5  6  7  8  9 10 11 12


x = as.integer(c(3,4,4,5))
y = as.integer(c(3,4,5))
setequal(x,y)
#> [1] TRUE
merge_setequal(x,y)
#> [1] TRUE
merge_setequal(x,y, method="exact")
#> [1] FALSE


For set complement there is also a merge_range* function:

x = as.integer(c(0,1,2,2,3,3,3))
y = as.integer(c(1,2,3))
setdiff(x,y)
#> [1] 0
merge_setdiff(x,y)
#> [1] 0
merge_setdiff(x,y, method="exact")
#> [1] 0 2 3 3
merge_rangediff(c(0L,4L),y)
#> [1] 0 4
merge_rangediff(c(0L,4L),c(-3L,-2L)) # y has no effect due to different sign
#> [1] 0 1 2 3 4
merge_rangediff(c(0L,4L),c(-3L,-2L), revy=TRUE)
#> [1] 0 1 4
merge_rangediff(c(0L,4L),c(-3L,-2L), revx=TRUE)
#> [1] -4 -1  0


merge_symdiff for symmetric set complement is used similar (without a merge_range* function), as is merge_intersect, where the latter is accompanied by a merge_range* function:

x = -2:1
y = -1:2
setdiff(x,y)
#> [1] -2
union(setdiff(x,y),setdiff(y,x))
#> [1] -2  2
merge_symdiff(x,y)
#> [1] -2  2
merge_intersect(x,y)
#> [1] -1  0  1
merge_rangesect(c(-2L,1L),y)
#> [1] -1  0  1


The merge_union function has a third method all which behaves like c but keeps the output sorted

x = as.integer(c(1,2,2,3,3,3))
y = 2:4
union(x,y)
#> [1] 1 2 3 4
merge_union(x,y, method="unique")
#> [1] 1 2 3 4
merge_union(x,y, method="exact")
#> [1] 1 2 2 3 3 3 4
merge_union(x,y, method="all")
#> [1] 1 2 2 2 3 3 3 3 4
sort(c(x,y))
#> [1] 1 2 2 2 3 3 3 3 4
c(x,y)
#> [1] 1 2 2 3 3 3 2 3 4


Unlike the bit_* functions the merge_* functions have a merge_match function:

x = 2:4
y = as.integer(c(0,1,2,2,3,3,3))
match(x,y)
#> [1]  3  5 NA
merge_match(x,y)
#> [1]  3  5 NA


and unlike R's %in% operator the following functions are directly implemented, not on top of merge, and hence save extra passes over the data:

x %in% y
#> [1]  TRUE  TRUE FALSE
merge_in(x,y)
#> [1]  TRUE  TRUE FALSE
merge_notin(x,y)
#> [1] FALSE FALSE  TRUE


The range versions extract logical vectors from y, but only for the range in rx.

x <- c(2L,4L)
merge_rangein(x,y)
#> [1]  TRUE  TRUE FALSE
merge_rangenotin(x,y)
#> [1] FALSE FALSE  TRUE


Compare this to merge_rangesect above. merge_rangein is useful in the context of chunked processing, see the any, all, sum and [ methods of class bitwhich.

The functions merge_first and merge_last give first and last element of a sorted set. By default that is min and max, however these functions also have an revx argument. There are also functions that deliver the first resp. last elements of a certain range that are in or not in a certain set.

x <- bit_sort(sample(1000,10))
merge_first(x)
#> [1] 104
merge_last(x)
#> [1] 983
merge_firstnotin(c(300L,600L), x)
#> [1] 300
merge_firstin(c(300L,600L), x)
#> [1] 326
merge_lastin(c(300L,600L), x)
#> [1] 499
merge_lastnotin(c(300L,600L), x)
#> [1] 600