Usage of the bit package

The bit package provides the following S3 functionality:

knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
require(bit)
.ff.version <- try(packageVersion("ff"), silent = TRUE)
.ff.is.available <- !inherits(.ff.version, "try-error") && .ff.version >= "4.0.0" && require(ff)
# rmarkdown::render("vignettes/bit-usage.Rmd")
# devtools::build_vignettes()

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

Note the following features

Note the following warnings


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()
bit()
bitwhich()

or vectors of a certain length initialized to FALSE

logical(3)
bit(3)
bitwhich(3)

bitwhich can be created initialized to all elements TRUE with

bitwhich(3, TRUE)

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

bitwhich(3, 2)
bitwhich(3, -2)

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

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

b <- bit(3)
length(b) <- 6
b

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
w <- bitwhich(3,-2)
length(w) <- 6
w

Vector subscripting non-existing elements returns NA

l <- logical(3); l[6]
b <- bit(3); b[6]
w <- bitwhich(3); w[6]

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

l[6] <- NA; l
b[6] <- NA; b
w[6] <- NA; w

As usual list subscripting is only allowed for existing elements

l[[6]]
b[[6]]
w[[6]]

while assignments to non-existing elements do increase length.

l[[9]] <- TRUE
b[[9]] <- TRUE
w[[9]] <- TRUE
l
b
w

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)

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
as.logical(w)

coercing back to logical fails using just which

l <- c(FALSE, TRUE, FALSE)
w <- which(l)
w
as.logical(w)  # does not coerce back

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

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

as.integer(w)

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)

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

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))

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)

Better than lazily relying on automatic propagation is

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

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)

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

c(b,l)
c(l, as.logical(b))

and of course the most efficient is

c(as.bit(l), b)

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

c.booltype(l, b)

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

b <- as.bit(c(FALSE,TRUE))
rev(b)
rep(b, 3)
rep(b, length.out=6)

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)
anyNA(b)
any(b)
all(b)
sum(b)
min(b)
max(b)
range(b)
summary(b)

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))
# minimum  position of first TRUE
min.booltype(c(FALSE,TRUE))

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))

and analyzing all such chunks

sapply(chunk(b, by=3e5, method="seq"), function(i)summary(b, range=i))

or better balanced

sapply(chunk(b, by=3e5), function(i)summary(b, range=i))

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[])

we can process the ff vector in chunks

sapply(chunk(x, by=3e5), function(i)summary(x[i]))

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)]))

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 |


Methods using random access to bit vectors

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
y <- sample(n, replace=TRUE); y
x %in% y
bit_in(x,y)
bit_in(x,y, retFUN=as.logical)

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)
bit_duplicated(x, retFUN=as.logical)
bit_duplicated(x, na.rm=NA, retFUN=as.logical)

duplicated(x, incomparables = NA)
bit_duplicated(x, na.rm=FALSE, retFUN=as.logical)

bit_duplicated(x, na.rm=TRUE, retFUN=as.logical)

The bit_unique function can also handle NA in three different ways

x <- c(NA,NA,1L,1L,2L,3L)
unique(x)
bit_unique(x)

unique(x, incomparables = NA)
bit_unique(x, na.rm=FALSE)

bit_unique(x, na.rm=TRUE)

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)
bit_union(x,y)

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)
bit_intersect(x,y)

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)
bit_setdiff(x,y)

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))
bit_symdiff(x,y)

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)
bit_setequal(x,y)

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))
bit_rangediff(c(7L,1L), (3:5))
bit_rangediff(c(1L,7L), -(3:5), revy=TRUE)
bit_rangediff(c(1L,7L), -(3:5), revx=TRUE)

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))
bit_rangediff(c(1L,7L), -(1:7))
bit_rangediff(c(1L,7L), (1:7), revy=TRUE)

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

(1:9)[-7]
bit_rangediff(c(1L,9L), -7L, revy=TRUE)

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
     anyDuplicated(x)   # early termination of hash work, returns 0 or position of first duplicate
any(bit_duplicated(x))  # full bit work, returns FALSE or TRUE
 bit_anyDuplicated(x)   # early termination of bit work, returns 0 or position of first duplicate

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
sum(bit_duplicated(x))  # full bit work, returns FALSE or TRUE
 bit_sumDuplicated(x)   # early termination of bit work, returns 0 or position of first duplicated

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))
sort(unique(x))
bit_sort_unique(x)

What if duplicates shall be kept?

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)
bit_sort(x)

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)
merge_unique(bit_sort(x))
bit_sort_unique(x)

For binary functions let's start with set equality:

x = as.integer(c(3,4,4,5))
y = as.integer(c(3,4,5))
setequal(x,y)
merge_setequal(x,y)
merge_setequal(x,y, method="exact")

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)
merge_setdiff(x,y)
merge_setdiff(x,y, method="exact")
merge_rangediff(c(0L,4L),y)
merge_rangediff(c(0L,4L),c(-3L,-2L)) # y has no effect due to different sign
merge_rangediff(c(0L,4L),c(-3L,-2L), revy=TRUE)
merge_rangediff(c(0L,4L),c(-3L,-2L), revx=TRUE)

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)
union(setdiff(x,y),setdiff(y,x))
merge_symdiff(x,y)
merge_intersect(x,y)
merge_rangesect(c(-2L,1L),y)

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)
merge_union(x,y, method="unique")
merge_union(x,y, method="exact")
merge_union(x,y, method="all")
sort(c(x,y))
c(x,y)

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)
merge_match(x,y)

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
merge_in(x,y)
merge_notin(x,y)

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

x <- c(2L,4L)
merge_rangein(x,y)
merge_rangenotin(x,y)

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)
merge_last(x)
merge_firstnotin(c(300L,600L), x)
merge_firstin(c(300L,600L), x)
merge_lastin(c(300L,600L), x)
merge_lastnotin(c(300L,600L), x)



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bit documentation built on Aug. 4, 2020, 9:06 a.m.