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std.algorithm.iteration

This is a submodule of std.algorithm. It contains generic iteration algorithms.

Cheat Sheet
Function Name Description
cache Eagerly evaluates and caches another range's front.
cacheBidirectional As above, but also provides back and popBack.
chunkBy chunkBy!((a,b) => a[1] == b[1])([[1, 1], [1, 2], [2, 2], [2, 1]]) returns a range containing 3 subranges: the first with just [1, 1]; the second with the elements [1, 2] and [2, 2]; and the third with just [2, 1].
cumulativeFold cumulativeFold!((a, b) => a + b)([1, 2, 3, 4]) returns a lazily-evaluated range containing the successive reduced values 1, 3, 6, 10.
each each!writeln([1, 2, 3]) eagerly prints the numbers 1, 2 and 3 on their own lines.
filter filter!(a => a > 0)([1, -1, 2, 0, -3]) iterates over elements 1 and 2.
filterBidirectional Similar to filter, but also provides back and popBack at a small increase in cost.
fold fold!((a, b) => a + b)([1, 2, 3, 4]) returns 10.
group group([5, 2, 2, 3, 3]) returns a range containing the tuples tuple(5, 1), tuple(2, 2), and tuple(3, 2).
joiner joiner(["hello", "world!"], "; ") returns a range that iterates over the characters "hello; world!". No new string is created - the existing inputs are iterated.
map map!(a => a * 2)([1, 2, 3]) lazily returns a range with the numbers 2, 4, 6.
mean Colloquially known as the average, mean([1, 2, 3]) returns 2.
permutations Lazily computes all permutations using Heap's algorithm.
reduce reduce!((a, b) => a + b)([1, 2, 3, 4]) returns 10. This is the old implementation of fold.
splitter Lazily splits a range by a separator.
substitute [1, 2].substitute(1, 0.1) returns [0.1, 2].
sum Same as fold, but specialized for accurate summation.
uniq Iterates over the unique elements in a range, which is assumed sorted.
License:
Boost License 1.0.
Authors:
Andrei Alexandrescu
Source
std/algorithm/iteration.d
auto cache(Range)(Range range)
Constraints: if (isInputRange!Range);

auto cacheBidirectional(Range)(Range range)
Constraints: if (isBidirectionalRange!Range);

cache eagerly evaluates front of range on each construction or call to popFront, to store the result in a cache. The result is then directly returned when front is called, rather than re-evaluated.

This can be a useful function to place in a chain, after functions that have expensive evaluation, as a lazy alternative to std.array.array. In particular, it can be placed after a call to map, or before a call std.range.filter or std.range.tee

cache may provide bidirectional range iteration if needed, but since this comes at an increased cost, it must be explicitly requested via the call to cacheBidirectional. Furthermore, a bidirectional cache will evaluate the "center" element twice, when there is only one element left in the range.

cache does not provide random access primitives, as cache would be unable to cache the random accesses. If Range provides slicing primitives, then cache will provide the same slicing primitives, but hasSlicing!Cache will not yield true (as the std.range.primitives.hasSlicing trait also checks for random access).

Parameters:
Range range an input range
Returns:
An input range with the cached values of range
Examples:
import std.algorithm.comparison : equal;
import std.range, std.stdio;
import std.typecons : tuple;

ulong counter = 0;
double fun(int x)
{
    ++counter;
    // http://en.wikipedia.org/wiki/Quartic_function
    return ( (x + 4.0) * (x + 1.0) * (x - 1.0) * (x - 3.0) ) / 14.0 + 0.5;
}
// Without cache, with array (greedy)
auto result1 = iota(-4, 5).map!(a =>tuple(a, fun(a)))()
                         .filter!(a => a[1] < 0)()
                         .map!(a => a[0])()
                         .array();

// the values of x that have a negative y are:
assert(equal(result1, [-3, -2, 2]));

// Check how many times fun was evaluated.
// As many times as the number of items in both source and result.
writeln(counter); // iota(-4, 5).length + result1.length

counter = 0;
// Without array, with cache (lazy)
auto result2 = iota(-4, 5).map!(a =>tuple(a, fun(a)))()
                         .cache()
                         .filter!(a => a[1] < 0)()
                         .map!(a => a[0])();

// the values of x that have a negative y are:
assert(equal(result2, [-3, -2, 2]));

// Check how many times fun was evaluated.
// Only as many times as the number of items in source.
writeln(counter); // iota(-4, 5).length
Examples:
Tip: cache is eager when evaluating elements. If calling front on the underlying range has a side effect, it will be observable before calling front on the actual cached range. Furthermore, care should be taken composing cache with std.range.take. By placing take before cache, then cache will be "aware" of when the range ends, and correctly stop caching elements when needed. If calling front has no side effect though, placing take after cache may yield a faster range. Either way, the resulting ranges will be equivalent, but maybe not at the same cost or side effects. 
import std.algorithm.comparison : equal;
import std.range;
int i = 0;

auto r = iota(0, 4).tee!((a){i = a;}, No.pipeOnPop);
auto r1 = r.take(3).cache();
auto r2 = r.cache().take(3);

assert(equal(r1, [0, 1, 2]));
assert(i == 2); //The last "seen" element was 2. The data in cache has been cleared.

assert(equal(r2, [0, 1, 2]));
assert(i == 3); //cache has accessed 3. It is still stored internally by cache.
template map(fun...) if (fun.length >= 1)

Implements the homonym function (also known as transform) present in many languages of functional flavor. The call map!(fun)(range) returns a range of which elements are obtained by applying fun(a) left to right for all elements a in range. The original ranges are not changed. Evaluation is done lazily.

Parameters:
fun one or more transformation functions
See Also:
Map (higher-order function)
Examples:
import std.algorithm.comparison : equal;
import std.range : chain;
int[] arr1 = [ 1, 2, 3, 4 ];
int[] arr2 = [ 5, 6 ];
auto squares = map!(a => a * a)(chain(arr1, arr2));
assert(equal(squares, [ 1, 4, 9, 16, 25, 36 ]));
Examples:
Multiple functions can be passed to map. In that case, the element type of map is a tuple containing one element for each function. 
auto sums = [2, 4, 6, 8];
auto products = [1, 4, 9, 16];

size_t i = 0;
foreach (result; [ 1, 2, 3, 4 ].map!("a + a", "a * a"))
{
    writeln(result[0]); // sums[i]
    writeln(result[1]); // products[i]
    ++i;
}
Examples:
You may alias map with some function(s) to a symbol and use it separately: 
import std.algorithm.comparison : equal;
import std.conv : to;

alias stringize = map!(to!string);
assert(equal(stringize([ 1, 2, 3, 4 ]), [ "1", "2", "3", "4" ]));
auto map(Range)(Range r)
Constraints: if (isInputRange!(Unqual!Range));
Parameters:
Range r an input range
Returns:
A range with each fun applied to all the elements. If there is more than one fun, the element type will be Tuple containing one element for each fun.
template each(alias fun = "a")

Eagerly iterates over r and calls fun over each element.

If no function to call is specified, each defaults to doing nothing but consuming the entire range. r.front will be evaluated, but that can be avoided by specifying a lambda with a lazy parameter.

each also supports opApply-based types, so it works with e.g. std.parallelism.parallel.

Normally the entire range is iterated. If partial iteration (early stopping) is desired, fun needs to return a value of type std.typecons.Flag!"each" (Yes.each to continue iteration, or No.each to stop iteration).

Parameters:
fun function to apply to each element of the range
Range r range or iterable over which each iterates
Returns:
Yes.each if the entire range was iterated, No.each in case of early stopping.
See Also:
std.range.tee
Examples:
import std.range : iota;
import std.typecons : Flag, Yes, No;

long[] arr;
iota(5).each!(n => arr ~= n);
writeln(arr); // [0, 1, 2, 3, 4]
iota(5).each!((n) { arr ~= n; return No.each; });
writeln(arr); // [0, 1, 2, 3, 4, 0]

// If the range supports it, the value can be mutated in place
arr.each!((ref n) => n++);
writeln(arr); // [1, 2, 3, 4, 5, 1]

arr.each!"a++";
writeln(arr); // [2, 3, 4, 5, 6, 2]

// by-ref lambdas are not allowed for non-ref ranges
static assert(!is(typeof(arr.map!(n => n).each!((ref n) => n++))));

// The default predicate consumes the range
auto m = arr.map!(n => n);
(&m).each();
assert(m.empty);

// Indexes are also available for in-place mutations
arr[] = 0;
arr.each!"a=i"();
writeln(arr); // [0, 1, 2, 3, 4, 5]

// opApply iterators work as well
static class S
{
    int x;
    int opApply(scope int delegate(ref int _x) dg) { return dg(x); }
}

auto s = new S;
s.each!"a++";
writeln(s.x); // 1
Examples:
each works with iterable objects which provide an index variable, along with each element 
import std.range : iota, lockstep;

auto arr = [1, 2, 3, 4];

// 1 ref parameter
arr.each!((ref e) => e = 0);
writeln(arr.sum); // 0

// 1 ref parameter and index
arr.each!((i, ref e) => e = cast(int) i);
writeln(arr.sum); // 4.iota.sum
Flag!"each" each(Range)(Range r)
Constraints: if (!isForeachIterable!Range && (isRangeIterable!Range || __traits(compiles, typeof(r.front).length)));

Flag!"each" each(Iterable)(auto ref Iterable r)
Constraints: if (isForeachIterable!Iterable || __traits(compiles, Parameters!(Parameters!(r.opApply))));
Parameters:
Range r range or iterable over which each iterates
template filter(alias predicate) if (is(typeof(unaryFun!predicate)))

Implements the higher order filter function. The predicate is passed to std.functional.unaryFun, and can either accept a string, or any callable that can be executed via pred(element).

Parameters:
predicate Function to apply to each element of range
Returns:
filter!(predicate)(range) returns a new range containing only elements x in range for which predicate(x) returns true.
See Also:
Filter (higher-order function)
Examples:
import std.algorithm.comparison : equal;
import std.math : approxEqual;
import std.range;

int[] arr = [ 1, 2, 3, 4, 5 ];

// Filter below 3
auto small = filter!(a => a < 3)(arr);
assert(equal(small, [ 1, 2 ]));

// Filter again, but with Uniform Function Call Syntax (UFCS)
auto sum = arr.filter!(a => a < 3);
assert(equal(sum, [ 1, 2 ]));

// In combination with chain() to span multiple ranges
int[] a = [ 3, -2, 400 ];
int[] b = [ 100, -101, 102 ];
auto r = chain(a, b).filter!(a => a > 0);
assert(equal(r, [ 3, 400, 100, 102 ]));

// Mixing convertible types is fair game, too
double[] c = [ 2.5, 3.0 ];
auto r1 = chain(c, a, b).filter!(a => cast(int) a != a);
assert(approxEqual(r1, [ 2.5 ]));
auto filter(Range)(Range range)
Constraints: if (isInputRange!(Unqual!Range));
Parameters:
Range range An input range of elements
Returns:
A range containing only elements x in range for which predicate(x) returns true.
template filterBidirectional(alias pred)

Similar to filter, except it defines a bidirectional range. There is a speed disadvantage - the constructor spends time finding the last element in the range that satisfies the filtering condition (in addition to finding the first one). The advantage is that the filtered range can be spanned from both directions. Also, std.range.retro can be applied against the filtered range.

The predicate is passed to std.functional.unaryFun, and can either accept a string, or any callable that can be executed via pred(element).

Parameters:
pred Function to apply to each element of range
Examples:
import std.algorithm.comparison : equal;
import std.range;

int[] arr = [ 1, 2, 3, 4, 5 ];
auto small = filterBidirectional!("a < 3")(arr);
static assert(isBidirectionalRange!(typeof(small)));
writeln(small.back); // 2
assert(equal(small, [ 1, 2 ]));
assert(equal(retro(small), [ 2, 1 ]));
// In combination with chain() to span multiple ranges
int[] a = [ 3, -2, 400 ];
int[] b = [ 100, -101, 102 ];
auto r = filterBidirectional!("a > 0")(chain(a, b));
writeln(r.back); // 102
auto filterBidirectional(Range)(Range r)
Constraints: if (isBidirectionalRange!(Unqual!Range));
Parameters:
Range r Bidirectional range of elements
Returns:
A range containing only the elements in r for which pred returns true.
Group!(pred, Range) group(alias pred = "a == b", Range)(Range r);

struct Group(alias pred, R) if (isInputRange!R);

Groups consecutively equivalent elements into a single tuple of the element and the number of its repetitions.

Similarly to uniq, group produces a range that iterates over unique consecutive elements of the given range. Each element of this range is a tuple of the element and the number of times it is repeated in the original range. Equivalence of elements is assessed by using the predicate pred, which defaults to "a == b". The predicate is passed to std.functional.binaryFun, and can either accept a string, or any callable that can be executed via pred(element, element).

Parameters:
pred Binary predicate for determining equivalence of two elements.
R The range type
Range r The input range to iterate over.
Returns:
A range of elements of type Tuple!(ElementType!R, uint), representing each consecutively unique element and its respective number of occurrences in that run. This will be an input range if R is an input range, and a forward range in all other cases.
See Also:
chunkBy, which chunks an input range into subranges of equivalent adjacent elements.
Examples:
import std.algorithm.comparison : equal;
import std.typecons : tuple, Tuple;

int[] arr = [ 1, 2, 2, 2, 2, 3, 4, 4, 4, 5 ];
assert(equal(group(arr), [ tuple(1, 1u), tuple(2, 4u), tuple(3, 1u),
    tuple(4, 3u), tuple(5, 1u) ][]));
Examples:
Using group, an associative array can be easily generated with the count of each unique element in the range. 
import std.algorithm.sorting : sort;
import std.array : assocArray;

uint[string] result;
auto range = ["a", "b", "a", "c", "b", "c", "c", "d", "e"];
result = range.sort!((a, b) => a < b)
    .group
    .assocArray;

writeln(result); // ["a":2U, "b":2U, "c":3U, "d":1U, "e":1U]
auto chunkBy(alias pred, Range)(Range r)
Constraints: if (isInputRange!Range);

Chunks an input range into subranges of equivalent adjacent elements. In other languages this is often called partitionBy, groupBy or sliceWhen.

Equivalence is defined by the predicate pred, which can be either binary, which is passed to std.functional.binaryFun, or unary, which is passed to std.functional.unaryFun. In the binary form, two range elements a and b are considered equivalent if pred(a,b) is true. In unary form, two elements are considered equivalent if pred(a) == pred(b) is true.

This predicate must be an equivalence relation, that is, it must be reflexive (pred(x,x) is always true), symmetric (pred(x,y) == pred(y,x)), and transitive (pred(x,y) && pred(y,z) implies pred(x,z)). If this is not the case, the range returned by chunkBy may assert at runtime or behave erratically.

Parameters:
pred Predicate for determining equivalence.
Range r An input range to be chunked.
Returns:
With a binary predicate, a range of ranges is returned in which all elements in a given subrange are equivalent under the given predicate. With a unary predicate, a range of tuples is returned, with the tuple consisting of the result of the unary predicate for each subrange, and the subrange itself.
Notes
Equivalent elements separated by an intervening non-equivalent element will appear in separate subranges; this function only considers adjacent equivalence. Elements in the subranges will always appear in the same order they appear in the original range.
See Also:
group, which collapses adjacent equivalent elements into a single element.
Examples:
Showing usage with binary predicate: 
import std.algorithm.comparison : equal;

// Grouping by particular attribute of each element:
auto data = [
    [1, 1],
    [1, 2],
    [2, 2],
    [2, 3]
];

auto r1 = data.chunkBy!((a,b) => a[0] == b[0]);
assert(r1.equal!equal([
    [[1, 1], [1, 2]],
    [[2, 2], [2, 3]]
]));

auto r2 = data.chunkBy!((a,b) => a[1] == b[1]);
assert(r2.equal!equal([
    [[1, 1]],
    [[1, 2], [2, 2]],
    [[2, 3]]
]));
Examples:
Showing usage with unary predicate: 
import std.algorithm.comparison : equal;
import std.range.primitives;
import std.typecons : tuple;

// Grouping by particular attribute of each element:
auto range =
[
    [1, 1],
    [1, 1],
    [1, 2],
    [2, 2],
    [2, 3],
    [2, 3],
    [3, 3]
];

auto byX = chunkBy!(a => a[0])(range);
auto expected1 =
[
    tuple(1, [[1, 1], [1, 1], [1, 2]]),
    tuple(2, [[2, 2], [2, 3], [2, 3]]),
    tuple(3, [[3, 3]])
];
foreach (e; byX)
{
    assert(!expected1.empty);
    writeln(e[0]); // expected1.front[0]
    assert(e[1].equal(expected1.front[1]));
    expected1.popFront();
}

auto byY = chunkBy!(a => a[1])(range);
auto expected2 =
[
    tuple(1, [[1, 1], [1, 1]]),
    tuple(2, [[1, 2], [2, 2]]),
    tuple(3, [[2, 3], [2, 3], [3, 3]])
];
foreach (e; byY)
{
    assert(!expected2.empty);
    writeln(e[0]); // expected2.front[0]
    assert(e[1].equal(expected2.front[1]));
    expected2.popFront();
}
auto joiner(RoR, Separator)(RoR r, Separator sep)
Constraints: if (isInputRange!RoR && isInputRange!(ElementType!RoR) && isForwardRange!Separator && is(ElementType!Separator : ElementType!(ElementType!RoR)));

auto joiner(RoR)(RoR r)
Constraints: if (isInputRange!RoR && isInputRange!(ElementType!RoR));

Lazily joins a range of ranges with a separator. The separator itself is a range. If a separator is not provided, then the ranges are joined directly without anything in between them (often called flatten in other languages).

Parameters:
RoR r An input range of input ranges to be joined.
Separator sep A forward range of element(s) to serve as separators in the joined range.
Returns:
A range of elements in the joined range. This will be a forward range if both outer and inner ranges of RoR are forward ranges; otherwise it will be only an input range. The range bidirectionality is propagated if no separator is specified.
See Also:
std.range.chain, which chains a sequence of ranges with compatible elements into a single range.
Examples:
import std.algorithm.comparison : equal;
import std.conv : text;

assert(["abc", "def"].joiner.equal("abcdef"));
assert(["Mary", "has", "a", "little", "lamb"]
    .joiner("...")
    .equal("Mary...has...a...little...lamb"));
assert(["", "abc"].joiner("xyz").equal("xyzabc"));
assert([""].joiner("xyz").equal(""));
assert(["", ""].joiner("xyz").equal("xyz"));
Examples:
import std.algorithm.comparison : equal;
import std.range : repeat;

assert([""].joiner.equal(""));
assert(["", ""].joiner.equal(""));
assert(["", "abc"].joiner.equal("abc"));
assert(["abc", ""].joiner.equal("abc"));
assert(["abc", "def"].joiner.equal("abcdef"));
assert(["Mary", "has", "a", "little", "lamb"].joiner.equal("Maryhasalittlelamb"));
assert("abc".repeat(3).joiner.equal("abcabcabc"));
Examples:
joiner allows in-place mutation! 
import std.algorithm.comparison : equal;
auto a = [ [1, 2, 3], [42, 43] ];
auto j = joiner(a);
j.front = 44;
writeln(a); // [[44, 2, 3], [42, 43]]
assert(equal(j, [44, 2, 3, 42, 43]));
Examples:
insert characters fully lazily into a string 
import std.algorithm.comparison : equal;
import std.range : chain, cycle, iota, only, retro, take, zip;
import std.format : format;

static immutable number = "12345678";
static immutable delimiter = ",";
auto formatted = number.retro
    .zip(3.iota.cycle.take(number.length))
    .map!(z => chain(z[0].only, z[1] == 2 ? delimiter : null))
    .joiner
    .retro;
static immutable expected = "12,345,678";
assert(formatted.equal(expected));
Examples:
joiner can be bidirectional 
import std.algorithm.comparison : equal;
import std.range : retro;

auto a = [[1, 2, 3], [4, 5]];
auto j = a.joiner;
j.back = 44;
writeln(a); // [[1, 2, 3], [4, 44]]
assert(equal(j.retro, [44, 4, 3, 2, 1]));
template reduce(fun...) if (fun.length >= 1)

Implements the homonym function (also known as accumulate, compress, inject, or foldl) present in various programming languages of functional flavor. There is also fold which does the same thing but with the opposite parameter order. The call reduce!(fun)(seed, range) first assigns seed to an internal variable result, also called the accumulator. Then, for each element x in range, result = fun(result, x) gets evaluated. Finally, result is returned. The one-argument version reduce!(fun)(range) works similarly, but it uses the first element of the range as the seed (the range must be non-empty).

Returns:
the accumulated result
Parameters:
fun one or more functions
See Also:
Fold (higher-order function) fold is functionally equivalent to reduce with the argument order reversed, and without the need to use tuple for multiple seeds. This makes it easier to use in UFCS chains. sum is similar to reduce!((a, b) => a + b) that offers pairwise summing of floating point numbers.
Examples:
Many aggregate range operations turn out to be solved with reduce quickly and easily. The example below illustrates reduce's remarkable power and flexibility. 
import std.algorithm.comparison : max, min;
import std.math : approxEqual;
import std.range;

int[] arr = [ 1, 2, 3, 4, 5 ];
// Sum all elements
auto sum = reduce!((a,b) => a + b)(0, arr);
writeln(sum); // 15

// Sum again, using a string predicate with "a" and "b"
sum = reduce!"a + b"(0, arr);
writeln(sum); // 15

// Compute the maximum of all elements
auto largest = reduce!(max)(arr);
writeln(largest); // 5

// Max again, but with Uniform Function Call Syntax (UFCS)
largest = arr.reduce!(max);
writeln(largest); // 5

// Compute the number of odd elements
auto odds = reduce!((a,b) => a + (b & 1))(0, arr);
writeln(odds); // 3

// Compute the sum of squares
auto ssquares = reduce!((a,b) => a + b * b)(0, arr);
writeln(ssquares); // 55

// Chain multiple ranges into seed
int[] a = [ 3, 4 ];
int[] b = [ 100 ];
auto r = reduce!("a + b")(chain(a, b));
writeln(r); // 107

// Mixing convertible types is fair game, too
double[] c = [ 2.5, 3.0 ];
auto r1 = reduce!("a + b")(chain(a, b, c));
assert(approxEqual(r1, 112.5));

// To minimize nesting of parentheses, Uniform Function Call Syntax can be used
auto r2 = chain(a, b, c).reduce!("a + b");
assert(approxEqual(r2, 112.5));
Examples:
Sometimes it is very useful to compute multiple aggregates in one pass. One advantage is that the computation is faster because the looping overhead is shared. That's why reduce accepts multiple functions. If two or more functions are passed, reduce returns a std.typecons.Tuple object with one member per passed-in function. The number of seeds must be correspondingly increased. 
import std.algorithm.comparison : max, min;
import std.math : approxEqual, sqrt;
import std.typecons : tuple, Tuple;

double[] a = [ 3.0, 4, 7, 11, 3, 2, 5 ];
// Compute minimum and maximum in one pass
auto r = reduce!(min, max)(a);
// The type of r is Tuple!(int, int)
assert(approxEqual(r[0], 2));  // minimum
assert(approxEqual(r[1], 11)); // maximum

// Compute sum and sum of squares in one pass
r = reduce!("a + b", "a + b * b")(tuple(0.0, 0.0), a);
assert(approxEqual(r[0], 35));  // sum
assert(approxEqual(r[1], 233)); // sum of squares
// Compute average and standard deviation from the above
auto avg = r[0] / a.length;
writeln(avg); // 5
auto stdev = sqrt(r[1] / a.length - avg * avg);
writeln(cast(int)stdev); // 2
auto reduce(R)(R r)
Constraints: if (isIterable!R);

No-seed version. The first element of r is used as the seed's value.

For each function f in fun, the corresponding seed type S is Unqual!(typeof(f(e, e))), where e is an element of r: ElementType!R for ranges, and ForeachType!R otherwise.

Once S has been determined, then S s = e; and s = f(s, e); must both be legal.

Parameters:
R r an iterable value as defined by isIterable
Returns:
the final result of the accumulator applied to the iterable
Throws:
Exception if r is empty
auto reduce(S, R)(S seed, R r)
Constraints: if (isIterable!R);

Seed version. The seed should be a single value if fun is a single function. If fun is multiple functions, then seed should be a std.typecons.Tuple, with one field per function in f.

For convenience, if the seed is const, or has qualified fields, then reduce will operate on an unqualified copy. If this happens then the returned type will not perfectly match S.

Use fold instead of reduce to use the seed version in a UFCS chain.

Parameters:
S seed the initial value of the accumulator
R r an iterable value as defined by isIterable
Returns:
the final result of the accumulator applied to the iterable
template fold(fun...) if (fun.length >= 1)

Implements the homonym function (also known as accumulate, compress, inject, or foldl) present in various programming languages of functional flavor. The call fold!(fun)(range, seed) first assigns seed to an internal variable result, also called the accumulator. Then, for each element x in range, result = fun(result, x) gets evaluated. Finally, result is returned. The one-argument version fold!(fun)(range) works similarly, but it uses the first element of the range as the seed (the range must be non-empty).

Parameters:
fun the predicate function(s) to apply to the elements
See Also:
Fold (higher-order function) sum is similar to fold!((a, b) => a + b) that offers precise summing of floating point numbers. This is functionally equivalent to reduce with the argument order reversed, and without the need to use tuple for multiple seeds.
Examples:
immutable arr = [1, 2, 3, 4, 5];

// Sum all elements
writeln(arr.fold!( (a, b) => a + b)); // 15

// Sum all elements with explicit seed
writeln(arr.fold!( (a, b) => a + b)(6)); // 21

import std.algorithm.comparison : min, max;
import std.typecons : tuple;

// Compute minimum and maximum at the same time
writeln(arr.fold!(min, max)); // tuple(1, 5)

// Compute minimum and maximum at the same time with seeds
writeln(arr.fold!(min, max)(0, 7)); // tuple(0, 7)

// Can be used in a UFCS chain
writeln(arr.map!(a => a + 1).fold!( (a, b) => a + b)); // 20

// Return the last element of any range
writeln(arr.fold!( (a, b) => b)); // 5
auto fold(R, S...)(R r, S seed);
Parameters:
R r the input range to fold
S seed the initial value of the accumulator
Returns:
the accumulated result
template cumulativeFold(fun...) if (fun.length >= 1)

Similar to fold, but returns a range containing the successive reduced values. The call cumulativeFold!(fun)(range, seed) first assigns seed to an internal variable result, also called the accumulator. The returned range contains the values result = fun(result, x) lazily evaluated for each element x in range. Finally, the last element has the same value as fold!(fun)(seed, range). The one-argument version cumulativeFold!(fun)(range) works similarly, but it returns the first element unchanged and uses it as seed for the next elements. This function is also known as partial_sum, accumulate, scan, Cumulative Sum.

Parameters:
fun one or more functions to use as fold operation
Returns:
The function returns a range containing the consecutive reduced values. If there is more than one fun, the element type will be std.typecons.Tuple containing one element for each fun.
See Also:
Prefix Sum
Note
In functional programming languages this is typically called scan, scanl, scanLeft or reductions.
Examples:
import std.algorithm.comparison : max, min;
import std.array : array;
import std.math : approxEqual;
import std.range : chain;

int[] arr = [1, 2, 3, 4, 5];
// Partial sum of all elements
auto sum = cumulativeFold!((a, b) => a + b)(arr, 0);
writeln(sum.array); // [1, 3, 6, 10, 15]

// Partial sum again, using a string predicate with "a" and "b"
auto sum2 = cumulativeFold!"a + b"(arr, 0);
writeln(sum2.array); // [1, 3, 6, 10, 15]

// Compute the partial maximum of all elements
auto largest = cumulativeFold!max(arr);
writeln(largest.array); // [1, 2, 3, 4, 5]

// Partial max again, but with Uniform Function Call Syntax (UFCS)
largest = arr.cumulativeFold!max;
writeln(largest.array); // [1, 2, 3, 4, 5]

// Partial count of odd elements
auto odds = arr.cumulativeFold!((a, b) => a + (b & 1))(0);
writeln(odds.array); // [1, 1, 2, 2, 3]

// Compute the partial sum of squares
auto ssquares = arr.cumulativeFold!((a, b) => a + b * b)(0);
writeln(ssquares.array); // [1, 5, 14, 30, 55]

// Chain multiple ranges into seed
int[] a = [3, 4];
int[] b = [100];
auto r = cumulativeFold!"a + b"(chain(a, b));
writeln(r.array); // [3, 7, 107]

// Mixing convertible types is fair game, too
double[] c = [2.5, 3.0];
auto r1 = cumulativeFold!"a + b"(chain(a, b, c));
assert(approxEqual(r1, [3, 7, 107, 109.5, 112.5]));

// To minimize nesting of parentheses, Uniform Function Call Syntax can be used
auto r2 = chain(a, b, c).cumulativeFold!"a + b";
assert(approxEqual(r2, [3, 7, 107, 109.5, 112.5]));
Examples:
Sometimes it is very useful to compute multiple aggregates in one pass. One advantage is that the computation is faster because the looping overhead is shared. That's why cumulativeFold accepts multiple functions. If two or more functions are passed, cumulativeFold returns a std.typecons.Tuple object with one member per passed-in function. The number of seeds must be correspondingly increased. 
import std.algorithm.comparison : max, min;
import std.algorithm.iteration : map;
import std.math : approxEqual;
import std.typecons : tuple;

double[] a = [3.0, 4, 7, 11, 3, 2, 5];
// Compute minimum and maximum in one pass
auto r = a.cumulativeFold!(min, max);
// The type of r is Tuple!(int, int)
assert(approxEqual(r.map!"a[0]", [3, 3, 3, 3, 3, 2, 2]));     // minimum
assert(approxEqual(r.map!"a[1]", [3, 4, 7, 11, 11, 11, 11])); // maximum

// Compute sum and sum of squares in one pass
auto r2 = a.cumulativeFold!("a + b", "a + b * b")(tuple(0.0, 0.0));
assert(approxEqual(r2.map!"a[0]", [3, 7, 14, 25, 28, 30, 35]));      // sum
assert(approxEqual(r2.map!"a[1]", [9, 25, 74, 195, 204, 208, 233])); // sum of squares
auto cumulativeFold(R)(R range)
Constraints: if (isInputRange!(Unqual!R));

No-seed version. The first element of r is used as the seed's value. For each function f in fun, the corresponding seed type S is Unqual!(typeof(f(e, e))), where e is an element of r: ElementType!R. Once S has been determined, then S s = e; and s = f(s, e); must both be legal.

Parameters:
R range An input range
Returns:
a range containing the consecutive reduced values.
auto cumulativeFold(R, S)(R range, S seed)
Constraints: if (isInputRange!(Unqual!R));

Seed version. The seed should be a single value if fun is a single function. If fun is multiple functions, then seed should be a std.typecons.Tuple, with one field per function in f. For convenience, if the seed is const, or has qualified fields, then cumulativeFold will operate on an unqualified copy. If this happens then the returned type will not perfectly match S.

Parameters:
R range An input range
S seed the initial value of the accumulator
Returns:
a range containing the consecutive reduced values.
auto splitter(alias pred = "a == b", Range, Separator)(Range r, Separator s)
Constraints: if (is(typeof(binaryFun!pred(r.front, s)) : bool) && (hasSlicing!Range && hasLength!Range || isNarrowString!Range));

auto splitter(alias pred = "a == b", Range, Separator)(Range r, Separator s)
Constraints: if (is(typeof(binaryFun!pred(r.front, s.front)) : bool) && (hasSlicing!Range || isNarrowString!Range) && isForwardRange!Separator && (hasLength!Separator || isNarrowString!Separator));

auto splitter(alias isTerminator, Range)(Range r)
Constraints: if (isForwardRange!Range && is(typeof(unaryFun!isTerminator(r.front))));

Lazily splits a range using an element or range as a separator. Separator ranges can be any narrow string type or sliceable range type.

Two adjacent separators are considered to surround an empty element in the split range. Use filter!(a => !a.empty) on the result to compress empty elements.

The predicate is passed to std.functional.binaryFun and accepts any callable function that can be executed via pred(element, s).

Notes
If splitting a string on whitespace and token compression is desired, consider using splitter without specifying a separator.
If no separator is passed, the predicate isTerminator decides whether to accept an element of r.
Parameters:
pred The predicate for comparing each element with the separator, defaulting to "a == b".
Range r The input range to be split. Must support slicing and .length or be a narrow string type.
Separator s The element (or range) to be treated as the separator between range segments to be split.
isTerminator The predicate for deciding where to split the range when no separator is passed
Constraints
The predicate pred needs to accept an element of r and the separator s.
Returns:
An input range of the subranges of elements between separators. If r is a forward range or bidirectional range, the returned range will be likewise. When a range is used a separator, bidirectionality isn't possible. If an empty range is given, the result is an empty range. If a range with one separator is given, the result is a range with two empty elements.
See Also:
std.regex.splitter for a version that splits using a regular expression defined separator and std.array.split for a version that splits eagerly.
Examples:
Basic splitting with characters and numbers. 
import std.algorithm.comparison : equal;

assert("a|bc|def".splitter('|').equal([ "a", "bc", "def" ]));

int[] a = [1, 0, 2, 3, 0, 4, 5, 6];
int[][] w = [ [1], [2, 3], [4, 5, 6] ];
assert(a.splitter(0).equal(w));
Examples:
Adjacent separators. 
import std.algorithm.comparison : equal;

assert("|ab|".splitter('|').equal([ "", "ab", "" ]));
assert("ab".splitter('|').equal([ "ab" ]));

assert("a|b||c".splitter('|').equal([ "a", "b", "", "c" ]));
assert("hello  world".splitter(' ').equal([ "hello", "", "world" ]));

auto a = [ 1, 2, 0, 0, 3, 0, 4, 5, 0 ];
auto w = [ [1, 2], [], [3], [4, 5], [] ];
assert(a.splitter(0).equal(w));
Examples:
Empty and separator-only ranges. 
import std.algorithm.comparison : equal;
import std.range : empty;

assert("".splitter('|').empty);
assert("|".splitter('|').equal([ "", "" ]));
assert("||".splitter('|').equal([ "", "", "" ]));
Examples:
Use a range for splitting 
import std.algorithm.comparison : equal;

assert("a=>bc=>def".splitter("=>").equal([ "a", "bc", "def" ]));
assert("a|b||c".splitter("||").equal([ "a|b", "c" ]));
assert("hello  world".splitter("  ").equal([ "hello", "world" ]));

int[] a = [ 1, 2, 0, 0, 3, 0, 4, 5, 0 ];
int[][] w = [ [1, 2], [3, 0, 4, 5, 0] ];
assert(a.splitter([0, 0]).equal(w));

a = [ 0, 0 ];
assert(a.splitter([0, 0]).equal([ (int[]).init, (int[]).init ]));

a = [ 0, 0, 1 ];
assert(a.splitter([0, 0]).equal([ [], [1] ]));
Examples:
Custom predicate functions. 
import std.algorithm.comparison : equal;
import std.ascii : toLower;

assert("abXcdxef".splitter!"a.toLower == b"('x').equal(
             [ "ab", "cd", "ef" ]));

auto w = [ [0], [1], [2] ];
assert(w.splitter!"a.front == b"(1).equal([ [[0]], [[2]] ]));
Examples:
Use splitter without a separator 
import std.algorithm.comparison : equal;
import std.range.primitives : front;

assert(equal(splitter!(a => a == '|')("a|bc|def"), [ "a", "bc", "def" ]));
assert(equal(splitter!(a => a == ' ')("hello  world"), [ "hello", "", "world" ]));

int[] a = [ 1, 2, 0, 0, 3, 0, 4, 5, 0 ];
int[][] w = [ [1, 2], [], [3], [4, 5], [] ];
assert(equal(splitter!(a => a == 0)(a), w));

a = [ 0 ];
assert(equal(splitter!(a => a == 0)(a), [ (int[]).init, (int[]).init ]));

a = [ 0, 1 ];
assert(equal(splitter!(a => a == 0)(a), [ [], [1] ]));

w = [ [0], [1], [2] ];
assert(equal(splitter!(a => a.front == 1)(w), [ [[0]], [[2]] ]));
Examples:
Leading separators, trailing separators, or no separators. 
import std.algorithm.comparison : equal;

assert("|ab|".splitter('|').equal([ "", "ab", "" ]));
assert("ab".splitter('|').equal([ "ab" ]));
Examples:
Splitter returns bidirectional ranges if the delimiter is a single element 
import std.algorithm.comparison : equal;
import std.range : retro;
assert("a|bc|def".splitter('|').retro.equal([ "def", "bc", "a" ]));
Examples:
Splitting by word lazily 
import std.ascii : isWhite;
import std.algorithm.comparison : equal;
import std.algorithm.iteration : splitter;

string str = "Hello World!";
assert(str.splitter!(isWhite).equal(["Hello", "World!"]));
auto splitter(Range)(Range s)
Constraints: if (isSomeString!Range || isRandomAccessRange!Range && hasLength!Range && hasSlicing!Range && !isConvertibleToString!Range && isSomeChar!(ElementEncodingType!Range));

Lazily splits the character-based range s into words, using whitespace as the delimiter.

This function is character-range specific and, contrary to splitter!(std.uni.isWhite), runs of whitespace will be merged together (no empty tokens will be produced).

Parameters:
Range s The character-based range to be split. Must be a string, or a random-access range of character types.
Returns:
An input range of slices of the original range split by whitespace.
Examples:
import std.algorithm.comparison : equal;
auto a = " a     bcd   ef gh ";
assert(equal(splitter(a), ["a", "bcd", "ef", "gh"][]));
template substitute(substs...) if (substs.length >= 2 && isExpressions!substs)

auto substitute(alias pred = (a, b) => a == b, R, Substs...)(R r, Substs substs)
Constraints: if (isInputRange!R && (Substs.length >= 2) && !is(CommonType!Substs == void));

Returns a range with all occurrences of substs in r. replaced with their substitution.

Single value replacements ('ö'.substitute!('ä', 'a', 'ö', 'o', 'ü', 'u)) are supported as well and in Ο(1).

Parameters:
R r an input range
Value value a single value which can be substituted in Ο(1)
Substs substs a set of replacements/substitutions
pred the equality function to test if element(s) are equal to a substitution
Returns:
a range with the substitutions replaced.
See Also:
std.array.replace for an eager replace algorithm or std.string.translate, and std.string.tr for string algorithms with translation tables.
Examples:
import std.algorithm.comparison : equal;

// substitute single elements
assert("do_it".substitute('_', ' ').equal("do it"));

// substitute multiple, single elements
assert("do_it".substitute('_', ' ',
                           'd', 'g',
                           'i', 't',
                           't', 'o')
              .equal("go to"));

// substitute subranges
assert("do_it".substitute("_", " ",
                          "do", "done")
              .equal("done it"));

// substitution works for any ElementType
int[] x = [1, 2, 3];
auto y = x.substitute(1, 0.1);
assert(y.equal([0.1, 2, 3]));
static assert(is(typeof(y.front) == double));

import std.range : retro;
assert([1, 2, 3].substitute(1, 0.1).retro.equal([3, 2, 0.1]));
Examples:
Use the faster compile-time overload 
import std.algorithm.comparison : equal;

// substitute subranges of a range
assert("apple_tree".substitute!("apple", "banana",
                                "tree", "shrub").equal("banana_shrub"));

// substitute subranges of a range
assert("apple_tree".substitute!('a', 'b',
                                't', 'f').equal("bpple_free"));

// substitute values
writeln('a'.substitute!('a', 'b', 't', 'f')); // 'b'
Examples:
Multiple substitutes 
import std.algorithm.comparison : equal;
import std.range.primitives : ElementType;

int[3] x = [1, 2, 3];
auto y = x[].substitute(1, 0.1)
            .substitute(0.1, 0.2);
static assert(is(typeof(y.front) == double));
assert(y.equal([0.2, 2, 3]));

auto z = "42".substitute('2', '3')
             .substitute('3', '1');
static assert(is(ElementType!(typeof(z)) == dchar));
assert(equal(z, "41"));
auto substitute(Value)(Value value)
Constraints: if (isInputRange!Value || !is(CommonType!(Value, typeof(substs[0])) == void));

Substitute single values with compile-time substitution mappings.

Complexity
Ο(1) due to D's switch guaranteeing Ο(1);
auto sum(R)(R r)
Constraints: if (isInputRange!R && !isInfinite!R && is(typeof(r.front + r.front)));

auto sum(R, E)(R r, E seed)
Constraints: if (isInputRange!R && !isInfinite!R && is(typeof(seed = seed + r.front)));

Sums elements of r, which must be a finite input range. Although conceptually sum(r) is equivalent to fold!((a, b) => a + b)(r, 0), sum uses specialized algorithms to maximize accuracy, as follows.



For floating point inputs, calculations are made in real precision for real inputs and in double precision otherwise (Note this is a special case that deviates from fold's behavior, which would have kept float precision for a float range). For all other types, the calculations are done in the same type obtained from from adding two elements of the range, which may be a different type from the elements themselves (for example, in case of integral promotion).

A seed may be passed to sum. Not only will this seed be used as an initial value, but its type will override all the above, and determine the algorithm and precision used for summation.

Note that these specialized summing algorithms execute more primitive operations than vanilla summation. Therefore, if in certain cases maximum speed is required at expense of precision, one can use fold!((a, b) => a + b)(r, 0), which is not specialized for summation.
Parameters:
E seed the initial value of the summation
R r a finite input range
Returns:
The sum of all the elements in the range r.
Examples:
Ditto 
import std.range;

//simple integral sumation
writeln(sum([1, 2, 3, 4])); // 10

//with integral promotion
writeln(sum([false, true, true, false, true])); // 3
writeln(sum(ubyte.max.repeat(100))); // 25500

//The result may overflow
writeln(uint.max.repeat(3).sum()); // 4294967293U
//But a seed can be used to change the sumation primitive
writeln(uint.max.repeat(3).sum(ulong.init)); // 12884901885UL

//Floating point sumation
writeln(sum([1.0, 2.0, 3.0, 4.0])); // 10

//Floating point operations have double precision minimum
static assert(is(typeof(sum([1F, 2F, 3F, 4F])) == double));
writeln(sum([1F, 2, 3, 4])); // 10

//Force pair-wise floating point sumation on large integers
import std.math : approxEqual;
assert(iota(ulong.max / 2, ulong.max / 2 + 4096).sum(0.0)
           .approxEqual((ulong.max / 2) * 4096.0 + 4096^^2 / 2));
T mean(T = double, R)(R r)
Constraints: if (isInputRange!R && isNumeric!(ElementType!R) && !isInfinite!R);

auto mean(R, T)(R r, T seed)
Constraints: if (isInputRange!R && !isNumeric!(ElementType!R) && is(typeof(r.front + seed)) && is(typeof(r.front / size_t(1))) && !isInfinite!R);

Finds the mean (colloquially known as the average) of a range.

For built-in numerical types, accurate Knuth & Welford mean calculation is used. For user-defined types, element by element summation is used. Additionally an extra parameter seed is needed in order to correctly seed the summation with the equivalent to 0.

The first overload of this function will return T.init if the range is empty. However, the second overload will return seed on empty ranges.

This function is Ο(r.length).

Parameters:
T The type of the return value.
R r An input range
T seed For user defined types. Should be equivalent to 0.
Returns:
The mean of r when r is non-empty.
Examples:
import std.math : approxEqual, isNaN;

static immutable arr1 = [1, 2, 3];
static immutable arr2 = [1.5, 2.5, 12.5];

assert(arr1.mean.approxEqual(2));
assert(arr2.mean.approxEqual(5.5));

assert(arr1[0 .. 0].mean.isNaN);
auto uniq(alias pred = "a == b", Range)(Range r)
Constraints: if (isInputRange!Range && is(typeof(binaryFun!pred(r.front, r.front)) == bool));

Lazily iterates unique consecutive elements of the given range (functionality akin to the uniq system utility). Equivalence of elements is assessed by using the predicate pred, by default "a == b". The predicate is passed to std.functional.binaryFun, and can either accept a string, or any callable that can be executed via pred(element, element). If the given range is bidirectional, uniq also yields a bidirectional range.

Parameters:
pred Predicate for determining equivalence between range elements.
Range r An input range of elements to filter.
Returns:
An input range of consecutively unique elements in the original range. If r is also a forward range or bidirectional range, the returned range will be likewise.
Examples:
import std.algorithm.comparison : equal;
import std.algorithm.mutation : copy;

int[] arr = [ 1, 2, 2, 2, 2, 3, 4, 4, 4, 5 ];
assert(equal(uniq(arr), [ 1, 2, 3, 4, 5 ][]));

// Filter duplicates in-place using copy
arr.length -= arr.uniq().copy(arr).length;
writeln(arr); // [1, 2, 3, 4, 5]

// Note that uniqueness is only determined consecutively; duplicated
// elements separated by an intervening different element will not be
// eliminated:
assert(equal(uniq([ 1, 1, 2, 1, 1, 3, 1]), [1, 2, 1, 3, 1]));
Permutations!Range permutations(Range)(Range r)
Constraints: if (isRandomAccessRange!Range && hasLength!Range);

struct Permutations(Range) if (isRandomAccessRange!Range && hasLength!Range);

Lazily computes all permutations of r using Heap's algorithm.

Parameters:
Range the range type
Range r the random access range to find the permutations for.
Returns:
A forward range of elements of which are an std.range.indexed view into r.
See Also:
std.algorithm.sorting.nextPermutation.
Examples:
import std.algorithm.comparison : equal;
import std.range : iota;
assert(equal!equal(iota(3).permutations,
    [[0, 1, 2],
     [1, 0, 2],
     [2, 0, 1],
     [0, 2, 1],
     [1, 2, 0],
     [2, 1, 0]]));

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Licensed under the Boost License 1.0.
https://dlang.org/phobos/std_algorithm_iteration.html