java/10 : java.base/java/util/stream/package-info.java

java.util.stream
https://openjdk.java.net/
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Classes to support functional-style operations on streams of elements, such
as map-reduce transformations on collections.  For example:

    int sum = widgets.stream()
                     .filter(b -> b.getColor() == RED)
                     .mapToInt(b -> b.getWeight())
                     .sum();

Here we use widgets, a Collection<Widget>, as a source for a stream, and then perform a filter-map-reduce on the stream to obtain the sum of the weights of the red widgets. (Summation is an example of a reduction
operation.)
The key abstraction introduced in this package is stream. The classes Stream, IntStream, LongStream, and DoubleStream are streams over objects and the primitive int, long and double types. Streams differ from collections in several ways: 

    No storage.  A stream is not a data structure that stores elements;
    instead, it conveys elements from a source such as a data structure,
    an array, a generator function, or an I/O channel, through a pipeline of
    computational operations.
    Functional in nature. An operation on a stream produces a result, but does not modify its source. For example, filtering a Stream obtained from a collection produces a new Stream without the filtered elements, rather than removing elements from the source collection.
    Laziness-seeking. Many stream operations, such as filtering, mapping, or duplicate removal, can be implemented lazily, exposing opportunities for optimization. For example, "find the first String with three consecutive vowels" need not examine all the input strings. Stream operations are divided into intermediate (Stream-producing) operations and terminal (value- or side-effect-producing) operations. Intermediate operations are always lazy.
    Possibly unbounded. While collections have a finite size, streams need not. Short-circuiting operations such as limit(n) or findFirst() can allow computations on infinite streams to complete in finite time.
    Consumable. The elements of a stream are only visited once during the life of a stream. Like an Iterator, a new stream must be generated to revisit the same elements of the source. 

Streams can be obtained in a number of ways. Some examples include:

    From a Collection via the stream() and parallelStream() methods;
    From an array via Arrays.stream(Object[]);
    From static factory methods on the stream classes, such as Stream.of(Object[]), IntStream.range(int, int) or Stream.iterate(Object, UnaryOperator<Object>);
    The lines of a file can be obtained from BufferedReader.lines();
    Streams of file paths can be obtained from methods in Files;
    Streams of random numbers can be obtained from Random.ints();
    Numerous other stream-bearing methods in the JDK, including BitSet.stream(), Pattern.splitAsStream(CharSequence), and JarFile.stream().

Additional stream sources can be provided by third-party libraries using
these techniques.
Stream operations and pipelines
Stream operations are divided into intermediate and
terminal operations, and are combined to form stream
pipelines. A stream pipeline consists of a source (such as a Collection, an array, a generator function, or an I/O channel); followed by zero or more intermediate operations such as Stream.filter or Stream.map; and a terminal operation such as Stream.forEach or Stream.reduce. 
Intermediate operations return a new stream.  They are always
lazy; executing an intermediate operation such as filter() does not actually perform any filtering, but instead creates a new stream that, when traversed, contains the elements of the initial stream that match the given predicate. Traversal of the pipeline source does not begin until the terminal operation of the pipeline is executed. 
Terminal operations, such as Stream.forEach or IntStream.sum, may traverse the stream to produce a result or a side-effect. After the terminal operation is performed, the stream pipeline is considered consumed, and can no longer be used; if you need to traverse the same data source again, you must return to the data source to get a new stream. In almost all cases, terminal operations are eager, completing their traversal of the data source and processing of the pipeline before returning. Only the terminal operations iterator() and spliterator() are not; these are provided as an "escape hatch" to enable arbitrary client-controlled pipeline traversals in the event that the existing operations are not sufficient to the task. 
 Processing streams lazily allows for significant efficiencies; in a
pipeline such as the filter-map-sum example above, filtering, mapping, and
summing can be fused into a single pass on the data, with minimal
intermediate state. Laziness also allows avoiding examining all the data
when it is not necessary; for operations such as "find the first string
longer than 1000 characters", it is only necessary to examine just enough
strings to find one that has the desired characteristics without examining
all of the strings available from the source. (This behavior becomes even
more important when the input stream is infinite and not merely large.)
Intermediate operations are further divided into stateless
and stateful operations. Stateless operations, such as filter and map, retain no state from previously seen element when processing a new element -- each element can be processed independently of operations on other elements. Stateful operations, such as distinct and sorted, may incorporate state from previously seen elements when processing new elements. 
Stateful operations may need to process the entire input
before producing a result.  For example, one cannot produce any results from
sorting a stream until one has seen all elements of the stream.  As a result,
under parallel computation, some pipelines containing stateful intermediate
operations may require multiple passes on the data or may need to buffer
significant data.  Pipelines containing exclusively stateless intermediate
operations can be processed in a single pass, whether sequential or parallel,
with minimal data buffering.
Further, some operations are deemed short-circuiting operations.
An intermediate operation is short-circuiting if, when presented with
infinite input, it may produce a finite stream as a result.  A terminal
operation is short-circuiting if, when presented with infinite input, it may
terminate in finite time.  Having a short-circuiting operation in the pipeline
is a necessary, but not sufficient, condition for the processing of an infinite
stream to terminate normally in finite time.
Parallelism
Processing elements with an explicit for-loop is inherently serial. Streams facilitate parallel execution by reframing the computation as a pipeline of aggregate operations, rather than as imperative operations on each individual element. All streams operations can execute either in serial or in parallel. The stream implementations in the JDK create serial streams unless parallelism is explicitly requested. For example, Collection has methods Collection.stream and Collection.parallelStream, which produce sequential and parallel streams respectively; other stream-bearing methods such as IntStream.range(int, int) produce sequential streams but these streams can be efficiently parallelized by invoking their BaseStream.parallel() method. To execute the prior "sum of weights of widgets" query in parallel, we would do: 

    int sumOfWeights = widgets.parallelStream()
                              .filter(b -> b.getColor() == RED)
                              .mapToInt(b -> b.getWeight())
                              .sum();

The only difference between the serial and parallel versions of this example is the creation of the initial stream, using "parallelStream()" instead of "stream()". The stream pipeline is executed sequentially or in parallel depending on the mode of the stream on which the terminal operation is invoked. The sequential or parallel mode of a stream can be determined with the BaseStream.isParallel() method, and the stream's mode can be modified with the BaseStream.sequential() and BaseStream.parallel() operations. The most recent sequential or parallel mode setting applies to the execution of the entire stream pipeline. 
Except for operations identified as explicitly nondeterministic, such as findAny(), whether a stream executes sequentially or in parallel should not change the result of the computation. 
Most stream operations accept parameters that describe user-specified
behavior, which are often lambda expressions.  To preserve correct behavior,
these behavioral parameters must be non-interfering, and in
most cases must be stateless.  Such parameters are always instances
of a functional interface such as Function, and are often lambda expressions or method references. 
Non-interference
 Streams enable you to execute possibly-parallel aggregate operations over a variety of data sources, including even non-thread-safe collections such as ArrayList. This is possible only if we can prevent interference with the data source during the execution of a stream pipeline. Except for the escape-hatch operations iterator() and spliterator(), execution begins when the terminal operation is invoked, and ends when the terminal operation completes. For most data sources, preventing interference means ensuring that the data source is not modified at all during the execution of the stream pipeline. The notable exception to this are streams whose sources are concurrent collections, which are specifically designed to handle concurrent modification. Concurrent stream sources are those whose Spliterator reports the CONCURRENT characteristic. Accordingly, behavioral parameters in stream pipelines whose source might
not be concurrent should never modify the stream's data source.
A behavioral parameter is said to interfere with a non-concurrent
data source if it modifies, or causes to be
modified, the stream's data source.  The need for non-interference applies
to all pipelines, not just parallel ones.  Unless the stream source is
concurrent, modifying a stream's data source during execution of a stream
pipeline can cause exceptions, incorrect answers, or nonconformant behavior.
For well-behaved stream sources, the source can be modified before the
terminal operation commences and those modifications will be reflected in
the covered elements.  For example, consider the following code:

    List<String> l = new ArrayList(Arrays.asList("one", "two"));
    Stream<String> sl = l.stream();
    l.add("three");
    String s = sl.collect(joining(" "));
 First a list is created consisting of two strings: "one"; and "two". Then a stream is created from that list. Next the list is modified by adding a third string: "three". Finally the elements of the stream are collected and joined together. Since the list was modified before the terminal collect operation commenced the result will be a string of "one two three". All the streams returned from JDK collections, and most other JDK classes, are well-behaved in this manner; for streams generated by other libraries, see Low-level stream
construction for requirements for building well-behaved streams.
Stateless behaviors
Stream pipeline results may be nondeterministic or incorrect if the behavioral
parameters to the stream operations are stateful. A stateful lambda (or other object implementing the appropriate functional interface) is one whose result depends on any state which might change during the execution of the stream pipeline. An example of a stateful lambda is the parameter to map() in: 
    Set<Integer> seen = Collections.synchronizedSet(new HashSet<>());
    stream.parallel().map(e -> { if (seen.add(e)) return 0; else return e; })...

Here, if the mapping operation is performed in parallel, the results for the
same input could vary from run to run, due to thread scheduling differences,
whereas, with a stateless lambda expression the results would always be the
same.
Note also that attempting to access mutable state from behavioral parameters
presents you with a bad choice with respect to safety and performance; if
you do not synchronize access to that state, you have a data race and
therefore your code is broken, but if you do synchronize access to that
state, you risk having contention undermine the parallelism you are seeking
to benefit from.  The best approach is to avoid stateful behavioral
parameters to stream operations entirely; there is usually a way to
restructure the stream pipeline to avoid statefulness.
Side-effects
Side-effects in behavioral parameters to stream operations are, in general,
discouraged, as they can often lead to unwitting violations of the
statelessness requirement, as well as other thread-safety hazards.
If the behavioral parameters do have side-effects, unless explicitly
stated, there are no guarantees as to:

   the 
   visibility of those side-effects to other threads;
   that different operations on the "same" element within the same stream
   pipeline are executed in the same thread; and
   that behavioral parameters are always invoked, since a stream
   implementation is free to elide operations (or entire stages) from a
   stream pipeline if it can prove that it would not affect the result of the
   computation.
   

The ordering of side-effects may be surprising.  Even when a pipeline is
constrained to produce a result that is consistent with the encounter order of the stream source (for example, IntStream.range(0,5).parallel().map(x -> x*2).toArray() must produce [0, 2, 4, 6, 8]), no guarantees are made as to the order in which the mapper function is applied to individual elements, or in what thread any behavioral parameter is executed for a given element. 
The eliding of side-effects may also be surprising. With the exception of terminal operations forEach and forEachOrdered, side-effects of behavioral parameters may not always be executed when the stream implementation can optimize away the execution of behavioral parameters without affecting the result of the computation. (For a specific example see the API note documented on the count operation.) 
Many computations where one might be tempted to use side effects can be more
safely and efficiently expressed without side-effects, such as using
reduction instead of mutable accumulators. However, side-effects such as using println() for debugging purposes are usually harmless. A small number of stream operations, such as forEach() and peek(), can operate only via side-effects; these should be used with care. 
As an example of how to transform a stream pipeline that inappropriately
uses side-effects to one that does not, the following code searches a stream
of strings for those matching a given regular expression, and puts the
matches in a list.

    ArrayList<String> results = new ArrayList<>();
    stream.filter(s -> pattern.matcher(s).matches())
          .forEach(s -> results.add(s));  // Unnecessary use of side-effects!
 This code unnecessarily uses side-effects. If executed in parallel, the non-thread-safety of ArrayList would cause incorrect results, and adding needed synchronization would cause contention, undermining the benefit of parallelism. Furthermore, using side-effects here is completely unnecessary; the forEach() can simply be replaced with a reduction operation that is safer, more efficient, and more amenable to parallelization: 
    List<String>results =
        stream.filter(s -> pattern.matcher(s).matches())
              .collect(Collectors.toList());  // No side-effects!

Ordering
Streams may or may not have a defined encounter order. Whether or not a stream has an encounter order depends on the source and the intermediate operations. Certain stream sources (such as List or arrays) are intrinsically ordered, whereas others (such as HashSet) are not. Some intermediate operations, such as sorted(), may impose an encounter order on an otherwise unordered stream, and others may render an ordered stream unordered, such as BaseStream.unordered(). Further, some terminal operations may ignore encounter order, such as forEach(). 
If a stream is ordered, most operations are constrained to operate on the elements in their encounter order; if the source of a stream is a List containing [1, 2, 3], then the result of executing map(x -> x*2) must be [2, 4, 6]. However, if the source has no defined encounter order, then any permutation of the values [2, 4, 6] would be a valid result. 
For sequential streams, the presence or absence of an encounter order does
not affect performance, only determinism.  If a stream is ordered, repeated
execution of identical stream pipelines on an identical source will produce
an identical result; if it is not ordered, repeated execution might produce
different results.
For parallel streams, relaxing the ordering constraint can sometimes enable more efficient execution. Certain aggregate operations, such as filtering duplicates (distinct()) or grouped reductions (Collectors.groupingBy()) can be implemented more efficiently if ordering of elements is not relevant. Similarly, operations that are intrinsically tied to encounter order, such as limit(), may require buffering to ensure proper ordering, undermining the benefit of parallelism. In cases where the stream has an encounter order, but the user does not particularly care about that encounter order, explicitly de-ordering the stream with unordered() may improve parallel performance for some stateful or terminal operations. However, most stream pipelines, such as the "sum of weight of blocks" example above, still parallelize efficiently even under ordering constraints. 
Reduction operations
A reduction operation (also called a fold) takes a sequence of input elements and combines them into a single summary result by repeated application of a combining operation, such as finding the sum or maximum of a set of numbers, or accumulating elements into a list. The streams classes have multiple forms of general reduction operations, called reduce() and collect(), as well as multiple specialized reduction forms such as sum(), max(), or count(). Of course, such operations can be readily implemented as simple sequential
loops, as in:

   int sum = 0;
   for (int x : numbers) {
      sum += x;
   }

However, there are good reasons to prefer a reduce operation
over a mutative accumulation such as the above.  Not only is a reduction
"more abstract" -- it operates on the stream as a whole rather than individual
elements -- but a properly constructed reduce operation is inherently
parallelizable, so long as the function(s) used to process the elements
are associative and
stateless.
For example, given a stream of numbers for which we want to find the sum, we
can write:

   int sum = numbers.stream().reduce(0, (x,y) -> x+y);

or:

   int sum = numbers.stream().reduce(0, Integer::sum);

These reduction operations can run safely in parallel with almost no
modification:

   int sum = numbers.parallelStream().reduce(0, Integer::sum);

Reduction parallellizes well because the implementation can operate on subsets of the data in parallel, and then combine the intermediate results to get the final correct answer. (Even if the language had a "parallel for-each" construct, the mutative accumulation approach would still required the developer to provide thread-safe updates to the shared accumulating variable sum, and the required synchronization would then likely eliminate any performance gain from parallelism.) Using reduce() instead removes all of the burden of parallelizing the reduction operation, and the library can provide an efficient parallel implementation with no additional synchronization required. 
The "widgets" examples shown earlier shows how reduction combines with other operations to replace for loops with bulk operations. If widgets is a collection of Widget objects, which have a getWeight method, we can find the heaviest widget with: 

    OptionalInt heaviest = widgets.parallelStream()
                                  .mapToInt(Widget::getWeight)
                                  .max();

In its more general form, a reduce operation on elements of type <T> yielding a result of type <U> requires three parameters: 

<U> U reduce(U identity,
             BiFunction<U, ? super T, U> accumulator,
             BinaryOperator<U> combiner);

Here, the identity element is both an initial seed value for the reduction
and a default result if there are no input elements. The accumulator
function takes a partial result and the next element, and produces a new
partial result. The combiner function combines two partial results
to produce a new partial result.  (The combiner is necessary in parallel
reductions, where the input is partitioned, a partial accumulation computed
for each partition, and then the partial results are combined to produce a
final result.)
More formally, the identity value must be an identity for the combiner function. This means that for all u, combiner.apply(identity, u) is equal to u. Additionally, the combiner function must be associative and must be compatible with the accumulator function: for all u and t, combiner.apply(u, accumulator.apply(identity, t)) must be equals() to accumulator.apply(u, t). 
The three-argument form is a generalization of the two-argument form,
incorporating a mapping step into the accumulation step.  We could
re-cast the simple sum-of-weights example using the more general form as
follows:

    int sumOfWeights = widgets.stream()
                              .reduce(0,
                                      (sum, b) -> sum + b.getWeight(),
                                      Integer::sum);

though the explicit map-reduce form is more readable and therefore should
usually be preferred. The generalized form is provided for cases where
significant work can be optimized away by combining mapping and reducing
into a single function.
Mutable reduction
A mutable reduction operation accumulates input elements into a mutable result container, such as a Collection or StringBuilder, as it processes the elements in the stream. If we wanted to take a stream of strings and concatenate them into a
single long string, we could achieve this with ordinary reduction:

    String concatenated = strings.reduce("", String::concat)

We would get the desired result, and it would even work in parallel.  However,
we might not be happy about the performance!  Such an implementation would do
a great deal of string copying, and the run time would be O(n^2) in the number of characters. A more performant approach would be to accumulate the results into a StringBuilder, which is a mutable container for accumulating strings. We can use the same technique to parallelize mutable reduction as we do with ordinary reduction. 
The mutable reduction operation is called collect(), as it collects together the desired results into a result container such as a Collection. A collect operation requires three functions: a supplier function to construct new instances of the result container, an accumulator function to incorporate an input element into a result container, and a combining function to merge the contents of one result container into another. The form of this is very similar to the general form of ordinary reduction: 

<R> R collect(Supplier<R> supplier,
              BiConsumer<R, ? super T> accumulator,
              BiConsumer<R, R> combiner);

As with reduce(), a benefit of expressing collect in this abstract way is that it is directly amenable to parallelization: we can accumulate partial results in parallel and then combine them, so long as the accumulation and combining functions satisfy the appropriate requirements. For example, to collect the String representations of the elements in a stream into an ArrayList, we could write the obvious sequential for-each form: 

    ArrayList<String> strings = new ArrayList<>();
    for (T element : stream) {
        strings.add(element.toString());
    }

Or we could use a parallelizable collect form:

    ArrayList<String> strings = stream.collect(() -> new ArrayList<>(),
                                               (c, e) -> c.add(e.toString()),
                                               (c1, c2) -> c1.addAll(c2));

or, pulling the mapping operation out of the accumulator function, we could
express it more succinctly as:

    List<String> strings = stream.map(Object::toString)
                                 .collect(ArrayList::new, ArrayList::add, ArrayList::addAll);
 Here, our supplier is just the 
ArrayList constructor, the accumulator adds the stringified element to an ArrayList, and the combiner simply uses addAll to copy the strings from one container into the other. The three aspects of collect -- supplier, accumulator, and combiner -- are tightly coupled. We can use the abstraction of a Collector to capture all three aspects. The above example for collecting strings into a List can be rewritten using a standard Collector as: 

    List<String> strings = stream.map(Object::toString)
                                 .collect(Collectors.toList());

Packaging mutable reductions into a Collector has another advantage: composability. The class Collectors contains a number of predefined factories for collectors, including combinators that transform one collector into another. For example, suppose we have a collector that computes the sum of the salaries of a stream of employees, as follows: 

    Collector<Employee, ?, Integer> summingSalaries
        = Collectors.summingInt(Employee::getSalary);
 (The ? for the second type parameter merely indicates that we don't care about the intermediate representation used by this collector.) If we wanted to create a collector to tabulate the sum of salaries by department, we could reuse summingSalaries using groupingBy: 
    Map<Department, Integer> salariesByDept
        = employees.stream().collect(Collectors.groupingBy(Employee::getDepartment,
                                                           summingSalaries));

As with the regular reduction operation, collect() operations can only be parallelized if appropriate conditions are met. For any partially accumulated result, combining it with an empty result container must produce an equivalent result. That is, for a partially accumulated result p that is the result of any series of accumulator and combiner invocations, p must be equivalent to combiner.apply(p, supplier.get()). 
Further, however the computation is split, it must produce an equivalent result. For any input elements t1 and t2, the results r1 and r2 in the computation below must be equivalent: 

    A a1 = supplier.get();
    accumulator.accept(a1, t1);
    accumulator.accept(a1, t2);
    R r1 = finisher.apply(a1);  // result without splitting
    A a2 = supplier.get();
    accumulator.accept(a2, t1);
    A a3 = supplier.get();
    accumulator.accept(a3, t2);
    R r2 = finisher.apply(combiner.apply(a2, a3));  // result with splitting

Here, equivalence generally means according to Object.equals(Object). but in some cases equivalence may be relaxed to account for differences in order. 
Reduction, concurrency, and ordering
 With some complex reduction operations, for example a collect() that produces a Map, such as: 
    Map<Buyer, List<Transaction>> salesByBuyer
        = txns.parallelStream()
              .collect(Collectors.groupingBy(Transaction::getBuyer));
 it may actually be counterproductive to perform the operation in parallel. This is because the combining step (merging one Map into another by key) can be expensive for some Map implementations. Suppose, however, that the result container used in this reduction was a concurrently modifiable collection -- such as a ConcurrentHashMap. In that case, the parallel invocations of the accumulator could actually deposit their results concurrently into the same shared result container, eliminating the need for the combiner to merge distinct result containers. This potentially provides a boost to the parallel execution performance. We call this a concurrent reduction.
A Collector that supports concurrent reduction is marked with the Characteristics.CONCURRENT characteristic. However, a concurrent collection also has a downside. If multiple threads are depositing results concurrently into a shared container, the order in which results are deposited is non-deterministic. Consequently, a concurrent reduction is only possible if ordering is not important for the stream being processed. The Stream.collect(Collector) implementation will only perform a concurrent reduction if 

The stream is parallel;
The collector has the Characteristics.CONCURRENT characteristic, and;
Either the stream is unordered, or the collector has the Characteristics.UNORDERED characteristic. 
 You can ensure the stream is unordered by using the BaseStream.unordered() method. For example: 
    Map<Buyer, List<Transaction>> salesByBuyer
        = txns.parallelStream()
              .unordered()
              .collect(groupingByConcurrent(Transaction::getBuyer));
 (where Collectors.groupingByConcurrent is the concurrent equivalent of groupingBy). Note that if it is important that the elements for a given key appear in
the order they appear in the source, then we cannot use a concurrent
reduction, as ordering is one of the casualties of concurrent insertion.
We would then be constrained to implement either a sequential reduction or
a merge-based parallel reduction.
Associativity
 An operator or function op is associative if the following
holds:

    (a op b) op c == a op (b op c)

The importance of this to parallel evaluation can be seen if we expand this
to four terms:

    a op b op c op d == (a op b) op (c op d)
 So we can evaluate (a op b) in parallel with (c op d), and then invoke op on the results. Examples of associative operations include numeric addition, min, and
max, and string concatenation.
Low-level stream construction
 So far, all the stream examples have used methods like Collection.stream() or Arrays.stream(Object[]) to obtain a stream. How are those stream-bearing methods implemented? The class StreamSupport has a number of low-level methods for creating a stream, all using some form of a Spliterator. A spliterator is the parallel analogue of an Iterator; it describes a (possibly infinite) collection of elements, with support for sequentially advancing, bulk traversal, and splitting off some portion of the input into another spliterator which can be processed in parallel. At the lowest level, all streams are driven by a spliterator. 
There are a number of implementation choices in implementing a spliterator, nearly all of which are tradeoffs between simplicity of implementation and runtime performance of streams using that spliterator. The simplest, but least performant, way to create a spliterator is to create one from an iterator using Spliterators.spliteratorUnknownSize(Iterator<? extends Object>, int). While such a spliterator will work, it will likely offer poor parallel performance, since we have lost sizing information (how big is the underlying data set), as well as being constrained to a simplistic splitting algorithm. 
A higher-quality spliterator will provide balanced and known-size splits, accurate sizing information, and a number of other characteristics of the spliterator or data that can be used by implementations to optimize execution. 
Spliterators for mutable data sources have an additional challenge; timing of binding to the data, since the data could change between the time the spliterator is created and the time the stream pipeline is executed. Ideally, a spliterator for a stream would report a characteristic of IMMUTABLE or CONCURRENT; if not it should be late-binding. If a source cannot directly supply a recommended spliterator, it may indirectly supply a spliterator using a Supplier, and construct a stream via the Supplier-accepting versions of stream(). The spliterator is obtained from the supplier only after the terminal operation of the stream pipeline commences. 
These requirements significantly reduce the scope of potential
interference between mutations of the stream source and execution of stream
pipelines. Streams based on spliterators with the desired characteristics,
or those using the Supplier-based factory forms, are immune to
modifications of the data source prior to commencement of the terminal
operation (provided the behavioral parameters to the stream operations meet
the required criteria for non-interference and statelessness).  See
Non-Interference
for more details.
Since: 1.8/**
 * Classes to support functional-style operations on streams of elements, such
 * as map-reduce transformations on collections.  For example:
 *
 * <pre>{@code
 *     int sum = widgets.stream()
 *                      .filter(b -> b.getColor() == RED)
 *                      .mapToInt(b -> b.getWeight())
 *                      .sum();
 * }</pre>
 *
 * <p>Here we use {@code widgets}, a {@code Collection<Widget>},
 * as a source for a stream, and then perform a filter-map-reduce on the stream
 * to obtain the sum of the weights of the red widgets.  (Summation is an
 * example of a <a href="package-summary.html#Reduction">reduction</a>
 * operation.)
 *
 * <p>The key abstraction introduced in this package is <em>stream</em>.  The
 * classes {@link java.util.stream.Stream}, {@link java.util.stream.IntStream},
 * {@link java.util.stream.LongStream}, and {@link java.util.stream.DoubleStream}
 * are streams over objects and the primitive {@code int}, {@code long} and
 * {@code double} types.  Streams differ from collections in several ways:
 *
 * <ul>
 *     <li>No storage.  A stream is not a data structure that stores elements;
 *     instead, it conveys elements from a source such as a data structure,
 *     an array, a generator function, or an I/O channel, through a pipeline of
 *     computational operations.</li>
 *     <li>Functional in nature.  An operation on a stream produces a result,
 *     but does not modify its source.  For example, filtering a {@code Stream}
 *     obtained from a collection produces a new {@code Stream} without the
 *     filtered elements, rather than removing elements from the source
 *     collection.</li>
 *     <li>Laziness-seeking.  Many stream operations, such as filtering, mapping,
 *     or duplicate removal, can be implemented lazily, exposing opportunities
 *     for optimization.  For example, "find the first {@code String} with
 *     three consecutive vowels" need not examine all the input strings.
 *     Stream operations are divided into intermediate ({@code Stream}-producing)
 *     operations and terminal (value- or side-effect-producing) operations.
 *     Intermediate operations are always lazy.</li>
 *     <li>Possibly unbounded.  While collections have a finite size, streams
 *     need not.  Short-circuiting operations such as {@code limit(n)} or
 *     {@code findFirst()} can allow computations on infinite streams to
 *     complete in finite time.</li>
 *     <li>Consumable. The elements of a stream are only visited once during
 *     the life of a stream. Like an {@link java.util.Iterator}, a new stream
 *     must be generated to revisit the same elements of the source.
 *     </li>
 * </ul>
 *
 * Streams can be obtained in a number of ways. Some examples include:
 * <ul>
 *     <li>From a {@link java.util.Collection} via the {@code stream()} and
 *     {@code parallelStream()} methods;</li>
 *     <li>From an array via {@link java.util.Arrays#stream(Object[])};</li>
 *     <li>From static factory methods on the stream classes, such as
 *     {@link java.util.stream.Stream#of(Object[])},
 *     {@link java.util.stream.IntStream#range(int, int)}
 *     or {@link java.util.stream.Stream#iterate(Object, UnaryOperator)};</li>
 *     <li>The lines of a file can be obtained from {@link java.io.BufferedReader#lines()};</li>
 *     <li>Streams of file paths can be obtained from methods in {@link java.nio.file.Files};</li>
 *     <li>Streams of random numbers can be obtained from {@link java.util.Random#ints()};</li>
 *     <li>Numerous other stream-bearing methods in the JDK, including
 *     {@link java.util.BitSet#stream()},
 *     {@link java.util.regex.Pattern#splitAsStream(java.lang.CharSequence)},
 *     and {@link java.util.jar.JarFile#stream()}.</li>
 * </ul>
 *
 * <p>Additional stream sources can be provided by third-party libraries using
 * <a href="package-summary.html#StreamSources">these techniques</a>.
 *
 * <h2><a id="StreamOps">Stream operations and pipelines</a></h2>
 *
 * <p>Stream operations are divided into <em>intermediate</em> and
 * <em>terminal</em> operations, and are combined to form <em>stream
 * pipelines</em>.  A stream pipeline consists of a source (such as a
 * {@code Collection}, an array, a generator function, or an I/O channel);
 * followed by zero or more intermediate operations such as
 * {@code Stream.filter} or {@code Stream.map}; and a terminal operation such
 * as {@code Stream.forEach} or {@code Stream.reduce}.
 *
 * <p>Intermediate operations return a new stream.  They are always
 * <em>lazy</em>; executing an intermediate operation such as
 * {@code filter()} does not actually perform any filtering, but instead
 * creates a new stream that, when traversed, contains the elements of
 * the initial stream that match the given predicate.  Traversal
 * of the pipeline source does not begin until the terminal operation of the
 * pipeline is executed.
 *
 * <p>Terminal operations, such as {@code Stream.forEach} or
 * {@code IntStream.sum}, may traverse the stream to produce a result or a
 * side-effect. After the terminal operation is performed, the stream pipeline
 * is considered consumed, and can no longer be used; if you need to traverse
 * the same data source again, you must return to the data source to get a new
 * stream.  In almost all cases, terminal operations are <em>eager</em>,
 * completing their traversal of the data source and processing of the pipeline
 * before returning.  Only the terminal operations {@code iterator()} and
 * {@code spliterator()} are not; these are provided as an "escape hatch" to enable
 * arbitrary client-controlled pipeline traversals in the event that the
 * existing operations are not sufficient to the task.
 *
 * <p> Processing streams lazily allows for significant efficiencies; in a
 * pipeline such as the filter-map-sum example above, filtering, mapping, and
 * summing can be fused into a single pass on the data, with minimal
 * intermediate state. Laziness also allows avoiding examining all the data
 * when it is not necessary; for operations such as "find the first string
 * longer than 1000 characters", it is only necessary to examine just enough
 * strings to find one that has the desired characteristics without examining
 * all of the strings available from the source. (This behavior becomes even
 * more important when the input stream is infinite and not merely large.)
 *
 * <p>Intermediate operations are further divided into <em>stateless</em>
 * and <em>stateful</em> operations. Stateless operations, such as {@code filter}
 * and {@code map}, retain no state from previously seen element when processing
 * a new element -- each element can be processed
 * independently of operations on other elements.  Stateful operations, such as
 * {@code distinct} and {@code sorted}, may incorporate state from previously
 * seen elements when processing new elements.
 *
 * <p>Stateful operations may need to process the entire input
 * before producing a result.  For example, one cannot produce any results from
 * sorting a stream until one has seen all elements of the stream.  As a result,
 * under parallel computation, some pipelines containing stateful intermediate
 * operations may require multiple passes on the data or may need to buffer
 * significant data.  Pipelines containing exclusively stateless intermediate
 * operations can be processed in a single pass, whether sequential or parallel,
 * with minimal data buffering.
 *
 * <p>Further, some operations are deemed <em>short-circuiting</em> operations.
 * An intermediate operation is short-circuiting if, when presented with
 * infinite input, it may produce a finite stream as a result.  A terminal
 * operation is short-circuiting if, when presented with infinite input, it may
 * terminate in finite time.  Having a short-circuiting operation in the pipeline
 * is a necessary, but not sufficient, condition for the processing of an infinite
 * stream to terminate normally in finite time.
 *
 * <h3><a id="Parallelism">Parallelism</a></h3>
 *
 * <p>Processing elements with an explicit {@code for-}loop is inherently serial.
 * Streams facilitate parallel execution by reframing the computation as a pipeline of
 * aggregate operations, rather than as imperative operations on each individual
 * element.  All streams operations can execute either in serial or in parallel.
 * The stream implementations in the JDK create serial streams unless parallelism is
 * explicitly requested.  For example, {@code Collection} has methods
 * {@link java.util.Collection#stream} and {@link java.util.Collection#parallelStream},
 * which produce sequential and parallel streams respectively; other
 * stream-bearing methods such as {@link java.util.stream.IntStream#range(int, int)}
 * produce sequential streams but these streams can be efficiently parallelized by
 * invoking their {@link java.util.stream.BaseStream#parallel()} method.
 * To execute the prior "sum of weights of widgets" query in parallel, we would
 * do:
 *
 * <pre>{@code
 *     int sumOfWeights = widgets.}<code><b>parallelStream()</b></code>{@code
 *                               .filter(b -> b.getColor() == RED)
 *                               .mapToInt(b -> b.getWeight())
 *                               .sum();
 * }</pre>
 *
 * <p>The only difference between the serial and parallel versions of this
 * example is the creation of the initial stream, using "{@code parallelStream()}"
 * instead of "{@code stream()}". The stream pipeline is executed sequentially or
 * in parallel depending on the mode of the stream on which the terminal operation
 * is invoked. The sequential or parallel mode of a stream can be determined with the
 * {@link java.util.stream.BaseStream#isParallel()} method, and the
 * stream's mode can be modified with the
 * {@link java.util.stream.BaseStream#sequential()} and
 * {@link java.util.stream.BaseStream#parallel()} operations.
 * The most recent sequential or parallel mode setting applies to the
 * execution of the entire stream pipeline.
 *
 * <p>Except for operations identified as explicitly nondeterministic, such
 * as {@code findAny()}, whether a stream executes sequentially or in parallel
 * should not change the result of the computation.
 *
 * <p>Most stream operations accept parameters that describe user-specified
 * behavior, which are often lambda expressions.  To preserve correct behavior,
 * these <em>behavioral parameters</em> must be <em>non-interfering</em>, and in
 * most cases must be <em>stateless</em>.  Such parameters are always instances
 * of a <a href="../function/package-summary.html">functional interface</a> such
 * as {@link java.util.function.Function}, and are often lambda expressions or
 * method references.
 *
 * <h3><a id="NonInterference">Non-interference</a></h3>
 *
 * Streams enable you to execute possibly-parallel aggregate operations over a
 * variety of data sources, including even non-thread-safe collections such as
 * {@code ArrayList}. This is possible only if we can prevent
 * <em>interference</em> with the data source during the execution of a stream
 * pipeline.  Except for the escape-hatch operations {@code iterator()} and
 * {@code spliterator()}, execution begins when the terminal operation is
 * invoked, and ends when the terminal operation completes.  For most data
 * sources, preventing interference means ensuring that the data source is
 * <em>not modified at all</em> during the execution of the stream pipeline.
 * The notable exception to this are streams whose sources are concurrent
 * collections, which are specifically designed to handle concurrent modification.
 * Concurrent stream sources are those whose {@code Spliterator} reports the
 * {@code CONCURRENT} characteristic.
 *
 * <p>Accordingly, behavioral parameters in stream pipelines whose source might
 * not be concurrent should never modify the stream's data source.
 * A behavioral parameter is said to <em>interfere</em> with a non-concurrent
 * data source if it modifies, or causes to be
 * modified, the stream's data source.  The need for non-interference applies
 * to all pipelines, not just parallel ones.  Unless the stream source is
 * concurrent, modifying a stream's data source during execution of a stream
 * pipeline can cause exceptions, incorrect answers, or nonconformant behavior.
 *
 * For well-behaved stream sources, the source can be modified before the
 * terminal operation commences and those modifications will be reflected in
 * the covered elements.  For example, consider the following code:
 *
 * <pre>{@code
 *     List<String> l = new ArrayList(Arrays.asList("one", "two"));
 *     Stream<String> sl = l.stream();
 *     l.add("three");
 *     String s = sl.collect(joining(" "));
 * }</pre>
 *
 * First a list is created consisting of two strings: "one"; and "two". Then a
 * stream is created from that list. Next the list is modified by adding a third
 * string: "three". Finally the elements of the stream are collected and joined
 * together. Since the list was modified before the terminal {@code collect}
 * operation commenced the result will be a string of "one two three". All the
 * streams returned from JDK collections, and most other JDK classes,
 * are well-behaved in this manner; for streams generated by other libraries, see
 * <a href="package-summary.html#StreamSources">Low-level stream
 * construction</a> for requirements for building well-behaved streams.
 *
 * <h3><a id="Statelessness">Stateless behaviors</a></h3>
 *
 * Stream pipeline results may be nondeterministic or incorrect if the behavioral
 * parameters to the stream operations are <em>stateful</em>.  A stateful lambda
 * (or other object implementing the appropriate functional interface) is one
 * whose result depends on any state which might change during the execution
 * of the stream pipeline.  An example of a stateful lambda is the parameter
 * to {@code map()} in:
 *
 * <pre>{@code
 *     Set<Integer> seen = Collections.synchronizedSet(new HashSet<>());
 *     stream.parallel().map(e -> { if (seen.add(e)) return 0; else return e; })...
 * }</pre>
 *
 * Here, if the mapping operation is performed in parallel, the results for the
 * same input could vary from run to run, due to thread scheduling differences,
 * whereas, with a stateless lambda expression the results would always be the
 * same.
 *
 * <p>Note also that attempting to access mutable state from behavioral parameters
 * presents you with a bad choice with respect to safety and performance; if
 * you do not synchronize access to that state, you have a data race and
 * therefore your code is broken, but if you do synchronize access to that
 * state, you risk having contention undermine the parallelism you are seeking
 * to benefit from.  The best approach is to avoid stateful behavioral
 * parameters to stream operations entirely; there is usually a way to
 * restructure the stream pipeline to avoid statefulness.
 *
 * <h3><a id="SideEffects">Side-effects</a></h3>
 *
 * Side-effects in behavioral parameters to stream operations are, in general,
 * discouraged, as they can often lead to unwitting violations of the
 * statelessness requirement, as well as other thread-safety hazards.
 *
 * <p>If the behavioral parameters do have side-effects, unless explicitly
 * stated, there are no guarantees as to:
 * <ul>
 *    <li>the <a href="../concurrent/package-summary.html#MemoryVisibility">
 *    <i>visibility</i></a> of those side-effects to other threads;</li>
 *    <li>that different operations on the "same" element within the same stream
 *    pipeline are executed in the same thread; and</li>
 *    <li>that behavioral parameters are always invoked, since a stream
 *    implementation is free to elide operations (or entire stages) from a
 *    stream pipeline if it can prove that it would not affect the result of the
 *    computation.
 *    </li>
 * </ul>
 * <p>The ordering of side-effects may be surprising.  Even when a pipeline is
 * constrained to produce a <em>result</em> that is consistent with the
 * encounter order of the stream source (for example,
 * {@code IntStream.range(0,5).parallel().map(x -> x*2).toArray()}
 * must produce {@code [0, 2, 4, 6, 8]}), no guarantees are made as to the order
 * in which the mapper function is applied to individual elements, or in what
 * thread any behavioral parameter is executed for a given element.
 *
 * <p>The eliding of side-effects may also be surprising.  With the exception of
 * terminal operations {@link java.util.stream.Stream#forEach forEach} and
 * {@link java.util.stream.Stream#forEachOrdered forEachOrdered}, side-effects
 * of behavioral parameters may not always be executed when the stream
 * implementation can optimize away the execution of behavioral parameters
 * without affecting the result of the computation.  (For a specific example
 * see the API note documented on the {@link java.util.stream.Stream#count count}
 * operation.)
 *
 * <p>Many computations where one might be tempted to use side effects can be more
 * safely and efficiently expressed without side-effects, such as using
 * <a href="package-summary.html#Reduction">reduction</a> instead of mutable
 * accumulators. However, side-effects such as using {@code println()} for debugging
 * purposes are usually harmless.  A small number of stream operations, such as
 * {@code forEach()} and {@code peek()}, can operate only via side-effects;
 * these should be used with care.
 *
 * <p>As an example of how to transform a stream pipeline that inappropriately
 * uses side-effects to one that does not, the following code searches a stream
 * of strings for those matching a given regular expression, and puts the
 * matches in a list.
 *
 * <pre>{@code
 *     ArrayList<String> results = new ArrayList<>();
 *     stream.filter(s -> pattern.matcher(s).matches())
 *           .forEach(s -> results.add(s));  // Unnecessary use of side-effects!
 * }</pre>
 *
 * This code unnecessarily uses side-effects.  If executed in parallel, the
 * non-thread-safety of {@code ArrayList} would cause incorrect results, and
 * adding needed synchronization would cause contention, undermining the
 * benefit of parallelism.  Furthermore, using side-effects here is completely
 * unnecessary; the {@code forEach()} can simply be replaced with a reduction
 * operation that is safer, more efficient, and more amenable to
 * parallelization:
 *
 * <pre>{@code
 *     List<String>results =
 *         stream.filter(s -> pattern.matcher(s).matches())
 *               .collect(Collectors.toList());  // No side-effects!
 * }</pre>
 *
 * <h3><a id="Ordering">Ordering</a></h3>
 *
 * <p>Streams may or may not have a defined <em>encounter order</em>.  Whether
 * or not a stream has an encounter order depends on the source and the
 * intermediate operations.  Certain stream sources (such as {@code List} or
 * arrays) are intrinsically ordered, whereas others (such as {@code HashSet})
 * are not.  Some intermediate operations, such as {@code sorted()}, may impose
 * an encounter order on an otherwise unordered stream, and others may render an
 * ordered stream unordered, such as {@link java.util.stream.BaseStream#unordered()}.
 * Further, some terminal operations may ignore encounter order, such as
 * {@code forEach()}.
 *
 * <p>If a stream is ordered, most operations are constrained to operate on the
 * elements in their encounter order; if the source of a stream is a {@code List}
 * containing {@code [1, 2, 3]}, then the result of executing {@code map(x -> x*2)}
 * must be {@code [2, 4, 6]}.  However, if the source has no defined encounter
 * order, then any permutation of the values {@code [2, 4, 6]} would be a valid
 * result.
 *
 * <p>For sequential streams, the presence or absence of an encounter order does
 * not affect performance, only determinism.  If a stream is ordered, repeated
 * execution of identical stream pipelines on an identical source will produce
 * an identical result; if it is not ordered, repeated execution might produce
 * different results.
 *
 * <p>For parallel streams, relaxing the ordering constraint can sometimes enable
 * more efficient execution.  Certain aggregate operations,
 * such as filtering duplicates ({@code distinct()}) or grouped reductions
 * ({@code Collectors.groupingBy()}) can be implemented more efficiently if ordering of elements
 * is not relevant.  Similarly, operations that are intrinsically tied to encounter order,
 * such as {@code limit()}, may require
 * buffering to ensure proper ordering, undermining the benefit of parallelism.
 * In cases where the stream has an encounter order, but the user does not
 * particularly <em>care</em> about that encounter order, explicitly de-ordering
 * the stream with {@link java.util.stream.BaseStream#unordered() unordered()} may
 * improve parallel performance for some stateful or terminal operations.
 * However, most stream pipelines, such as the "sum of weight of blocks" example
 * above, still parallelize efficiently even under ordering constraints.
 *
 * <h2><a id="Reduction">Reduction operations</a></h2>
 *
 * A <em>reduction</em> operation (also called a <em>fold</em>) takes a sequence
 * of input elements and combines them into a single summary result by repeated
 * application of a combining operation, such as finding the sum or maximum of
 * a set of numbers, or accumulating elements into a list.  The streams classes have
 * multiple forms of general reduction operations, called
 * {@link java.util.stream.Stream#reduce(java.util.function.BinaryOperator) reduce()}
 * and {@link java.util.stream.Stream#collect(java.util.stream.Collector) collect()},
 * as well as multiple specialized reduction forms such as
 * {@link java.util.stream.IntStream#sum() sum()}, {@link java.util.stream.IntStream#max() max()},
 * or {@link java.util.stream.IntStream#count() count()}.
 *
 * <p>Of course, such operations can be readily implemented as simple sequential
 * loops, as in:
 * <pre>{@code
 *    int sum = 0;
 *    for (int x : numbers) {
 *       sum += x;
 *    }
 * }</pre>
 * However, there are good reasons to prefer a reduce operation
 * over a mutative accumulation such as the above.  Not only is a reduction
 * "more abstract" -- it operates on the stream as a whole rather than individual
 * elements -- but a properly constructed reduce operation is inherently
 * parallelizable, so long as the function(s) used to process the elements
 * are <a href="package-summary.html#Associativity">associative</a> and
 * <a href="package-summary.html#Statelessness">stateless</a>.
 * For example, given a stream of numbers for which we want to find the sum, we
 * can write:
 * <pre>{@code
 *    int sum = numbers.stream().reduce(0, (x,y) -> x+y);
 * }</pre>
 * or:
 * <pre>{@code
 *    int sum = numbers.stream().reduce(0, Integer::sum);
 * }</pre>
 *
 * <p>These reduction operations can run safely in parallel with almost no
 * modification:
 * <pre>{@code
 *    int sum = numbers.parallelStream().reduce(0, Integer::sum);
 * }</pre>
 *
 * <p>Reduction parallellizes well because the implementation
 * can operate on subsets of the data in parallel, and then combine the
 * intermediate results to get the final correct answer.  (Even if the language
 * had a "parallel for-each" construct, the mutative accumulation approach would
 * still required the developer to provide
 * thread-safe updates to the shared accumulating variable {@code sum}, and
 * the required synchronization would then likely eliminate any performance gain from
 * parallelism.)  Using {@code reduce()} instead removes all of the
 * burden of parallelizing the reduction operation, and the library can provide
 * an efficient parallel implementation with no additional synchronization
 * required.
 *
 * <p>The "widgets" examples shown earlier shows how reduction combines with
 * other operations to replace for loops with bulk operations.  If {@code widgets}
 * is a collection of {@code Widget} objects, which have a {@code getWeight} method,
 * we can find the heaviest widget with:
 * <pre>{@code
 *     OptionalInt heaviest = widgets.parallelStream()
 *                                   .mapToInt(Widget::getWeight)
 *                                   .max();
 * }</pre>
 *
 * <p>In its more general form, a {@code reduce} operation on elements of type
 * {@code <T>} yielding a result of type {@code <U>} requires three parameters:
 * <pre>{@code
 * <U> U reduce(U identity,
 *              BiFunction<U, ? super T, U> accumulator,
 *              BinaryOperator<U> combiner);
 * }</pre>
 * Here, the <em>identity</em> element is both an initial seed value for the reduction
 * and a default result if there are no input elements. The <em>accumulator</em>
 * function takes a partial result and the next element, and produces a new
 * partial result. The <em>combiner</em> function combines two partial results
 * to produce a new partial result.  (The combiner is necessary in parallel
 * reductions, where the input is partitioned, a partial accumulation computed
 * for each partition, and then the partial results are combined to produce a
 * final result.)
 *
 * <p>More formally, the {@code identity} value must be an <em>identity</em> for
 * the combiner function. This means that for all {@code u},
 * {@code combiner.apply(identity, u)} is equal to {@code u}. Additionally, the
 * {@code combiner} function must be <a href="package-summary.html#Associativity">associative</a> and
 * must be compatible with the {@code accumulator} function: for all {@code u}
 * and {@code t}, {@code combiner.apply(u, accumulator.apply(identity, t))} must
 * be {@code equals()} to {@code accumulator.apply(u, t)}.
 *
 * <p>The three-argument form is a generalization of the two-argument form,
 * incorporating a mapping step into the accumulation step.  We could
 * re-cast the simple sum-of-weights example using the more general form as
 * follows:
 * <pre>{@code
 *     int sumOfWeights = widgets.stream()
 *                               .reduce(0,
 *                                       (sum, b) -> sum + b.getWeight(),
 *                                       Integer::sum);
 * }</pre>
 * though the explicit map-reduce form is more readable and therefore should
 * usually be preferred. The generalized form is provided for cases where
 * significant work can be optimized away by combining mapping and reducing
 * into a single function.
 *
 * <h3><a id="MutableReduction">Mutable reduction</a></h3>
 *
 * A <em>mutable reduction operation</em> accumulates input elements into a
 * mutable result container, such as a {@code Collection} or {@code StringBuilder},
 * as it processes the elements in the stream.
 *
 * <p>If we wanted to take a stream of strings and concatenate them into a
 * single long string, we <em>could</em> achieve this with ordinary reduction:
 * <pre>{@code
 *     String concatenated = strings.reduce("", String::concat)
 * }</pre>
 *
 * <p>We would get the desired result, and it would even work in parallel.  However,
 * we might not be happy about the performance!  Such an implementation would do
 * a great deal of string copying, and the run time would be <em>O(n^2)</em> in
 * the number of characters.  A more performant approach would be to accumulate
 * the results into a {@link java.lang.StringBuilder}, which is a mutable
 * container for accumulating strings.  We can use the same technique to
 * parallelize mutable reduction as we do with ordinary reduction.
 *
 * <p>The mutable reduction operation is called
 * {@link java.util.stream.Stream#collect(Collector) collect()},
 * as it collects together the desired results into a result container such
 * as a {@code Collection}.
 * A {@code collect} operation requires three functions:
 * a supplier function to construct new instances of the result container, an
 * accumulator function to incorporate an input element into a result
 * container, and a combining function to merge the contents of one result
 * container into another.  The form of this is very similar to the general
 * form of ordinary reduction:
 * <pre>{@code
 * <R> R collect(Supplier<R> supplier,
 *               BiConsumer<R, ? super T> accumulator,
 *               BiConsumer<R, R> combiner);
 * }</pre>
 * <p>As with {@code reduce()}, a benefit of expressing {@code collect} in this
 * abstract way is that it is directly amenable to parallelization: we can
 * accumulate partial results in parallel and then combine them, so long as the
 * accumulation and combining functions satisfy the appropriate requirements.
 * For example, to collect the String representations of the elements in a
 * stream into an {@code ArrayList}, we could write the obvious sequential
 * for-each form:
 * <pre>{@code
 *     ArrayList<String> strings = new ArrayList<>();
 *     for (T element : stream) {
 *         strings.add(element.toString());
 *     }
 * }</pre>
 * Or we could use a parallelizable collect form:
 * <pre>{@code
 *     ArrayList<String> strings = stream.collect(() -> new ArrayList<>(),
 *                                                (c, e) -> c.add(e.toString()),
 *                                                (c1, c2) -> c1.addAll(c2));
 * }</pre>
 * or, pulling the mapping operation out of the accumulator function, we could
 * express it more succinctly as:
 * <pre>{@code
 *     List<String> strings = stream.map(Object::toString)
 *                                  .collect(ArrayList::new, ArrayList::add, ArrayList::addAll);
 * }</pre>
 * Here, our supplier is just the {@link java.util.ArrayList#ArrayList()
 * ArrayList constructor}, the accumulator adds the stringified element to an
 * {@code ArrayList}, and the combiner simply uses {@link java.util.ArrayList#addAll addAll}
 * to copy the strings from one container into the other.
 *
 * <p>The three aspects of {@code collect} -- supplier, accumulator, and
 * combiner -- are tightly coupled.  We can use the abstraction of a
 * {@link java.util.stream.Collector} to capture all three aspects.  The
 * above example for collecting strings into a {@code List} can be rewritten
 * using a standard {@code Collector} as:
 * <pre>{@code
 *     List<String> strings = stream.map(Object::toString)
 *                                  .collect(Collectors.toList());
 * }</pre>
 *
 * <p>Packaging mutable reductions into a Collector has another advantage:
 * composability.  The class {@link java.util.stream.Collectors} contains a
 * number of predefined factories for collectors, including combinators
 * that transform one collector into another.  For example, suppose we have a
 * collector that computes the sum of the salaries of a stream of
 * employees, as follows:
 *
 * <pre>{@code
 *     Collector<Employee, ?, Integer> summingSalaries
 *         = Collectors.summingInt(Employee::getSalary);
 * }</pre>
 *
 * (The {@code ?} for the second type parameter merely indicates that we don't
 * care about the intermediate representation used by this collector.)
 * If we wanted to create a collector to tabulate the sum of salaries by
 * department, we could reuse {@code summingSalaries} using
 * {@link java.util.stream.Collectors#groupingBy(java.util.function.Function, java.util.stream.Collector) groupingBy}:
 *
 * <pre>{@code
 *     Map<Department, Integer> salariesByDept
 *         = employees.stream().collect(Collectors.groupingBy(Employee::getDepartment,
 *                                                            summingSalaries));
 * }</pre>
 *
 * <p>As with the regular reduction operation, {@code collect()} operations can
 * only be parallelized if appropriate conditions are met.  For any partially
 * accumulated result, combining it with an empty result container must
 * produce an equivalent result.  That is, for a partially accumulated result
 * {@code p} that is the result of any series of accumulator and combiner
 * invocations, {@code p} must be equivalent to
 * {@code combiner.apply(p, supplier.get())}.
 *
 * <p>Further, however the computation is split, it must produce an equivalent
 * result.  For any input elements {@code t1} and {@code t2}, the results
 * {@code r1} and {@code r2} in the computation below must be equivalent:
 * <pre>{@code
 *     A a1 = supplier.get();
 *     accumulator.accept(a1, t1);
 *     accumulator.accept(a1, t2);
 *     R r1 = finisher.apply(a1);  // result without splitting
 *
 *     A a2 = supplier.get();
 *     accumulator.accept(a2, t1);
 *     A a3 = supplier.get();
 *     accumulator.accept(a3, t2);
 *     R r2 = finisher.apply(combiner.apply(a2, a3));  // result with splitting
 * }</pre>
 *
 * <p>Here, equivalence generally means according to {@link java.lang.Object#equals(Object)}.
 * but in some cases equivalence may be relaxed to account for differences in
 * order.
 *
 * <h3><a id="ConcurrentReduction">Reduction, concurrency, and ordering</a></h3>
 *
 * With some complex reduction operations, for example a {@code collect()} that
 * produces a {@code Map}, such as:
 * <pre>{@code
 *     Map<Buyer, List<Transaction>> salesByBuyer
 *         = txns.parallelStream()
 *               .collect(Collectors.groupingBy(Transaction::getBuyer));
 * }</pre>
 * it may actually be counterproductive to perform the operation in parallel.
 * This is because the combining step (merging one {@code Map} into another by
 * key) can be expensive for some {@code Map} implementations.
 *
 * <p>Suppose, however, that the result container used in this reduction
 * was a concurrently modifiable collection -- such as a
 * {@link java.util.concurrent.ConcurrentHashMap}. In that case, the parallel
 * invocations of the accumulator could actually deposit their results
 * concurrently into the same shared result container, eliminating the need for
 * the combiner to merge distinct result containers. This potentially provides
 * a boost to the parallel execution performance. We call this a
 * <em>concurrent</em> reduction.
 *
 * <p>A {@link java.util.stream.Collector} that supports concurrent reduction is
 * marked with the {@link java.util.stream.Collector.Characteristics#CONCURRENT}
 * characteristic.  However, a concurrent collection also has a downside.  If
 * multiple threads are depositing results concurrently into a shared container,
 * the order in which results are deposited is non-deterministic. Consequently,
 * a concurrent reduction is only possible if ordering is not important for the
 * stream being processed. The {@link java.util.stream.Stream#collect(Collector)}
 * implementation will only perform a concurrent reduction if
 * <ul>
 * <li>The stream is parallel;</li>
 * <li>The collector has the
 * {@link java.util.stream.Collector.Characteristics#CONCURRENT} characteristic,
 * and;</li>
 * <li>Either the stream is unordered, or the collector has the
 * {@link java.util.stream.Collector.Characteristics#UNORDERED} characteristic.
 * </ul>
 * You can ensure the stream is unordered by using the
 * {@link java.util.stream.BaseStream#unordered()} method.  For example:
 * <pre>{@code
 *     Map<Buyer, List<Transaction>> salesByBuyer
 *         = txns.parallelStream()
 *               .unordered()
 *               .collect(groupingByConcurrent(Transaction::getBuyer));
 * }</pre>
 * (where {@link java.util.stream.Collectors#groupingByConcurrent} is the
 * concurrent equivalent of {@code groupingBy}).
 *
 * <p>Note that if it is important that the elements for a given key appear in
 * the order they appear in the source, then we cannot use a concurrent
 * reduction, as ordering is one of the casualties of concurrent insertion.
 * We would then be constrained to implement either a sequential reduction or
 * a merge-based parallel reduction.
 *
 * <h3><a id="Associativity">Associativity</a></h3>
 *
 * An operator or function {@code op} is <em>associative</em> if the following
 * holds:
 * <pre>{@code
 *     (a op b) op c == a op (b op c)
 * }</pre>
 * The importance of this to parallel evaluation can be seen if we expand this
 * to four terms:
 * <pre>{@code
 *     a op b op c op d == (a op b) op (c op d)
 * }</pre>
 * So we can evaluate {@code (a op b)} in parallel with {@code (c op d)}, and
 * then invoke {@code op} on the results.
 *
 * <p>Examples of associative operations include numeric addition, min, and
 * max, and string concatenation.
 *
 * <h2><a id="StreamSources">Low-level stream construction</a></h2>
 *
 * So far, all the stream examples have used methods like
 * {@link java.util.Collection#stream()} or {@link java.util.Arrays#stream(Object[])}
 * to obtain a stream.  How are those stream-bearing methods implemented?
 *
 * <p>The class {@link java.util.stream.StreamSupport} has a number of
 * low-level methods for creating a stream, all using some form of a
 * {@link java.util.Spliterator}. A spliterator is the parallel analogue of an
 * {@link java.util.Iterator}; it describes a (possibly infinite) collection of
 * elements, with support for sequentially advancing, bulk traversal, and
 * splitting off some portion of the input into another spliterator which can
 * be processed in parallel.  At the lowest level, all streams are driven by a
 * spliterator.
 *
 * <p>There are a number of implementation choices in implementing a
 * spliterator, nearly all of which are tradeoffs between simplicity of
 * implementation and runtime performance of streams using that spliterator.
 * The simplest, but least performant, way to create a spliterator is to
 * create one from an iterator using
 * {@link java.util.Spliterators#spliteratorUnknownSize(java.util.Iterator, int)}.
 * While such a spliterator will work, it will likely offer poor parallel
 * performance, since we have lost sizing information (how big is the
 * underlying data set), as well as being constrained to a simplistic
 * splitting algorithm.
 *
 * <p>A higher-quality spliterator will provide balanced and known-size
 * splits, accurate sizing information, and a number of other
 * {@link java.util.Spliterator#characteristics() characteristics} of the
 * spliterator or data that can be used by implementations to optimize
 * execution.
 *
 * <p>Spliterators for mutable data sources have an additional challenge;
 * timing of binding to the data, since the data could change between the time
 * the spliterator is created and the time the stream pipeline is executed.
 * Ideally, a spliterator for a stream would report a characteristic of

 * {@code IMMUTABLE} or {@code CONCURRENT}; if not it should be
 * <a href="../Spliterator.html#binding"><em>late-binding</em></a>. If a source
 * cannot directly supply a recommended spliterator, it may indirectly supply
 * a spliterator using a {@code Supplier}, and construct a stream via the
 * {@code Supplier}-accepting versions of
 * {@link java.util.stream.StreamSupport#stream(Supplier, int, boolean) stream()}.
 * The spliterator is obtained from the supplier only after the terminal
 * operation of the stream pipeline commences.
 *
 * <p>These requirements significantly reduce the scope of potential
 * interference between mutations of the stream source and execution of stream
 * pipelines. Streams based on spliterators with the desired characteristics,
 * or those using the Supplier-based factory forms, are immune to
 * modifications of the data source prior to commencement of the terminal
 * operation (provided the behavioral parameters to the stream operations meet
 * the required criteria for non-interference and statelessness).  See
 * <a href="package-summary.html#NonInterference">Non-Interference</a>
 * for more details.
 *
 * @since 1.8
 */
package java.util.stream;

import java.util.function.BinaryOperator;
import java.util.function.UnaryOperator;