A comparison function, which imposes a <i>total ordering</i> on some
collection of objects. Comparators can be passed to a sort method (such
as {@link Collections#sort(List,Comparator) Collections.sort} or {@link
Arrays#sort(Object[],Comparator) Arrays.sort}) to allow precise control
over the sort order. Comparators can also be used to control the order of
certain data structures (such as {@link SortedSet sorted sets} or {@link
SortedMap sorted maps}), or to provide an ordering for collections of
objects that don't have a {@link Comparable natural ordering}.<p>
The ordering imposed by a comparator <tt>c</tt> on a set of elements
<tt>S</tt> is said to be <i>consistent with equals</i> if and only if
<tt>c.compare(e1, e2)==0</tt> has the same bool value as
<tt>e1.equals(e2)</tt> for every <tt>e1</tt> and <tt>e2</tt> in
<tt>S</tt>.<p>
Caution should be exercised when using a comparator capable of imposing an
ordering inconsistent with equals to order a sorted set (or sorted map).
Suppose a sorted set (or sorted map) with an explicit comparator <tt>c</tt>
is used with elements (or keys) drawn from a set <tt>S</tt>. If the
ordering imposed by <tt>c</tt> on <tt>S</tt> is inconsistent with equals,
the sorted set (or sorted map) will behave "strangely." In particular the
sorted set (or sorted map) will violate the general contract for set (or
map), which is defined in terms of <tt>equals</tt>.<p>
For example, suppose one adds two elements {@code a} and {@code b} such that
{@code (a.equals(b) && c.compare(a, b) != 0)}
to an empty {@code TreeSet} with comparator {@code c}.
The second {@code add} operation will return
true (and the size of the tree set will increase) because {@code a} and
{@code b} are not equivalent from the tree set's perspective, even though
this is contrary to the specification of the
{@link Set#add Set.add} method.<p>
Note: It is generally a good idea for comparators to also implement
<tt>java.io.Serializable</tt>, as they may be used as ordering methods in
serializable data structures (like {@link TreeSet}, {@link TreeMap}). In
order for the data structure to serialize successfully, the comparator (if
provided) must implement <tt>Serializable</tt>.<p>
For the mathematically inclined, the <i>relation</i> that defines the
<i>imposed ordering</i> that a given comparator <tt>c</tt> imposes on a
given set of objects <tt>S</tt> is:<pre>
{(x, y) such that c.compare(x, y) <= 0}.
</pre> The <i>quotient</i> for this total order is:<pre>
{(x, y) such that c.compare(x, y) == 0}.
</pre>
It follows immediately from the contract for <tt>compare</tt> that the
quotient is an <i>equivalence relation</i> on <tt>S</tt>, and that the
imposed ordering is a <i>total order</i> on <tt>S</tt>. When we say that
the ordering imposed by <tt>c</tt> on <tt>S</tt> is <i>consistent with
equals</i>, we mean that the quotient for the ordering is the equivalence
relation defined by the objects' {@link Object#equals(Object)
equals(Object)} method(s):<pre>
{(x, y) such that x.equals(y)}. </pre>
<p>Unlike {@code Comparable}, a comparator may optionally permit
comparison of null arguments, while maintaining the requirements for
an equivalence relation.
<p>This interface is a member of the
<a href="{@docRoot}/../technotes/guides/collections/index.html">
Java Collections Framework</a>.
@param (T) the type of objects that may be compared by this comparator
Compares its two arguments for order. Returns a negative integer,
zero, or a positive integer as the first argument is less than, equal
to, or greater than the second.<p>
A comparison function, which imposes a <i>total ordering</i> on some collection of objects. Comparators can be passed to a sort method (such as {@link Collections#sort(List,Comparator) Collections.sort} or {@link Arrays#sort(Object[],Comparator) Arrays.sort}) to allow precise control over the sort order. Comparators can also be used to control the order of certain data structures (such as {@link SortedSet sorted sets} or {@link SortedMap sorted maps}), or to provide an ordering for collections of objects that don't have a {@link Comparable natural ordering}.<p>
The ordering imposed by a comparator <tt>c</tt> on a set of elements <tt>S</tt> is said to be <i>consistent with equals</i> if and only if <tt>c.compare(e1, e2)==0</tt> has the same bool value as <tt>e1.equals(e2)</tt> for every <tt>e1</tt> and <tt>e2</tt> in <tt>S</tt>.<p>
Caution should be exercised when using a comparator capable of imposing an ordering inconsistent with equals to order a sorted set (or sorted map). Suppose a sorted set (or sorted map) with an explicit comparator <tt>c</tt> is used with elements (or keys) drawn from a set <tt>S</tt>. If the ordering imposed by <tt>c</tt> on <tt>S</tt> is inconsistent with equals, the sorted set (or sorted map) will behave "strangely." In particular the sorted set (or sorted map) will violate the general contract for set (or map), which is defined in terms of <tt>equals</tt>.<p>
For example, suppose one adds two elements {@code a} and {@code b} such that {@code (a.equals(b) && c.compare(a, b) != 0)} to an empty {@code TreeSet} with comparator {@code c}. The second {@code add} operation will return true (and the size of the tree set will increase) because {@code a} and {@code b} are not equivalent from the tree set's perspective, even though this is contrary to the specification of the {@link Set#add Set.add} method.<p>
Note: It is generally a good idea for comparators to also implement <tt>java.io.Serializable</tt>, as they may be used as ordering methods in serializable data structures (like {@link TreeSet}, {@link TreeMap}). In order for the data structure to serialize successfully, the comparator (if provided) must implement <tt>Serializable</tt>.<p>
For the mathematically inclined, the <i>relation</i> that defines the <i>imposed ordering</i> that a given comparator <tt>c</tt> imposes on a given set of objects <tt>S</tt> is:<pre> {(x, y) such that c.compare(x, y) <= 0}. </pre> The <i>quotient</i> for this total order is:<pre> {(x, y) such that c.compare(x, y) == 0}. </pre>
It follows immediately from the contract for <tt>compare</tt> that the quotient is an <i>equivalence relation</i> on <tt>S</tt>, and that the imposed ordering is a <i>total order</i> on <tt>S</tt>. When we say that the ordering imposed by <tt>c</tt> on <tt>S</tt> is <i>consistent with equals</i>, we mean that the quotient for the ordering is the equivalence relation defined by the objects' {@link Object#equals(Object) equals(Object)} method(s):<pre> {(x, y) such that x.equals(y)}. </pre>
<p>Unlike {@code Comparable}, a comparator may optionally permit comparison of null arguments, while maintaining the requirements for an equivalence relation.
<p>This interface is a member of the <a href="{@docRoot}/../technotes/guides/collections/index.html"> Java Collections Framework</a>.
@param (T) the type of objects that may be compared by this comparator
@author Josh Bloch @author Neal Gafter @see Comparable @see java.io.Serializable