* The pairing-heap.plt PlaneT Package -*- outline -*-
* The _pairing-heap.plt_ PlaneT Package -*- outline -*-
The pairing-heap.plt PlaneT package provides a unit which, when
invoked with a suitable _SRFI-67_ compare function as an import,
exports functions to create and manipulate pairing heaps.
The pairing-heap.plt package is maintained by Will M. Farr
<firstname.lastname@example.org>. It is released under the GPL; see the License
section of this document for more details.
A _pairing heap_ is a _self-adjusting datastructure_ designed to
efficiently support the following operations on a collection of
- min: Get the smallest object in the collection (every other object
is greater than or equal to this object).
- insert: Insert a new object into the collection.
- merge: Combine two collections of objects.
- remove-min: Remove the smallest element from a collection.
The first three of these operations take O(1) time. The last takes
O(n) time in the worst case, but is probably (almost certainly,
really, but there is no mathematical proof of which I am aware)
O(log(n)) _amortized_ time, where n is the number of elements in the
Because the pairing-heap datastructure is "self-adjusting", the
remove-min operation becomes more efficient the longer a specific
access pattern goes on.
You can read more about pairing heaps at Wikipedia:
The pairing heaps created and manipulated by the pairing-heap.plt
package are _persistant_, meaning that no operation modifies a
pre-existing datastructure. Operations like _insert_ and _remove-min_
return a *new* heap which contains all the elements of the original
heap plus/minus one.
The pairing-heap.plt package exports a _unit_ (see "unit.ss" from
MzLib for a complete description) which imports a compare function
compatible with _SRFI-67_ (a function taking two objects and returning
-1, 0, or 1) and exports functions to manipulate pairing-heaps of
objects. The following hypothetical REPL interaction demonstrates
sorting five random integers using the pairing-heap.plt package:
(require (planet "pairing-heap.ss" ("wmfarr" "pairing-heap.plt" 1)))
(import (rename compare^ (integer-compare compare)))
(- (random 100) 50))
(let ((h (heap (random-int) (random-int) (random-int) (random-int) (random-int))))
(let loop ((ints '())
(if (empty? remaining-h)
(loop (cons (min remaining-h) ints) (remove-min remaining-h)))))
=> (-49 -39 -35 -19 -14)
** Package Exports
> compare^ : The input signature of the pairing-heap@ unit. Defines a
single name, _compare_, which is intended to be a comparison
function compatible with _SRFI-67_.
> pairing-heap^ : The output signature of the pairing-heap@ unit.
Defines _min_, _insert_, _remove-min_, _merge_, _empty?_, _heap_,
_heap?_, _fold_, _elements_.
> pairing-heap@ : A unit with the input signature compare^ and output
signature pairing-heap^, which provides the following functions:
> (heap? obj) : #f unless obj is a heap.
> (empty? h) : #f unless h is an empty heap. It is an error to apply
empty? to an object which is not a heap.
> (heap obj ...) : Produces a heap containing the given objects.
Takes time O(n), where n is the number of objects given.
> (min h) : Returns the minimum object (according to the linked
compare function given to the pairing-heap@ unit) in the heap h.
All other objects in h are greater than or equal to (min h). Takes
> (insert obj objs ... h) : Returns a heap which contains all the
objects of h in addition to the given objects. Takes time O(n),
where n is the number of objects given to the insert command.
> (remove-min h) : Returns a heap which contains all the elements of h
except for the minimum element. Takes worst-case time O(n), and
(probably---hasn't been proved yet, but plenty of experimental
evidence) O(log(n)) amortized time, where n is the number of
elements contained in h.
> (merge h1 hs ...) : Returns a heap which contains all the elements
from the given heaps. Takes time O(n), where n is the number
of *heaps* given to the function.
> (fold kons knil h) : Applies the procedure kons to the elements of h
in unspecified order, feeding the output of successive applications
as the second argument to kons. The first application uses knil for
the second argument. For example, the _elements_ function is
(define (elements h)
(fold cons '() h))
> (elements h) : Returns a list of all elements in the heap h, in no
> (sort list-or-vector) : Returns a new sequence of the same type
which contains the elements of list-or-vector in non-decreasing
order. (i.e. the predicate (<=? compare obj1 obj2) returns #t for
every pair of adjascent objects in the sorted sequence.)
This software is released under the LGPL, which appears below.
Because the LGPL refers to the GPL, that license also appears below.
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receives a license from the original licensors, to run, modify and
propagate that work, subject to this License. You are not responsible
for enforcing compliance by third parties with this License.
An "entity transaction" is a transaction transferring control of an
organization, or substantially all assets of one, or subdividing an
organization, or merging organizations. If propagation of a covered
work results from an entity transaction, each party to that
transaction who receives a copy of the work also receives whatever
licenses to the work the party's predecessor in interest had or could
give under the previous paragraph, plus a right to possession of the
Corresponding Source of the work from the predecessor in interest, if
the predecessor has it or can get it with reasonable efforts.
You may not impose any further restrictions on the exercise of the
rights granted or affirmed under this License. For example, you may
not impose a license fee, royalty, or other charge for exercise of
rights granted under this License, and you may not initiate litigation
(including a cross-claim or counterclaim in a lawsuit) alleging that
any patent claim is infringed by making, using, selling, offering for
sale, or importing the Program or any portion of it.
A "contributor" is a copyright holder who authorizes use under this
License of the Program or a work on which the Program is based. The
work thus licensed is called the contributor's "contributor version".
A contributor's "essential patent claims" are all patent claims
owned or controlled by the contributor, whether already acquired or
hereafter acquired, that would be infringed by some manner, permitted
by this License, of making, using, or selling its contributor version,
but do not include claims that would be infringed only as a
consequence of further modification of the contributor version. For
purposes of this definition, "control" includes the right to grant
patent sublicenses in a manner consistent with the requirements of
Each contributor grants you a non-exclusive, worldwide, royalty-free
patent license under the contributor's essential patent claims, to
make, use, sell, offer for sale, import and otherwise run, modify and
propagate the contents of its contributor version.
In the following three paragraphs, a "patent license" is any express
agreement or commitment, however denominated, not to enforce a patent
(such as an express permission to practice a patent or covenant not to
sue for patent infringement). To "grant" such a patent license to a
party means to make such an agreement or commitment not to enforce a
patent against the party.
If you convey a covered work, knowingly relying on a patent license,
and the Corresponding Source of the work is not available for anyone
to copy, free of charge and under the terms of this License, through a
publicly available network server or other readily accessible means,
then you must either (1) cause the Corresponding Source to be so
available, or (2) arrange to deprive yourself of the benefit of the
patent license for this particular work, or (3) arrange, in a manner
consistent with the requirements of this License, to extend the patent
license to downstream recipients. "Knowingly relying" means you have
actual knowledge that, but for the patent license, your conveying the
covered work in a country, or your recipient's use of the covered work
in a country, would infringe one or more identifiable patents in that
country that you have reason to believe are valid.
If, pursuant to or in connection with a single transaction or
arrangement, you convey, or propagate by procuring conveyance of, a
covered work, and grant a patent license to some of the parties
receiving the covered work authorizing them to use, propagate, modify
or convey a specific copy of the covered work, then the patent license
you grant is automatically extended to all recipients of the covered
work and works based on it.
A patent license is "discriminatory" if it does not include within
the scope of its coverage, prohibits the exercise of, or is
conditioned on the non-exercise of one or more of the rights that are
specifically granted under this License. You may not convey a covered
work if you are a party to an arrangement with a third party that is
in the business of distributing software, under which you make payment
to the third party based on the extent of your activity of conveying
the work, and under which the third party grants, to any of the
parties who would receive the covered work from you, a discriminatory
patent license (a) in connection with copies of the covered work
conveyed by you (or copies made from those copies), or (b) primarily
for and in connection with specific products or compilations that
contain the covered work, unless you entered into that arrangement,
or that patent license was granted, prior to 28 March 2007.
Nothing in this License shall be construed as excluding or limiting
any implied license or other defenses to infringement that may
otherwise be available to you under applicable patent law.
12. No Surrender of Others' Freedom.
If conditions are imposed on you (whether by court order, agreement or
otherwise) that contradict the conditions of this License, they do not
excuse you from the conditions of this License. If you cannot convey a
covered work so as to satisfy simultaneously your obligations under this
License and any other pertinent obligations, then as a consequence you may
not convey it at all. For example, if you agree to terms that obligate you
to collect a royalty for further conveying from those to whom you convey
the Program, the only way you could satisfy both those terms and this
License would be to refrain entirely from conveying the Program.
13. Use with the GNU Affero General Public License.
Notwithstanding any other provision of this License, you have
permission to link or combine any covered work with a work licensed
under version 3 of the GNU Affero General Public License into a single
combined work, and to convey the resulting work. The terms of this
License will continue to apply to the part which is the covered work,
but the special requirements of the GNU Affero General Public License,
section 13, concerning interaction through a network will apply to the
combination as such.
14. Revised Versions of this License.
The Free Software Foundation may publish revised and/or new versions of
the GNU General Public License from time to time. Such new versions will
be similar in spirit to the present version, but may differ in detail to
address new problems or concerns.
Each version is given a distinguishing version number. If the
Program specifies that a certain numbered version of the GNU General
Public License "or any later version" applies to it, you have the
option of following the terms and conditions either of that numbered
version or of any later version published by the Free Software
Foundation. If the Program does not specify a version number of the
GNU General Public License, you may choose any version ever published
by the Free Software Foundation.
If the Program specifies that a proxy can decide which future
versions of the GNU General Public License can be used, that proxy's
public statement of acceptance of a version permanently authorizes you
to choose that version for the Program.
Later license versions may give you additional or different
permissions. However, no additional obligations are imposed on any
author or copyright holder as a result of your choosing to follow a
15. Disclaimer of Warranty.
THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY
APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT
HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY
OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO,
THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM
IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF
ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
16. Limitation of Liability.
IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING
WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS
THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY
GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE
USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF
DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD
PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),
EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF
17. Interpretation of Sections 15 and 16.
If the disclaimer of warranty and limitation of liability provided
above cannot be given local legal effect according to their terms,
reviewing courts shall apply local law that most closely approximates
an absolute waiver of all civil liability in connection with the
Program, unless a warranty or assumption of liability accompanies a
copy of the Program in return for a fee.
END OF TERMS AND CONDITIONS
How to Apply These Terms to Your New Programs
If you develop a new program, and you want it to be of the greatest
possible use to the public, the best way to achieve this is to make it
free software which everyone can redistribute and change under these terms.
To do so, attach the following notices to the program. It is safest
to attach them to the start of each source file to most effectively
state the exclusion of warranty; and each file should have at least
the "copyright" line and a pointer to where the full notice is found.
<one line to give the program's name and a brief idea of what it does.>
Copyright (C) <year> <name of author>
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
Also add information on how to contact you by electronic and paper mail.
If the program does terminal interaction, make it output a short
notice like this when it starts in an interactive mode:
<program> Copyright (C) <year> <name of author>
This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
This is free software, and you are welcome to redistribute it
under certain conditions; type `show c' for details.
The hypothetical commands `show w' and `show c' should show the appropriate
parts of the General Public License. Of course, your program's commands
might be different; for a GUI interface, you would use an "about box".
You should also get your employer (if you work as a programmer) or school,
if any, to sign a "copyright disclaimer" for the program, if necessary.
For more information on this, and how to apply and follow the GNU GPL, see
The GNU General Public License does not permit incorporating your program
into proprietary programs. If your program is a subroutine library, you
may consider it more useful to permit linking proprietary applications with
the library. If this is what you want to do, use the GNU Lesser General
Public License instead of this License. But first, please read