PEP: 326
Title: A Case for Top and Bottom Values
Version: $Revision$
Last-Modified: $Date$
Author: Josiah Carlson <jcarlson@uci.edu>,
        Terry Reedy <tjreedy@udel.edu>
Status: Rejected
Type: Standards Track
Content-Type: text/x-rst
Created: 20-Dec-2003
Python-Version: 2.4
Post-History: 20-Dec-2003, 03-Jan-2004, 05-Jan-2004, 07-Jan-2004,
              21-Feb-2004

Results
=======

This PEP has been rejected by the BDFL [12]_.  As per the
pseudo-sunset clause [13]_, PEP 326 is being updated one last time
with the latest suggestions, code modifications, etc., and includes a
link to a module [14]_ that implements the behavior described in the
PEP.  Users who desire the behavior listed in this PEP are encouraged
to use the module for the reasons listed in
`Independent Implementations?`_.


Abstract
========

This PEP proposes two singleton constants that represent a top and
bottom [3]_ value: ``Max`` and ``Min`` (or two similarly suggestive
names [4]_; see `Open Issues`_).

As suggested by their names, ``Max`` and ``Min`` would compare higher
or lower than any other object (respectively).  Such behavior results
in easier to understand code and fewer special cases in which a
temporary minimum or maximum value is required, and an actual minimum
or maximum numeric value is not limited.


Rationale
=========

While ``None`` can be used as an absolute minimum that any value can
attain [1]_, this may be deprecated [4]_ in Python 3.0 and shouldn't
be relied upon.

As a replacement for ``None`` being used as an absolute minimum, as
well as the introduction of an absolute maximum, the introduction of
two singleton constants ``Max`` and ``Min`` address concerns for the
constants to be self-documenting.

What is commonly done to deal with absolute minimum or maximum values,
is to set a value that is larger than the script author ever expects
the input to reach, and hope that it isn't reached.

Guido has brought up [2]_ the fact that there exists two constants
that can be used in the interim for maximum values: sys.maxint and
floating point positive infinity (1e309 will evaluate to positive
infinity).  However, each has their drawbacks.

- On most architectures sys.maxint is arbitrarily small (2**31-1 or
  2**63-1) and can be easily eclipsed by large 'long' integers or
  floating point numbers.

- Comparing long integers larger than the largest floating point
  number representable against any float will result in an exception
  being raised::

        >>> cmp(1.0, 10**309)
        Traceback (most recent call last):
        File "<stdin>", line 1, in ?
        OverflowError: long int too large to convert to float

  Even when large integers are compared against positive infinity::

        >>> cmp(1e309, 10**309)
        Traceback (most recent call last):
        File "<stdin>", line 1, in ?
        OverflowError: long int too large to convert to float

- These same drawbacks exist when numbers are negative.

Introducing ``Max`` and ``Min`` that work as described above does not
take much effort.  A sample Python `reference implementation`_ of both
is included.


Motivation
==========

There are hundreds of algorithms that begin by initializing some set
of values to a logical (or numeric) infinity or negative infinity.
Python lacks either infinity that works consistently or really is the
most extreme value that can be attained.  By adding ``Max`` and
``Min``, Python would have a real maximum and minimum value, and such
algorithms can become clearer due to the reduction of special cases.

``Max`` Examples
---------------------

When testing various kinds of servers, it is sometimes necessary to
only serve a certain number of clients before exiting, which results
in code like the following::

    count = 5

    def counts(stop):
        i = 0
        while i < stop:
            yield i
            i += 1

    for client_number in counts(count):
        handle_one_client()

When using ``Max`` as the value assigned to count, our testing server
becomes a production server with minimal effort.

As another example, in Dijkstra's shortest path algorithm on a graph
with weighted edges (all positive).

1. Set distances to every node in the graph to infinity.
2. Set the distance to the start node to zero.
3. Set visited to be an empty mapping.
4. While shortest distance of a node that has not been visited is less
   than infinity and the destination has not been visited.

   a. Get the node with the shortest distance.
   b. Visit the node.
   c. Update neighbor distances and parent pointers if necessary for
      neighbors that have not been visited.

5. If the destination has been visited, step back through parent
   pointers to find the reverse of the path to be taken.

.. _DijkstraSP_table:

Below is an example of Dijkstra's shortest path algorithm on a graph
with weighted edges using a table (a faster version that uses a heap
is available, but this version is offered due to its similarity to the
description above, the heap version is available via older versions of
this document). ::

    def DijkstraSP_table(graph, S, T):
        table = {}                                                 #3
        for node in graph.iterkeys():
            #(visited, distance, node, parent)
            table[node] = (0, Max, node, None)                     #1
        table[S] = (0, 0, S, None)                                 #2
        cur = min(table.values())                                  #4a
        while (not cur[0]) and cur[1] < Max:                       #4
            (visited, distance, node, parent) = cur
            table[node] = (1, distance, node, parent)              #4b
            for cdist, child in graph[node]:                       #4c
                ndist = distance+cdist                             #|
                if not table[child][0] and ndist < table[child][1]:#|
                    table[child] = (0, ndist, child, node)         #|_
            cur = min(table.values())                              #4a
        if not table[T][0]:
            return None
        cur = T                                                    #5
        path = [T]                                                 #|
        while table[cur][3] is not None:                           #|
            path.append(table[cur][3])                             #|
            cur = path[-1]                                         #|
        path.reverse()                                             #|
        return path                                                #|_

Readers should note that replacing ``Max`` in the above code with an
arbitrarily large number does not guarantee that the shortest path
distance to a node will never exceed that number.  Well, with one
caveat: one could certainly sum up the weights of every edge in the
graph, and set the 'arbitrarily large number' to that total.  However,
doing so does not make the algorithm any easier to understand and has
potential problems with numeric overflows.

.. _DijkstraSP_table_node:

Gustavo Niemeyer [9]_ points out that using a more Pythonic data
structure than tuples, to store information about node distances,
increases readability.  Two equivalent node structures (one using
``None``, the other using ``Max``) and their use in a suitably
modified Dijkstra's shortest path algorithm is given below. ::

    class SuperNode:
        def __init__(self, node, parent, distance, visited):
            self.node = node
            self.parent = parent
            self.distance = distance
            self.visited = visited

    class MaxNode(SuperNode):
        def __init__(self, node, parent=None, distance=Max,
                     visited=False):
            SuperNode.__init__(self, node, parent, distance, visited)
        def __cmp__(self, other):
            return cmp((self.visited, self.distance),
                       (other.visited, other.distance))

    class NoneNode(SuperNode):
        def __init__(self, node, parent=None, distance=None,
                     visited=False):
            SuperNode.__init__(self, node, parent, distance, visited)
        def __cmp__(self, other):
            pair = ((self.visited, self.distance),
                    (other.visited, other.distance))
            if None in (self.distance, other.distance):
                return -cmp(*pair)
            return cmp(*pair)

    def DijkstraSP_table_node(graph, S, T, Node):
        table = {}                                                 #3
        for node in graph.iterkeys():
            table[node] = Node(node)                               #1
        table[S] = Node(S, distance=0)                             #2
        cur = min(table.values())                                  #4a
        sentinel = Node(None).distance
        while not cur.visited and cur.distance != sentinel:        #4
            cur.visited = True                                     #4b
            for cdist, child in graph[node]:                       #4c
                ndist = distance+cdist                             #|
                if not table[child].visited and\                   #|
                   ndist < table[child].distance:                  #|
                    table[child].distance = ndist                  #|_
            cur = min(table.values())                              #4a
        if not table[T].visited:
            return None
        cur = T                                                    #5
        path = [T]                                                 #|
        while table[cur].parent is not None:                       #|
            path.append(table[cur].parent)                         #|
            cur = path[-1]                                         #|
        path.reverse()                                             #|
        return path                                                #|_

In the above, passing in either NoneNode or MaxNode would be
sufficient to use either ``None`` or ``Max`` for the node distance
'infinity'.  Note the additional special case required for ``None``
being used as a sentinel in NoneNode in the __cmp__ method.

This example highlights the special case handling where ``None`` is
used as a sentinel value for maximum values "in the wild", even though
None itself compares smaller than any other object in the standard
distribution.

As an aside, it is not clear to the author that using Nodes as a
replacement for tuples has increased readability significantly, if at
all.


A ``Min`` Example
-----------------

An example of usage for ``Min`` is an algorithm that solves the
following problem [6]_:

    Suppose you are given a directed graph, representing a
    communication network.  The vertices are the nodes in the network,
    and each edge is a communication channel. Each edge ``(u, v)`` has
    an associated value ``r(u, v)``, with ``0 <= r(u, v) <= 1``, which
    represents the reliability of the channel from ``u`` to ``v``
    (i.e., the probability that the channel from ``u`` to ``v`` will
    **not** fail).  Assume that the reliability probabilities of the
    channels are independent.  (This implies that the reliability of
    any path is the product of the reliability of the edges along the
    path.)  Now suppose you are given two nodes in the graph, ``A``
    and ``B``.

Such an algorithm is a 7 line modification to the `DijkstraSP_table`_
algorithm given above (modified lines prefixed with `*`)::

    def DijkstraSP_table(graph, S, T):
        table = {}                                                 #3
        for node in graph.iterkeys():
            #(visited, distance, node, parent)
    *       table[node] = (0, Min, node, None)                     #1
    *   table[S] = (0, 1, S, None)                                 #2
    *   cur = max(table.values())                                  #4a
    *   while (not cur[0]) and cur[1] > Min:                       #4
            (visited, distance, node, parent) = cur
            table[node] = (1, distance, node, parent)              #4b
            for cdist, child in graph[node]:                       #4c
    *           ndist = distance*cdist                             #|
    *           if not table[child][0] and ndist > table[child][1]:#|
                    table[child] = (0, ndist, child, node)         #|_
    *       cur = max(table.values())                              #4a
        if not table[T][0]:
            return None
        cur = T                                                    #5
        path = [T]                                                 #|
        while table[cur][3] is not None:                           #|
            path.append(table[cur][3])                             #|
            cur = path[-1]                                         #|
        path.reverse()                                             #|
        return path                                                #|_

Note that there is a way of translating the graph to so that it can be
passed unchanged into the original `DijkstraSP_table`_ algorithm.
There also exists a handful of easy methods for constructing Node
objects that would work with `DijkstraSP_table_node`_.  Such
translations are left as an exercise to the reader.


Other Examples
--------------

Andrew P. Lentvorski, Jr. [7]_ has pointed out that various data
structures involving range searching have immediate use for ``Max``
and ``Min`` values.  More specifically; Segment trees, Range trees,
k-d trees and database keys:

    ...The issue is that a range can be open on one side and does not
    always have an initialized case.

    The solutions I have seen are to either overload None as the
    extremum or use an arbitrary large magnitude number.  Overloading
    None means that the built-ins can't really be used without special
    case checks to work around the undefined (or "wrongly defined")
    ordering of None.  These checks tend to swamp the nice performance
    of built-ins like max() and min().

    Choosing a large magnitude number throws away the ability of
    Python to cope with arbitrarily large integers and introduces a
    potential source of overrun/underrun bugs.

Further use examples of both ``Max`` and ``Min`` are available in the
realm of graph algorithms, range searching algorithms, computational
geometry algorithms, and others.


Independent Implementations?
----------------------------

Independent implementations of the ``Min``/``Max`` concept by users
desiring such functionality are not likely to be compatible, and
certainly will produce inconsistent orderings.  The following examples
seek to show how inconsistent they can be.

- Let us pretend we have created proper separate implementations of
  MyMax, MyMin, YourMax and YourMin with the same code as given in
  the sample implementation (with some minor renaming)::

    >>> lst = [YourMin, MyMin, MyMin, YourMin, MyMax, YourMin, MyMax,
    YourMax, MyMax]
    >>> lst.sort()
    >>> lst
    [YourMin, YourMin, MyMin, MyMin, YourMin, MyMax, MyMax, YourMax,
    MyMax]

  Notice that while all the "Min"s are before the "Max"s, there is no
  guarantee that all instances of YourMin will come before MyMin, the
  reverse, or the equivalent MyMax and YourMax.

- The problem is also evident when using the heapq module::

    >>> lst = [YourMin, MyMin, MyMin, YourMin, MyMax, YourMin, MyMax,
    YourMax, MyMax]
    >>> heapq.heapify(lst)  #not needed, but it can't hurt
    >>> while lst: print heapq.heappop(lst),
    ...
    YourMin MyMin YourMin YourMin MyMin MyMax MyMax YourMax MyMax

- Furthermore, the findmin_Max code and both versions of Dijkstra
  could result in incorrect output by passing in secondary versions of
  ``Max``.

It has been pointed out [9]_ that the reference implementation given
below would be incompatible with independent implementations of
``Max``/``Min``.  The point of this PEP is for the introduction of
"The One True Implementation" of "The One True Maximum" and "The One
True Minimum".  User-based implementations of ``Max`` and ``Min``
objects would thusly be discouraged, and use of "The One True
Implementation" would obviously be encouraged.  Ambiguous behavior
resulting from mixing users' implementations of ``Max`` and ``Min``
with "The One True Implementation" should be easy to discover through
variable and/or source code introspection.


Reference Implementation
========================

::

    class _ExtremeType(object):

        def __init__(self, cmpr, rep):
            object.__init__(self)
            self._cmpr = cmpr
            self._rep = rep

        def __cmp__(self, other):
            if isinstance(other, self.__class__) and\
               other._cmpr == self._cmpr:
                return 0
            return self._cmpr

        def __repr__(self):
            return self._rep

    Max = _ExtremeType(1, "Max")
    Min = _ExtremeType(-1, "Min")

Results of Test Run::

    >>> max(Max, 2**65536)
    Max
    >>> min(Max, 2**65536)
    20035299304068464649790...
    (lines removed for brevity)
    ...72339445587895905719156736L
    >>> min(Min, -2**65536)
    Min
    >>> max(Min, -2**65536)
    -2003529930406846464979...
    (lines removed for brevity)
    ...072339445587895905719156736L


Open Issues
===========

As the PEP was rejected, all open issues are now closed and
inconsequential.  The module will use the names ``UniversalMaximum``
and ``UniversalMinimum`` due to the fact that it would be very
difficult to mistake what each does.  For those who require a shorter
name, renaming the singletons during import is suggested::

    from extremes import UniversalMaximum as uMax,
                         UniversalMinimum as uMin


References
==========

.. [1] RE: [Python-Dev] Re: Got None. Maybe Some?, Peters, Tim
   (https://mail.python.org/pipermail/python-dev/2003-December/041374.html)

.. [2] Re: [Python-Dev] Got None. Maybe Some?, van Rossum, Guido
   (https://mail.python.org/pipermail/python-dev/2003-December/041352.html)

.. [3] RE: [Python-Dev] Got None. Maybe Some?, Peters, Tim
   (https://mail.python.org/pipermail/python-dev/2003-December/041332.html)

.. [4] [Python-Dev] Re: PEP 326 now online, Reedy, Terry
   (https://mail.python.org/pipermail/python-dev/2004-January/041685.html)

.. [5] [Python-Dev] PEP 326 now online, Chermside, Michael
   (https://mail.python.org/pipermail/python-dev/2004-January/041704.html)

.. [6] Homework 6, Problem 7, Dillencourt, Michael
   (link may not be valid in the future)
   (http://www.ics.uci.edu/~dillenco/ics161/hw/hw6.pdf)

.. [7] RE: [Python-Dev] PEP 326 now online, Lentvorski, Andrew P., Jr.
   (https://mail.python.org/pipermail/python-dev/2004-January/041727.html)

.. [8] Re: It's not really Some is it?, Ippolito, Bob
   (http://www.livejournal.com/users/chouyu_31/138195.html?thread=274643#t274643)

.. [9] [Python-Dev] Re: PEP 326 now online, Niemeyer, Gustavo
   (https://mail.python.org/pipermail/python-dev/2004-January/042261.html);
   [Python-Dev] Re: PEP 326 now online, Carlson, Josiah
   (https://mail.python.org/pipermail/python-dev/2004-January/042272.html)

.. [11] [Python-Dev] PEP 326 (quick location possibility), Carlson, Josiah
   (https://mail.python.org/pipermail/python-dev/2004-January/042275.html)

.. [12] [Python-Dev] PEP 326 (quick location possibility), van Rossum, Guido
   (https://mail.python.org/pipermail/python-dev/2004-January/042306.html)

.. [13] [Python-Dev] PEP 326 (quick location possibility), Carlson, Josiah
   (https://mail.python.org/pipermail/python-dev/2004-January/042300.html)

.. [14] Recommended standard implementation of PEP 326, extremes.py,
   Carlson, Josiah
   (http://www.ics.uci.edu/~jcarlson/pep326/extremes.py)


Changes
=======

- Added this section.

- Added Motivation_ section.

- Changed markup to reStructuredText.

- Clarified Abstract_, Motivation_, `Reference Implementation`_ and
  `Open Issues`_ based on the simultaneous concepts of ``Max`` and
  ``Min``.

- Added two implementations of Dijkstra's Shortest Path algorithm that
  show where ``Max`` can be used to remove special cases.

- Added an example of use for ``Min`` to Motivation_.

- Added an example and `Other Examples`_ subheading.

- Modified `Reference Implementation`_ to instantiate both items from
  a single class/type.

- Removed a large number of open issues that are not within the scope
  of this PEP.

- Replaced an example from `Max Examples`_, changed an example in
  `A Min Example`_.

- Added some `References`_.

- BDFL rejects [12]_ PEP 326


Copyright
=========

This document has been placed in the public domain.


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