:mod:`heapq` --- Heap queue algorithm
=====================================

.. module:: heapq
   :synopsis: Heap queue algorithm (a.k.a. priority queue).
.. moduleauthor:: Kevin O'Connor
.. sectionauthor:: Guido van Rossum <guido@python.org>
.. sectionauthor:: François Pinard
.. sectionauthor:: Raymond Hettinger

This module provides an implementation of the heap queue algorithm, also known
as the priority queue algorithm.

Heaps are binary trees for which every parent node has a value less than or
equal to any of its children.  This implementation uses arrays for which
``heap[k] <= heap[2*k+1]`` and ``heap[k] <= heap[2*k+2]`` for all *k*, counting
elements from zero.  For the sake of comparison, non-existing elements are
considered to be infinite.  The interesting property of a heap is that its
smallest element is always the root, ``heap[0]``.

The API below differs from textbook heap algorithms in two aspects: (a) We use
zero-based indexing.  This makes the relationship between the index for a node
and the indexes for its children slightly less obvious, but is more suitable
since Python uses zero-based indexing. (b) Our pop method returns the smallest
item, not the largest (called a "min heap" in textbooks; a "max heap" is more
common in texts because of its suitability for in-place sorting).

These two make it possible to view the heap as a regular Python list without
surprises: ``heap[0]`` is the smallest item, and ``heap.sort()`` maintains the
heap invariant!

To create a heap, use a list initialized to ``[]``, or you can transform a
populated list into a heap via function :func:`heapify`.

The following functions are provided:


.. function:: heappush(heap, item)

   Push the value *item* onto the *heap*, maintaining the heap invariant.


.. function:: heappop(heap)

   Pop and return the smallest item from the *heap*, maintaining the heap
   invariant.  If the heap is empty, :exc:`IndexError` is raised.


.. function:: heappushpop(heap, item)

   Push *item* on the heap, then pop and return the smallest item from the
   *heap*.  The combined action runs more efficiently than :func:`heappush`
   followed by a separate call to :func:`heappop`.


.. function:: heapify(x)

   Transform list *x* into a heap, in-place, in linear time.


.. function:: heapreplace(heap, item)

   Pop and return the smallest item from the *heap*, and also push the new *item*.
   The heap size doesn't change. If the heap is empty, :exc:`IndexError` is raised.

   This one step operation is more efficient than a :func:`heappop` followed by
   :func:`heappush` and can be more appropriate when using a fixed-size heap.
   The pop/push combination always returns an element from the heap and replaces
   it with *item*.

   The value returned may be larger than the *item* added.  If that isn't
   desired, consider using :func:`heappushpop` instead.  Its push/pop
   combination returns the smaller of the two values, leaving the larger value
   on the heap.


The module also offers three general purpose functions based on heaps.


.. function:: merge(*iterables)

   Merge multiple sorted inputs into a single sorted output (for example, merge
   timestamped entries from multiple log files).  Returns an :term:`iterator`
   over the sorted values.

   Similar to ``sorted(itertools.chain(*iterables))`` but returns an iterable, does
   not pull the data into memory all at once, and assumes that each of the input
   streams is already sorted (smallest to largest).


.. function:: nlargest(n, iterable, key=None)

   Return a list with the *n* largest elements from the dataset defined by
   *iterable*.  *key*, if provided, specifies a function of one argument that is
   used to extract a comparison key from each element in the iterable:
   ``key=str.lower`` Equivalent to:  ``sorted(iterable, key=key,
   reverse=True)[:n]``


.. function:: nsmallest(n, iterable, key=None)

   Return a list with the *n* smallest elements from the dataset defined by
   *iterable*.  *key*, if provided, specifies a function of one argument that is
   used to extract a comparison key from each element in the iterable:
   ``key=str.lower`` Equivalent to:  ``sorted(iterable, key=key)[:n]``


The latter two functions perform best for smaller values of *n*.  For larger
values, it is more efficient to use the :func:`sorted` function.  Also, when
``n==1``, it is more efficient to use the built-in :func:`min` and :func:`max`
functions.


Basic Examples
--------------

A `heapsort <http://en.wikipedia.org/wiki/Heapsort>`_ can be implemented by
pushing all values onto a heap and then popping off the smallest values one at a
time::

   >>> def heapsort(iterable):
   ...     'Equivalent to sorted(iterable)'
   ...     h = []
   ...     for value in iterable:
   ...         heappush(h, value)
   ...     return [heappop(h) for i in range(len(h))]
   ...
   >>> heapsort([1, 3, 5, 7, 9, 2, 4, 6, 8, 0])
   [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

Heap elements can be tuples.  This is useful for assigning comparison values
(such as task priorities) alongside the main record being tracked::

    >>> h = []
    >>> heappush(h, (5, 'write code'))
    >>> heappush(h, (7, 'release product'))
    >>> heappush(h, (1, 'write spec'))
    >>> heappush(h, (3, 'create tests'))
    >>> heappop(h)
    (1, 'write spec')


Priority Queue Implementation Notes
-----------------------------------

A `priority queue <http://en.wikipedia.org/wiki/Priority_queue>`_ is common use
for a heap, and it presents several implementation challenges:

* Sort stability:  how do you get two tasks with equal priorities to be returned
  in the order they were originally added?

* Tuple comparison breaks for (priority, task) pairs if the priorities are equal
  and the tasks do not have a default comparison order.

* If the priority of a task changes, how do you move it to a new position in
  the heap?

* Or if a pending task needs to be deleted, how do you find it and remove it
  from the queue?

A solution to the first two challenges is to store entries as 3-element list
including the priority, an entry count, and the task.  The entry count serves as
a tie-breaker so that two tasks with the same priority are returned in the order
they were added. And since no two entry counts are the same, the tuple
comparison will never attempt to directly compare two tasks.

The remaining challenges revolve around finding a pending task and making
changes to its priority or removing it entirely.  Finding a task can be done
with a dictionary pointing to an entry in the queue.

Removing the entry or changing its priority is more difficult because it would
break the heap structure invariants.  So, a possible solution is to mark an
entry as invalid and optionally add a new entry with the revised priority::

    pq = []                         # the priority queue list
    counter = itertools.count(1)    # unique sequence count
    task_finder = {}                # mapping of tasks to entries
    INVALID = 0                     # mark an entry as deleted

    def add_task(priority, task, count=None):
        if count is None:
            count = next(counter)
        entry = [priority, count, task]
        task_finder[task] = entry
        heappush(pq, entry)

    def get_top_priority():
        while True:
            priority, count, task = heappop(pq)
            del task_finder[task]
            if count is not INVALID:
                return task

    def delete_task(task):
        entry = task_finder[task]
        entry[1] = INVALID

    def reprioritize(priority, task):
        entry = task_finder[task]
        add_task(priority, task, entry[1])
        entry[1] = INVALID


Theory
------

Heaps are arrays for which ``a[k] <= a[2*k+1]`` and ``a[k] <= a[2*k+2]`` for all
*k*, counting elements from 0.  For the sake of comparison, non-existing
elements are considered to be infinite.  The interesting property of a heap is
that ``a[0]`` is always its smallest element.

The strange invariant above is meant to be an efficient memory representation
for a tournament.  The numbers below are *k*, not ``a[k]``::

                                  0

                 1                                 2

         3               4                5               6

     7       8       9       10      11      12      13      14

   15 16   17 18   19 20   21 22   23 24   25 26   27 28   29 30

In the tree above, each cell *k* is topping ``2*k+1`` and ``2*k+2``. In an usual
binary tournament we see in sports, each cell is the winner over the two cells
it tops, and we can trace the winner down the tree to see all opponents s/he
had.  However, in many computer applications of such tournaments, we do not need
to trace the history of a winner. To be more memory efficient, when a winner is
promoted, we try to replace it by something else at a lower level, and the rule
becomes that a cell and the two cells it tops contain three different items, but
the top cell "wins" over the two topped cells.

If this heap invariant is protected at all time, index 0 is clearly the overall
winner.  The simplest algorithmic way to remove it and find the "next" winner is
to move some loser (let's say cell 30 in the diagram above) into the 0 position,
and then percolate this new 0 down the tree, exchanging values, until the
invariant is re-established. This is clearly logarithmic on the total number of
items in the tree. By iterating over all items, you get an O(n log n) sort.

A nice feature of this sort is that you can efficiently insert new items while
the sort is going on, provided that the inserted items are not "better" than the
last 0'th element you extracted.  This is especially useful in simulation
contexts, where the tree holds all incoming events, and the "win" condition
means the smallest scheduled time.  When an event schedule other events for
execution, they are scheduled into the future, so they can easily go into the
heap.  So, a heap is a good structure for implementing schedulers (this is what
I used for my MIDI sequencer :-).

Various structures for implementing schedulers have been extensively studied,
and heaps are good for this, as they are reasonably speedy, the speed is almost
constant, and the worst case is not much different than the average case.
However, there are other representations which are more efficient overall, yet
the worst cases might be terrible.

Heaps are also very useful in big disk sorts.  You most probably all know that a
big sort implies producing "runs" (which are pre-sorted sequences, which size is
usually related to the amount of CPU memory), followed by a merging passes for
these runs, which merging is often very cleverly organised [#]_. It is very
important that the initial sort produces the longest runs possible.  Tournaments
are a good way to that.  If, using all the memory available to hold a
tournament, you replace and percolate items that happen to fit the current run,
you'll produce runs which are twice the size of the memory for random input, and
much better for input fuzzily ordered.

Moreover, if you output the 0'th item on disk and get an input which may not fit
in the current tournament (because the value "wins" over the last output value),
it cannot fit in the heap, so the size of the heap decreases.  The freed memory
could be cleverly reused immediately for progressively building a second heap,
which grows at exactly the same rate the first heap is melting.  When the first
heap completely vanishes, you switch heaps and start a new run.  Clever and
quite effective!

In a word, heaps are useful memory structures to know.  I use them in a few
applications, and I think it is good to keep a 'heap' module around. :-)

.. rubric:: Footnotes

.. [#] The disk balancing algorithms which are current, nowadays, are more annoying
   than clever, and this is a consequence of the seeking capabilities of the disks.
   On devices which cannot seek, like big tape drives, the story was quite
   different, and one had to be very clever to ensure (far in advance) that each
   tape movement will be the most effective possible (that is, will best
   participate at "progressing" the merge).  Some tapes were even able to read
   backwards, and this was also used to avoid the rewinding time. Believe me, real
   good tape sorts were quite spectacular to watch! From all times, sorting has
   always been a Great Art! :-)