293 lines
12 KiB
ReStructuredText
293 lines
12 KiB
ReStructuredText
: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
|
|
|
|
**Source code:** :source:`Lib/heapq.py`
|
|
|
|
--------------
|
|
|
|
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! :-)
|
|
|