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The docs for `heapq.merge` are a little misleading. Iterables passed into heapq.merge with the reversed flag enabled must be sorted from largest to smallest to achieve the desired sorting effect, but the previous paragraph states that they should be sorted from smallest to largest.
311 lines
13 KiB
ReStructuredText
311 lines
13 KiB
ReStructuredText
:mod:`heapq` --- Heap queue algorithm
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=====================================
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.. module:: heapq
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:synopsis: Heap queue algorithm (a.k.a. priority queue).
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.. moduleauthor:: Kevin O'Connor
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.. sectionauthor:: Guido van Rossum <guido@python.org>
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.. sectionauthor:: François Pinard
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.. sectionauthor:: Raymond Hettinger
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**Source code:** :source:`Lib/heapq.py`
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--------------
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This module provides an implementation of the heap queue algorithm, also known
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as the priority queue algorithm.
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Heaps are binary trees for which every parent node has a value less than or
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equal to any of its children. This implementation uses arrays for which
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``heap[k] <= heap[2*k+1]`` and ``heap[k] <= heap[2*k+2]`` for all *k*, counting
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elements from zero. For the sake of comparison, non-existing elements are
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considered to be infinite. The interesting property of a heap is that its
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smallest element is always the root, ``heap[0]``.
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The API below differs from textbook heap algorithms in two aspects: (a) We use
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zero-based indexing. This makes the relationship between the index for a node
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and the indexes for its children slightly less obvious, but is more suitable
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since Python uses zero-based indexing. (b) Our pop method returns the smallest
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item, not the largest (called a "min heap" in textbooks; a "max heap" is more
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common in texts because of its suitability for in-place sorting).
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These two make it possible to view the heap as a regular Python list without
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surprises: ``heap[0]`` is the smallest item, and ``heap.sort()`` maintains the
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heap invariant!
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To create a heap, use a list initialized to ``[]``, or you can transform a
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populated list into a heap via function :func:`heapify`.
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The following functions are provided:
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.. function:: heappush(heap, item)
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Push the value *item* onto the *heap*, maintaining the heap invariant.
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.. function:: heappop(heap)
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Pop and return the smallest item from the *heap*, maintaining the heap
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invariant. If the heap is empty, :exc:`IndexError` is raised. To access the
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smallest item without popping it, use ``heap[0]``.
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.. function:: heappushpop(heap, item)
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Push *item* on the heap, then pop and return the smallest item from the
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*heap*. The combined action runs more efficiently than :func:`heappush`
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followed by a separate call to :func:`heappop`.
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.. function:: heapify(x)
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Transform list *x* into a heap, in-place, in linear time.
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.. function:: heapreplace(heap, item)
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Pop and return the smallest item from the *heap*, and also push the new *item*.
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The heap size doesn't change. If the heap is empty, :exc:`IndexError` is raised.
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This one step operation is more efficient than a :func:`heappop` followed by
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:func:`heappush` and can be more appropriate when using a fixed-size heap.
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The pop/push combination always returns an element from the heap and replaces
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it with *item*.
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The value returned may be larger than the *item* added. If that isn't
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desired, consider using :func:`heappushpop` instead. Its push/pop
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combination returns the smaller of the two values, leaving the larger value
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on the heap.
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The module also offers three general purpose functions based on heaps.
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.. function:: merge(*iterables, key=None, reverse=False)
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Merge multiple sorted inputs into a single sorted output (for example, merge
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timestamped entries from multiple log files). Returns an :term:`iterator`
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over the sorted values.
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Similar to ``sorted(itertools.chain(*iterables))`` but returns an iterable, does
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not pull the data into memory all at once, and assumes that each of the input
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streams is already sorted (smallest to largest).
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Has two optional arguments which must be specified as keyword arguments.
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*key* specifies a :term:`key function` of one argument that is used to
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extract a comparison key from each input element. The default value is
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``None`` (compare the elements directly).
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*reverse* is a boolean value. If set to ``True``, then the input elements
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are merged as if each comparison were reversed. To achieve behavior similar
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to ``sorted(itertools.chain(*iterables), reverse=True)``, all iterables must
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be sorted from largest to smallest.
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.. versionchanged:: 3.5
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Added the optional *key* and *reverse* parameters.
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.. function:: nlargest(n, iterable, key=None)
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Return a list with the *n* largest elements from the dataset defined by
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*iterable*. *key*, if provided, specifies a function of one argument that is
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used to extract a comparison key from each element in the iterable:
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``key=str.lower`` Equivalent to: ``sorted(iterable, key=key,
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reverse=True)[:n]``
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.. function:: nsmallest(n, iterable, key=None)
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Return a list with the *n* smallest elements from the dataset defined by
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*iterable*. *key*, if provided, specifies a function of one argument that is
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used to extract a comparison key from each element in the iterable:
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``key=str.lower`` Equivalent to: ``sorted(iterable, key=key)[:n]``
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The latter two functions perform best for smaller values of *n*. For larger
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values, it is more efficient to use the :func:`sorted` function. Also, when
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``n==1``, it is more efficient to use the built-in :func:`min` and :func:`max`
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functions. If repeated usage of these functions is required, consider turning
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the iterable into an actual heap.
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Basic Examples
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--------------
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A `heapsort <https://en.wikipedia.org/wiki/Heapsort>`_ can be implemented by
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pushing all values onto a heap and then popping off the smallest values one at a
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time::
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>>> def heapsort(iterable):
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... h = []
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... for value in iterable:
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... heappush(h, value)
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... return [heappop(h) for i in range(len(h))]
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...
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>>> heapsort([1, 3, 5, 7, 9, 2, 4, 6, 8, 0])
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[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
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This is similar to ``sorted(iterable)``, but unlike :func:`sorted`, this
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implementation is not stable.
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Heap elements can be tuples. This is useful for assigning comparison values
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(such as task priorities) alongside the main record being tracked::
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>>> h = []
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>>> heappush(h, (5, 'write code'))
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>>> heappush(h, (7, 'release product'))
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>>> heappush(h, (1, 'write spec'))
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>>> heappush(h, (3, 'create tests'))
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>>> heappop(h)
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(1, 'write spec')
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Priority Queue Implementation Notes
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-----------------------------------
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A `priority queue <https://en.wikipedia.org/wiki/Priority_queue>`_ is common use
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for a heap, and it presents several implementation challenges:
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* Sort stability: how do you get two tasks with equal priorities to be returned
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in the order they were originally added?
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* Tuple comparison breaks for (priority, task) pairs if the priorities are equal
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and the tasks do not have a default comparison order.
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* If the priority of a task changes, how do you move it to a new position in
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the heap?
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* Or if a pending task needs to be deleted, how do you find it and remove it
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from the queue?
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A solution to the first two challenges is to store entries as 3-element list
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including the priority, an entry count, and the task. The entry count serves as
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a tie-breaker so that two tasks with the same priority are returned in the order
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they were added. And since no two entry counts are the same, the tuple
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comparison will never attempt to directly compare two tasks.
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The remaining challenges revolve around finding a pending task and making
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changes to its priority or removing it entirely. Finding a task can be done
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with a dictionary pointing to an entry in the queue.
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Removing the entry or changing its priority is more difficult because it would
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break the heap structure invariants. So, a possible solution is to mark the
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entry as removed and add a new entry with the revised priority::
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pq = [] # list of entries arranged in a heap
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entry_finder = {} # mapping of tasks to entries
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REMOVED = '<removed-task>' # placeholder for a removed task
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counter = itertools.count() # unique sequence count
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def add_task(task, priority=0):
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'Add a new task or update the priority of an existing task'
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if task in entry_finder:
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remove_task(task)
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count = next(counter)
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entry = [priority, count, task]
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entry_finder[task] = entry
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heappush(pq, entry)
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def remove_task(task):
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'Mark an existing task as REMOVED. Raise KeyError if not found.'
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entry = entry_finder.pop(task)
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entry[-1] = REMOVED
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def pop_task():
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'Remove and return the lowest priority task. Raise KeyError if empty.'
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while pq:
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priority, count, task = heappop(pq)
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if task is not REMOVED:
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del entry_finder[task]
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return task
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raise KeyError('pop from an empty priority queue')
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Theory
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------
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Heaps are arrays for which ``a[k] <= a[2*k+1]`` and ``a[k] <= a[2*k+2]`` for all
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*k*, counting elements from 0. For the sake of comparison, non-existing
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elements are considered to be infinite. The interesting property of a heap is
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that ``a[0]`` is always its smallest element.
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The strange invariant above is meant to be an efficient memory representation
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for a tournament. The numbers below are *k*, not ``a[k]``::
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0
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1 2
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3 4 5 6
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7 8 9 10 11 12 13 14
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15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
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In the tree above, each cell *k* is topping ``2*k+1`` and ``2*k+2``. In a usual
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binary tournament we see in sports, each cell is the winner over the two cells
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it tops, and we can trace the winner down the tree to see all opponents s/he
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had. However, in many computer applications of such tournaments, we do not need
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to trace the history of a winner. To be more memory efficient, when a winner is
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promoted, we try to replace it by something else at a lower level, and the rule
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becomes that a cell and the two cells it tops contain three different items, but
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the top cell "wins" over the two topped cells.
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If this heap invariant is protected at all time, index 0 is clearly the overall
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winner. The simplest algorithmic way to remove it and find the "next" winner is
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to move some loser (let's say cell 30 in the diagram above) into the 0 position,
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and then percolate this new 0 down the tree, exchanging values, until the
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invariant is re-established. This is clearly logarithmic on the total number of
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items in the tree. By iterating over all items, you get an O(n log n) sort.
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A nice feature of this sort is that you can efficiently insert new items while
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the sort is going on, provided that the inserted items are not "better" than the
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last 0'th element you extracted. This is especially useful in simulation
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contexts, where the tree holds all incoming events, and the "win" condition
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means the smallest scheduled time. When an event schedules other events for
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execution, they are scheduled into the future, so they can easily go into the
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heap. So, a heap is a good structure for implementing schedulers (this is what
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I used for my MIDI sequencer :-).
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Various structures for implementing schedulers have been extensively studied,
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and heaps are good for this, as they are reasonably speedy, the speed is almost
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constant, and the worst case is not much different than the average case.
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However, there are other representations which are more efficient overall, yet
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the worst cases might be terrible.
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Heaps are also very useful in big disk sorts. You most probably all know that a
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big sort implies producing "runs" (which are pre-sorted sequences, whose size is
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usually related to the amount of CPU memory), followed by a merging passes for
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these runs, which merging is often very cleverly organised [#]_. It is very
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important that the initial sort produces the longest runs possible. Tournaments
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are a good way to achieve that. If, using all the memory available to hold a
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tournament, you replace and percolate items that happen to fit the current run,
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you'll produce runs which are twice the size of the memory for random input, and
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much better for input fuzzily ordered.
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Moreover, if you output the 0'th item on disk and get an input which may not fit
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in the current tournament (because the value "wins" over the last output value),
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it cannot fit in the heap, so the size of the heap decreases. The freed memory
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could be cleverly reused immediately for progressively building a second heap,
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which grows at exactly the same rate the first heap is melting. When the first
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heap completely vanishes, you switch heaps and start a new run. Clever and
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quite effective!
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In a word, heaps are useful memory structures to know. I use them in a few
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applications, and I think it is good to keep a 'heap' module around. :-)
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.. rubric:: Footnotes
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.. [#] The disk balancing algorithms which are current, nowadays, are more annoying
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than clever, and this is a consequence of the seeking capabilities of the disks.
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On devices which cannot seek, like big tape drives, the story was quite
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different, and one had to be very clever to ensure (far in advance) that each
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tape movement will be the most effective possible (that is, will best
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participate at "progressing" the merge). Some tapes were even able to read
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backwards, and this was also used to avoid the rewinding time. Believe me, real
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good tape sorts were quite spectacular to watch! From all times, sorting has
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always been a Great Art! :-)
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