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Georg Brandl116aa622007-08-15 14:28:22 +00001:mod:`heapq` --- Heap queue algorithm
2=====================================
3
4.. module:: heapq
5 :synopsis: Heap queue algorithm (a.k.a. priority queue).
6.. moduleauthor:: Kevin O'Connor
7.. sectionauthor:: Guido van Rossum <guido@python.org>
8.. sectionauthor:: François Pinard
Raymond Hettinger0e833c32010-08-07 23:31:27 +00009.. sectionauthor:: Raymond Hettinger
Georg Brandl116aa622007-08-15 14:28:22 +000010
Raymond Hettinger10480942011-01-10 03:26:08 +000011**Source code:** :source:`Lib/heapq.py`
12
Raymond Hettinger4f707fd2011-01-10 19:54:11 +000013--------------
14
Georg Brandl116aa622007-08-15 14:28:22 +000015This module provides an implementation of the heap queue algorithm, also known
16as the priority queue algorithm.
17
Georg Brandl57410c12010-11-23 08:37:54 +000018Heaps are binary trees for which every parent node has a value less than or
19equal to any of its children. This implementation uses arrays for which
20``heap[k] <= heap[2*k+1]`` and ``heap[k] <= heap[2*k+2]`` for all *k*, counting
21elements from zero. For the sake of comparison, non-existing elements are
22considered to be infinite. The interesting property of a heap is that its
23smallest element is always the root, ``heap[0]``.
Georg Brandl116aa622007-08-15 14:28:22 +000024
25The API below differs from textbook heap algorithms in two aspects: (a) We use
26zero-based indexing. This makes the relationship between the index for a node
27and the indexes for its children slightly less obvious, but is more suitable
28since Python uses zero-based indexing. (b) Our pop method returns the smallest
29item, not the largest (called a "min heap" in textbooks; a "max heap" is more
30common in texts because of its suitability for in-place sorting).
31
32These two make it possible to view the heap as a regular Python list without
33surprises: ``heap[0]`` is the smallest item, and ``heap.sort()`` maintains the
34heap invariant!
35
36To create a heap, use a list initialized to ``[]``, or you can transform a
37populated list into a heap via function :func:`heapify`.
38
39The following functions are provided:
40
41
42.. function:: heappush(heap, item)
43
44 Push the value *item* onto the *heap*, maintaining the heap invariant.
45
46
47.. function:: heappop(heap)
48
49 Pop and return the smallest item from the *heap*, maintaining the heap
50 invariant. If the heap is empty, :exc:`IndexError` is raised.
51
Benjamin Peterson35e8c462008-04-24 02:34:53 +000052
Christian Heimesdd15f6c2008-03-16 00:07:10 +000053.. function:: heappushpop(heap, item)
54
55 Push *item* on the heap, then pop and return the smallest item from the
56 *heap*. The combined action runs more efficiently than :func:`heappush`
57 followed by a separate call to :func:`heappop`.
58
Georg Brandl116aa622007-08-15 14:28:22 +000059
60.. function:: heapify(x)
61
62 Transform list *x* into a heap, in-place, in linear time.
63
64
65.. function:: heapreplace(heap, item)
66
67 Pop and return the smallest item from the *heap*, and also push the new *item*.
68 The heap size doesn't change. If the heap is empty, :exc:`IndexError` is raised.
Georg Brandl116aa622007-08-15 14:28:22 +000069
Raymond Hettinger6f80b4c2010-09-01 21:27:31 +000070 This one step operation is more efficient than a :func:`heappop` followed by
71 :func:`heappush` and can be more appropriate when using a fixed-size heap.
72 The pop/push combination always returns an element from the heap and replaces
73 it with *item*.
Georg Brandl116aa622007-08-15 14:28:22 +000074
Raymond Hettinger6f80b4c2010-09-01 21:27:31 +000075 The value returned may be larger than the *item* added. If that isn't
76 desired, consider using :func:`heappushpop` instead. Its push/pop
77 combination returns the smaller of the two values, leaving the larger value
78 on the heap.
Georg Brandlaf265f42008-12-07 15:06:20 +000079
Georg Brandl48310cd2009-01-03 21:18:54 +000080
Georg Brandl116aa622007-08-15 14:28:22 +000081The module also offers three general purpose functions based on heaps.
82
83
84.. function:: merge(*iterables)
85
86 Merge multiple sorted inputs into a single sorted output (for example, merge
Georg Brandl9afde1c2007-11-01 20:32:30 +000087 timestamped entries from multiple log files). Returns an :term:`iterator`
Benjamin Peterson206e3072008-10-19 14:07:49 +000088 over the sorted values.
Georg Brandl116aa622007-08-15 14:28:22 +000089
90 Similar to ``sorted(itertools.chain(*iterables))`` but returns an iterable, does
91 not pull the data into memory all at once, and assumes that each of the input
92 streams is already sorted (smallest to largest).
93
Georg Brandl116aa622007-08-15 14:28:22 +000094
Georg Brandl036490d2009-05-17 13:00:36 +000095.. function:: nlargest(n, iterable, key=None)
Georg Brandl116aa622007-08-15 14:28:22 +000096
97 Return a list with the *n* largest elements from the dataset defined by
98 *iterable*. *key*, if provided, specifies a function of one argument that is
99 used to extract a comparison key from each element in the iterable:
100 ``key=str.lower`` Equivalent to: ``sorted(iterable, key=key,
101 reverse=True)[:n]``
102
Georg Brandl116aa622007-08-15 14:28:22 +0000103
Georg Brandl036490d2009-05-17 13:00:36 +0000104.. function:: nsmallest(n, iterable, key=None)
Georg Brandl116aa622007-08-15 14:28:22 +0000105
106 Return a list with the *n* smallest elements from the dataset defined by
107 *iterable*. *key*, if provided, specifies a function of one argument that is
108 used to extract a comparison key from each element in the iterable:
109 ``key=str.lower`` Equivalent to: ``sorted(iterable, key=key)[:n]``
110
Georg Brandl116aa622007-08-15 14:28:22 +0000111
112The latter two functions perform best for smaller values of *n*. For larger
113values, it is more efficient to use the :func:`sorted` function. Also, when
Georg Brandl22b34312009-07-26 14:54:51 +0000114``n==1``, it is more efficient to use the built-in :func:`min` and :func:`max`
Georg Brandl116aa622007-08-15 14:28:22 +0000115functions.
116
117
Raymond Hettinger6f80b4c2010-09-01 21:27:31 +0000118Basic Examples
119--------------
120
121A `heapsort <http://en.wikipedia.org/wiki/Heapsort>`_ can be implemented by
122pushing all values onto a heap and then popping off the smallest values one at a
123time::
124
125 >>> def heapsort(iterable):
126 ... 'Equivalent to sorted(iterable)'
127 ... h = []
128 ... for value in iterable:
129 ... heappush(h, value)
130 ... return [heappop(h) for i in range(len(h))]
131 ...
132 >>> heapsort([1, 3, 5, 7, 9, 2, 4, 6, 8, 0])
133 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
134
135Heap elements can be tuples. This is useful for assigning comparison values
136(such as task priorities) alongside the main record being tracked::
137
138 >>> h = []
139 >>> heappush(h, (5, 'write code'))
140 >>> heappush(h, (7, 'release product'))
141 >>> heappush(h, (1, 'write spec'))
142 >>> heappush(h, (3, 'create tests'))
143 >>> heappop(h)
144 (1, 'write spec')
145
146
Raymond Hettinger0e833c32010-08-07 23:31:27 +0000147Priority Queue Implementation Notes
148-----------------------------------
149
150A `priority queue <http://en.wikipedia.org/wiki/Priority_queue>`_ is common use
151for a heap, and it presents several implementation challenges:
152
153* Sort stability: how do you get two tasks with equal priorities to be returned
154 in the order they were originally added?
155
156* Tuple comparison breaks for (priority, task) pairs if the priorities are equal
157 and the tasks do not have a default comparison order.
158
Raymond Hettinger648e7252010-08-07 23:37:37 +0000159* If the priority of a task changes, how do you move it to a new position in
Raymond Hettinger0e833c32010-08-07 23:31:27 +0000160 the heap?
161
162* Or if a pending task needs to be deleted, how do you find it and remove it
163 from the queue?
164
165A solution to the first two challenges is to store entries as 3-element list
166including the priority, an entry count, and the task. The entry count serves as
167a tie-breaker so that two tasks with the same priority are returned in the order
168they were added. And since no two entry counts are the same, the tuple
169comparison will never attempt to directly compare two tasks.
170
171The remaining challenges revolve around finding a pending task and making
172changes to its priority or removing it entirely. Finding a task can be done
173with a dictionary pointing to an entry in the queue.
174
175Removing the entry or changing its priority is more difficult because it would
176break the heap structure invariants. So, a possible solution is to mark an
177entry as invalid and optionally add a new entry with the revised priority::
178
179 pq = [] # the priority queue list
180 counter = itertools.count(1) # unique sequence count
181 task_finder = {} # mapping of tasks to entries
182 INVALID = 0 # mark an entry as deleted
183
184 def add_task(priority, task, count=None):
185 if count is None:
186 count = next(counter)
187 entry = [priority, count, task]
188 task_finder[task] = entry
189 heappush(pq, entry)
190
191 def get_top_priority():
192 while True:
193 priority, count, task = heappop(pq)
194 del task_finder[task]
195 if count is not INVALID:
196 return task
197
198 def delete_task(task):
199 entry = task_finder[task]
200 entry[1] = INVALID
201
202 def reprioritize(priority, task):
203 entry = task_finder[task]
204 add_task(priority, task, entry[1])
205 entry[1] = INVALID
206
207
Georg Brandl116aa622007-08-15 14:28:22 +0000208Theory
209------
210
Georg Brandl116aa622007-08-15 14:28:22 +0000211Heaps are arrays for which ``a[k] <= a[2*k+1]`` and ``a[k] <= a[2*k+2]`` for all
212*k*, counting elements from 0. For the sake of comparison, non-existing
213elements are considered to be infinite. The interesting property of a heap is
214that ``a[0]`` is always its smallest element.
215
216The strange invariant above is meant to be an efficient memory representation
217for a tournament. The numbers below are *k*, not ``a[k]``::
218
219 0
220
221 1 2
222
223 3 4 5 6
224
225 7 8 9 10 11 12 13 14
226
227 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
228
229In the tree above, each cell *k* is topping ``2*k+1`` and ``2*k+2``. In an usual
230binary tournament we see in sports, each cell is the winner over the two cells
231it tops, and we can trace the winner down the tree to see all opponents s/he
232had. However, in many computer applications of such tournaments, we do not need
233to trace the history of a winner. To be more memory efficient, when a winner is
234promoted, we try to replace it by something else at a lower level, and the rule
235becomes that a cell and the two cells it tops contain three different items, but
236the top cell "wins" over the two topped cells.
237
238If this heap invariant is protected at all time, index 0 is clearly the overall
239winner. The simplest algorithmic way to remove it and find the "next" winner is
240to move some loser (let's say cell 30 in the diagram above) into the 0 position,
241and then percolate this new 0 down the tree, exchanging values, until the
242invariant is re-established. This is clearly logarithmic on the total number of
243items in the tree. By iterating over all items, you get an O(n log n) sort.
244
245A nice feature of this sort is that you can efficiently insert new items while
246the sort is going on, provided that the inserted items are not "better" than the
247last 0'th element you extracted. This is especially useful in simulation
248contexts, where the tree holds all incoming events, and the "win" condition
249means the smallest scheduled time. When an event schedule other events for
250execution, they are scheduled into the future, so they can easily go into the
251heap. So, a heap is a good structure for implementing schedulers (this is what
252I used for my MIDI sequencer :-).
253
254Various structures for implementing schedulers have been extensively studied,
255and heaps are good for this, as they are reasonably speedy, the speed is almost
256constant, and the worst case is not much different than the average case.
257However, there are other representations which are more efficient overall, yet
258the worst cases might be terrible.
259
260Heaps are also very useful in big disk sorts. You most probably all know that a
261big sort implies producing "runs" (which are pre-sorted sequences, which size is
262usually related to the amount of CPU memory), followed by a merging passes for
263these runs, which merging is often very cleverly organised [#]_. It is very
264important that the initial sort produces the longest runs possible. Tournaments
265are a good way to that. If, using all the memory available to hold a
266tournament, you replace and percolate items that happen to fit the current run,
267you'll produce runs which are twice the size of the memory for random input, and
268much better for input fuzzily ordered.
269
270Moreover, if you output the 0'th item on disk and get an input which may not fit
271in the current tournament (because the value "wins" over the last output value),
272it cannot fit in the heap, so the size of the heap decreases. The freed memory
273could be cleverly reused immediately for progressively building a second heap,
274which grows at exactly the same rate the first heap is melting. When the first
275heap completely vanishes, you switch heaps and start a new run. Clever and
276quite effective!
277
278In a word, heaps are useful memory structures to know. I use them in a few
279applications, and I think it is good to keep a 'heap' module around. :-)
280
281.. rubric:: Footnotes
282
283.. [#] The disk balancing algorithms which are current, nowadays, are more annoying
284 than clever, and this is a consequence of the seeking capabilities of the disks.
285 On devices which cannot seek, like big tape drives, the story was quite
286 different, and one had to be very clever to ensure (far in advance) that each
287 tape movement will be the most effective possible (that is, will best
288 participate at "progressing" the merge). Some tapes were even able to read
289 backwards, and this was also used to avoid the rewinding time. Believe me, real
290 good tape sorts were quite spectacular to watch! From all times, sorting has
291 always been a Great Art! :-)
292