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  1: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:: Franรงois Pinard
  9
 10.. versionadded:: 2.3
 11
 12This module provides an implementation of the heap queue algorithm, also known
 13as the priority queue algorithm.
 14
 15Heaps are arrays for which ``heap[k] <= heap[2*k+1]`` and ``heap[k] <=
 16heap[2*k+2]`` for all *k*, counting elements from zero.  For the sake of
 17comparison, non-existing elements are considered to be infinite.  The
 18interesting property of a heap is that ``heap[0]`` is always its smallest
 19element.
 20
 21The API below differs from textbook heap algorithms in two aspects: (a) We use
 22zero-based indexing.  This makes the relationship between the index for a node
 23and the indexes for its children slightly less obvious, but is more suitable
 24since Python uses zero-based indexing. (b) Our pop method returns the smallest
 25item, not the largest (called a "min heap" in textbooks; a "max heap" is more
 26common in texts because of its suitability for in-place sorting).
 27
 28These two make it possible to view the heap as a regular Python list without
 29surprises: ``heap[0]`` is the smallest item, and ``heap.sort()`` maintains the
 30heap invariant!
 31
 32To create a heap, use a list initialized to ``[]``, or you can transform a
 33populated list into a heap via function :func:`heapify`.
 34
 35The following functions are provided:
 36
 37
 38.. function:: heappush(heap, item)
 39
 40   Push the value *item* onto the *heap*, maintaining the heap invariant.
 41
 42
 43.. function:: heappop(heap)
 44
 45   Pop and return the smallest item from the *heap*, maintaining the heap
 46   invariant.  If the heap is empty, :exc:`IndexError` is raised.
 47
 48.. function:: heappushpop(heap, item)
 49
 50   Push *item* on the heap, then pop and return the smallest item from the
 51   *heap*.  The combined action runs more efficiently than :func:`heappush`
 52   followed by a separate call to :func:`heappop`.
 53
 54   .. versionadded:: 2.6
 55
 56.. function:: heapify(x)
 57
 58   Transform list *x* into a heap, in-place, in linear time.
 59
 60
 61.. function:: heapreplace(heap, item)
 62
 63   Pop and return the smallest item from the *heap*, and also push the new *item*.
 64   The heap size doesn't change. If the heap is empty, :exc:`IndexError` is raised.
 65   This is more efficient than :func:`heappop` followed by  :func:`heappush`, and
 66   can be more appropriate when using a fixed-size heap.  Note that the value
 67   returned may be larger than *item*!  That constrains reasonable uses of this
 68   routine unless written as part of a conditional replacement::
 69
 70      if item > heap[0]:
 71          item = heapreplace(heap, item)
 72
 73Example of use:
 74
 75   >>> from heapq import heappush, heappop
 76   >>> heap = []
 77   >>> data = [1, 3, 5, 7, 9, 2, 4, 6, 8, 0]
 78   >>> for item in data:
 79   ...     heappush(heap, item)
 80   ...
 81   >>> ordered = []
 82   >>> while heap:
 83   ...     ordered.append(heappop(heap))
 84   ...
 85   >>> print ordered
 86   [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
 87   >>> data.sort()
 88   >>> print data == ordered
 89   True
 90
 91Using a heap to insert items at the correct place in a priority queue:
 92
 93   >>> heap = []
 94   >>> data = [(1, 'J'), (4, 'N'), (3, 'H'), (2, 'O')]
 95   >>> for item in data:
 96   ...     heappush(heap, item)
 97   ...
 98   >>> while heap:
 99   ...     print heappop(heap)[1]
100   J
101   O
102   H
103   N
104
105
106The module also offers three general purpose functions based on heaps.
107
108
109.. function:: merge(*iterables)
110
111   Merge multiple sorted inputs into a single sorted output (for example, merge
112   timestamped entries from multiple log files).  Returns an :term:`iterator`
113   over the sorted values.
114
115   Similar to ``sorted(itertools.chain(*iterables))`` but returns an iterable, does
116   not pull the data into memory all at once, and assumes that each of the input
117   streams is already sorted (smallest to largest).
118
119   .. versionadded:: 2.6
120
121
122.. function:: nlargest(n, iterable[, key])
123
124   Return a list with the *n* largest elements from the dataset defined by
125   *iterable*.  *key*, if provided, specifies a function of one argument that is
126   used to extract a comparison key from each element in the iterable:
127   ``key=str.lower`` Equivalent to:  ``sorted(iterable, key=key,
128   reverse=True)[:n]``
129
130   .. versionadded:: 2.4
131
132   .. versionchanged:: 2.5
133      Added the optional *key* argument.
134
135
136.. function:: nsmallest(n, iterable[, key])
137
138   Return a list with the *n* smallest elements from the dataset defined by
139   *iterable*.  *key*, if provided, specifies a function of one argument that is
140   used to extract a comparison key from each element in the iterable:
141   ``key=str.lower`` Equivalent to:  ``sorted(iterable, key=key)[:n]``
142
143   .. versionadded:: 2.4
144
145   .. versionchanged:: 2.5
146      Added the optional *key* argument.
147
148The latter two functions perform best for smaller values of *n*.  For larger
149values, it is more efficient to use the :func:`sorted` function.  Also, when
150``n==1``, it is more efficient to use the builtin :func:`min` and :func:`max`
151functions.
152
153
154Theory
155------
156
157(This explanation is due to Franรงois Pinard.  The Python code for this module
158was contributed by Kevin O'Connor.)
159
160Heaps are arrays for which ``a[k] <= a[2*k+1]`` and ``a[k] <= a[2*k+2]`` for all
161*k*, counting elements from 0.  For the sake of comparison, non-existing
162elements are considered to be infinite.  The interesting property of a heap is
163that ``a[0]`` is always its smallest element.
164
165The strange invariant above is meant to be an efficient memory representation
166for a tournament.  The numbers below are *k*, not ``a[k]``::
167
168                                  0
169
170                 1                                 2
171
172         3               4                5               6
173
174     7       8       9       10      11      12      13      14
175
176   15 16   17 18   19 20   21 22   23 24   25 26   27 28   29 30
177
178In the tree above, each cell *k* is topping ``2*k+1`` and ``2*k+2``. In an usual
179binary tournament we see in sports, each cell is the winner over the two cells
180it tops, and we can trace the winner down the tree to see all opponents s/he
181had.  However, in many computer applications of such tournaments, we do not need
182to trace the history of a winner. To be more memory efficient, when a winner is
183promoted, we try to replace it by something else at a lower level, and the rule
184becomes that a cell and the two cells it tops contain three different items, but
185the top cell "wins" over the two topped cells.
186
187If this heap invariant is protected at all time, index 0 is clearly the overall
188winner.  The simplest algorithmic way to remove it and find the "next" winner is
189to move some loser (let's say cell 30 in the diagram above) into the 0 position,
190and then percolate this new 0 down the tree, exchanging values, until the
191invariant is re-established. This is clearly logarithmic on the total number of
192items in the tree. By iterating over all items, you get an O(n log n) sort.
193
194A nice feature of this sort is that you can efficiently insert new items while
195the sort is going on, provided that the inserted items are not "better" than the
196last 0'th element you extracted.  This is especially useful in simulation
197contexts, where the tree holds all incoming events, and the "win" condition
198means the smallest scheduled time.  When an event schedule other events for
199execution, they are scheduled into the future, so they can easily go into the
200heap.  So, a heap is a good structure for implementing schedulers (this is what
201I used for my MIDI sequencer :-).
202
203Various structures for implementing schedulers have been extensively studied,
204and heaps are good for this, as they are reasonably speedy, the speed is almost
205constant, and the worst case is not much different than the average case.
206However, there are other representations which are more efficient overall, yet
207the worst cases might be terrible.
208
209Heaps are also very useful in big disk sorts.  You most probably all know that a
210big sort implies producing "runs" (which are pre-sorted sequences, which size is
211usually related to the amount of CPU memory), followed by a merging passes for
212these runs, which merging is often very cleverly organised [#]_. It is very
213important that the initial sort produces the longest runs possible.  Tournaments
214are a good way to that.  If, using all the memory available to hold a
215tournament, you replace and percolate items that happen to fit the current run,
216you'll produce runs which are twice the size of the memory for random input, and
217much better for input fuzzily ordered.
218
219Moreover, if you output the 0'th item on disk and get an input which may not fit
220in the current tournament (because the value "wins" over the last output value),
221it cannot fit in the heap, so the size of the heap decreases.  The freed memory
222could be cleverly reused immediately for progressively building a second heap,
223which grows at exactly the same rate the first heap is melting.  When the first
224heap completely vanishes, you switch heaps and start a new run.  Clever and
225quite effective!
226
227In a word, heaps are useful memory structures to know.  I use them in a few
228applications, and I think it is good to keep a 'heap' module around. :-)
229
230.. rubric:: Footnotes
231
232.. [#] The disk balancing algorithms which are current, nowadays, are more annoying
233   than clever, and this is a consequence of the seeking capabilities of the disks.
234   On devices which cannot seek, like big tape drives, the story was quite
235   different, and one had to be very clever to ensure (far in advance) that each
236   tape movement will be the most effective possible (that is, will best
237   participate at "progressing" the merge).  Some tapes were even able to read
238   backwards, and this was also used to avoid the rewinding time. Believe me, real
239   good tape sorts were quite spectacular to watch! From all times, sorting has
240   always been a Great Art! :-)
241