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:: François Pinard

     9 

    10 .. versionadded:: 2.3

    11 

    12 This module provides an implementation of the heap queue algorithm, also known

    13 as the priority queue algorithm.

    14 

    15 Heaps are arrays for which ``heap[k] <= heap[2*k+1]`` and ``heap[k] <=

    16 heap[2*k+2]`` for all *k*, counting elements from zero.  For the sake of

    17 comparison, non-existing elements are considered to be infinite.  The

    18 interesting property of a heap is that ``heap[0]`` is always its smallest

    19 element.

    20 

    21 The API below differs from textbook heap algorithms in two aspects: (a) We use

    22 zero-based indexing.  This makes the relationship between the index for a node

    23 and the indexes for its children slightly less obvious, but is more suitable

    24 since Python uses zero-based indexing. (b) Our pop method returns the smallest

    25 item, not the largest (called a "min heap" in textbooks; a "max heap" is more

    26 common in texts because of its suitability for in-place sorting).

    27 

    28 These two make it possible to view the heap as a regular Python list without

    29 surprises: ``heap[0]`` is the smallest item, and ``heap.sort()`` maintains the

    30 heap invariant!

    31 

    32 To create a heap, use a list initialized to ``[]``, or you can transform a

    33 populated list into a heap via function :func:`heapify`.

    34 

    35 The 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 

    73 Example 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 

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

    92 

    93 

    94 .. function:: merge(*iterables)

    95 

    96    Merge multiple sorted inputs into a single sorted output (for example, merge

    97    timestamped entries from multiple log files).  Returns an :term:`iterator`

    98    over the sorted values.

    99 

   100    Similar to ``sorted(itertools.chain(*iterables))`` but returns an iterable, does

   101    not pull the data into memory all at once, and assumes that each of the input

   102    streams is already sorted (smallest to largest).

   103 

   104    .. versionadded:: 2.6

   105 

   106 

   107 .. function:: nlargest(n, iterable[, key])

   108 

   109    Return a list with the *n* largest elements from the dataset defined by

   110    *iterable*.  *key*, if provided, specifies a function of one argument that is

   111    used to extract a comparison key from each element in the iterable:

   112    ``key=str.lower`` Equivalent to:  ``sorted(iterable, key=key,

   113    reverse=True)[:n]``

   114 

   115    .. versionadded:: 2.4

   116 

   117    .. versionchanged:: 2.5

   118       Added the optional *key* argument.

   119 

   120 

   121 .. function:: nsmallest(n, iterable[, key])

   122 

   123    Return a list with the *n* smallest elements from the dataset defined by

   124    *iterable*.  *key*, if provided, specifies a function of one argument that is

   125    used to extract a comparison key from each element in the iterable:

   126    ``key=str.lower`` Equivalent to:  ``sorted(iterable, key=key)[:n]``

   127 

   128    .. versionadded:: 2.4

   129 

   130    .. versionchanged:: 2.5

   131       Added the optional *key* argument.

   132 

   133 The latter two functions perform best for smaller values of *n*.  For larger

   134 values, it is more efficient to use the :func:`sorted` function.  Also, when

   135 ``n==1``, it is more efficient to use the builtin :func:`min` and :func:`max`

   136 functions.

   137 

   138 

   139 Theory

   140 ------

   141 

   142 (This explanation is due to François Pinard.  The Python code for this module

   143 was contributed by Kevin O'Connor.)

   144 

   145 Heaps are arrays for which ``a[k] <= a[2*k+1]`` and ``a[k] <= a[2*k+2]`` for all

   146 *k*, counting elements from 0.  For the sake of comparison, non-existing

   147 elements are considered to be infinite.  The interesting property of a heap is

   148 that ``a[0]`` is always its smallest element.

   149 

   150 The strange invariant above is meant to be an efficient memory representation

   151 for a tournament.  The numbers below are *k*, not ``a[k]``::

   152 

   153                                   0

   154 

   155                  1                                 2

   156 

   157          3               4                5               6

   158 

   159      7       8       9       10      11      12      13      14

   160 

   161    15 16   17 18   19 20   21 22   23 24   25 26   27 28   29 30

   162 

   163 In the tree above, each cell *k* is topping ``2*k+1`` and ``2*k+2``. In an usual

   164 binary tournament we see in sports, each cell is the winner over the two cells

   165 it tops, and we can trace the winner down the tree to see all opponents s/he

   166 had.  However, in many computer applications of such tournaments, we do not need

   167 to trace the history of a winner. To be more memory efficient, when a winner is

   168 promoted, we try to replace it by something else at a lower level, and the rule

   169 becomes that a cell and the two cells it tops contain three different items, but

   170 the top cell "wins" over the two topped cells.

   171 

   172 If this heap invariant is protected at all time, index 0 is clearly the overall

   173 winner.  The simplest algorithmic way to remove it and find the "next" winner is

   174 to move some loser (let's say cell 30 in the diagram above) into the 0 position,

   175 and then percolate this new 0 down the tree, exchanging values, until the

   176 invariant is re-established. This is clearly logarithmic on the total number of

   177 items in the tree. By iterating over all items, you get an O(n log n) sort.

   178 

   179 A nice feature of this sort is that you can efficiently insert new items while

   180 the sort is going on, provided that the inserted items are not "better" than the

   181 last 0'th element you extracted.  This is especially useful in simulation

   182 contexts, where the tree holds all incoming events, and the "win" condition

   183 means the smallest scheduled time.  When an event schedule other events for

   184 execution, they are scheduled into the future, so they can easily go into the

   185 heap.  So, a heap is a good structure for implementing schedulers (this is what

   186 I used for my MIDI sequencer :-).

   187 

   188 Various structures for implementing schedulers have been extensively studied,

   189 and heaps are good for this, as they are reasonably speedy, the speed is almost

   190 constant, and the worst case is not much different than the average case.

   191 However, there are other representations which are more efficient overall, yet

   192 the worst cases might be terrible.

   193 

   194 Heaps are also very useful in big disk sorts.  You most probably all know that a

   195 big sort implies producing "runs" (which are pre-sorted sequences, which size is

   196 usually related to the amount of CPU memory), followed by a merging passes for

   197 these runs, which merging is often very cleverly organised [#]_. It is very

   198 important that the initial sort produces the longest runs possible.  Tournaments

   199 are a good way to that.  If, using all the memory available to hold a

   200 tournament, you replace and percolate items that happen to fit the current run,

   201 you'll produce runs which are twice the size of the memory for random input, and

   202 much better for input fuzzily ordered.

   203 

   204 Moreover, if you output the 0'th item on disk and get an input which may not fit

   205 in the current tournament (because the value "wins" over the last output value),

   206 it cannot fit in the heap, so the size of the heap decreases.  The freed memory

   207 could be cleverly reused immediately for progressively building a second heap,

   208 which grows at exactly the same rate the first heap is melting.  When the first

   209 heap completely vanishes, you switch heaps and start a new run.  Clever and

   210 quite effective!

   211 

   212 In a word, heaps are useful memory structures to know.  I use them in a few

   213 applications, and I think it is good to keep a 'heap' module around. :-)

   214 

   215 .. rubric:: Footnotes

   216 

   217 .. [#] The disk balancing algorithms which are current, nowadays, are more annoying

   218    than clever, and this is a consequence of the seeking capabilities of the disks.

   219    On devices which cannot seek, like big tape drives, the story was quite

   220    different, and one had to be very clever to ensure (far in advance) that each

   221    tape movement will be the most effective possible (that is, will best

   222    participate at "progressing" the merge).  Some tapes were even able to read

   223    backwards, and this was also used to avoid the rewinding time. Believe me, real

   224    good tape sorts were quite spectacular to watch! From all times, sorting has

   225    always been a Great Art! :-)

   226