"heapq" --- Heap queue algorithm
********************************

**Source code:** Lib/heapq.py

======================================================================

This module provides an implementation of the heap queue algorithm,
also known as the priority queue algorithm.

Min-heaps are binary trees for which every parent node has a value
less than or equal to any of its children. We refer to this condition
as the heap invariant.

For min-heaps, this implementation uses lists for which "heap[k] <=
heap[2*k+1]" and "heap[k] <= heap[2*k+2]" for all *k* for which the
compared elements exist.  Elements are counted from zero.  The
interesting property of a min-heap is that its smallest element is
always the root, "heap[0]".

Max-heaps satisfy the reverse invariant: every parent node has a value
*greater* than any of its children.  These are implemented as lists
for which "maxheap[2*k+1] <= maxheap[k]" and "maxheap[2*k+2] <=
maxheap[k]" for all *k* for which the compared elements exist. The
root, "maxheap[0]", contains the *largest* element;
"heap.sort(reverse=True)" maintains the max-heap invariant.

The "heapq" API 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) Textbooks often focus on max-heaps, due to their suitability for
in-place sorting. Our implementation favors min-heaps as they better
correspond to Python "lists".

These two aspects 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!

Like "list.sort()", this implementation uses only the "<" operator for
comparisons, for both min-heaps and max-heaps.

In the API below, and in this documentation, the unqualified term
*heap* generally refers to a min-heap. The API for max-heaps is named
using a "_max"  suffix.

To create a heap, use a list initialized as "[]", or transform an
existing list into a min-heap or max-heap using the "heapify()" or
"heapify_max()" functions, respectively.

The following functions are provided for min-heaps:

heapq.heapify(x)

   Transform list *x* into a min-heap, in-place, in linear time.

heapq.heappush(heap, item)

   Push the value *item* onto the *heap*, maintaining the min-heap
   invariant.

heapq.heappop(heap)

   Pop and return the smallest item from the *heap*, maintaining the
   min-heap invariant.  If the heap is empty, "IndexError" is raised.
   To access the smallest item without popping it, use "heap[0]".

heapq.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
   "heappush()" followed by a separate call to "heappop()".

heapq.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,
   "IndexError" is raised.

   This one step operation is more efficient than a "heappop()"
   followed by "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 "heappushpop()" instead.  Its
   push/pop combination returns the smaller of the two values, leaving
   the larger value on the heap.

For max-heaps, the following functions are provided:

heapq.heapify_max(x)

   Transform list *x* into a max-heap, in-place, in linear time.

   Added in version 3.14.

heapq.heappush_max(heap, item)

   Push the value *item* onto the max-heap *heap*, maintaining the
   max-heap invariant.

   Added in version 3.14.

heapq.heappop_max(heap)

   Pop and return the largest item from the max-heap *heap*,
   maintaining the max-heap invariant.  If the max-heap is empty,
   "IndexError" is raised. To access the largest item without popping
   it, use "maxheap[0]".

   Added in version 3.14.

heapq.heappushpop_max(heap, item)

   Push *item* on the max-heap *heap*, then pop and return the largest
   item from *heap*. The combined action runs more efficiently than
   "heappush_max()" followed by a separate call to "heappop_max()".

   Added in version 3.14.

heapq.heapreplace_max(heap, item)

   Pop and return the largest item from the max-heap *heap* and also
   push the new *item*. The max-heap size doesn't change. If the max-
   heap is empty, "IndexError" is raised.

   The value returned may be smaller than the *item* added.  Refer to
   the analogous function "heapreplace()" for detailed usage notes.

   Added in version 3.14.

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

heapq.merge(*iterables, key=None, reverse=False)

   Merge multiple sorted inputs into a single sorted output (for
   example, merge timestamped entries from multiple log files).
   Returns an *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).

   Has two optional arguments which must be specified as keyword
   arguments.

   *key* specifies a *key function* of one argument that is used to
   extract a comparison key from each input element.  The default
   value is "None" (compare the elements directly).

   *reverse* is a boolean value.  If set to "True", then the input
   elements are merged as if each comparison were reversed. To achieve
   behavior similar to "sorted(itertools.chain(*iterables),
   reverse=True)", all iterables must be sorted from largest to
   smallest.

   Changed in version 3.5: Added the optional *key* and *reverse*
   parameters.

heapq.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 *iterable* (for example, "key=str.lower").  Equivalent
   to:  "sorted(iterable, key=key, reverse=True)[:n]".

heapq.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 *iterable* (for example, "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 "sorted()" function.
Also, when "n==1", it is more efficient to use the built-in "min()"
and "max()" functions.  If repeated usage of these functions is
required, consider turning the iterable into an actual heap.


Basic Examples
==============

A 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):
   ...     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]

This is similar to "sorted(iterable)", but unlike "sorted()", this
implementation is not stable.

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')


Other Applications
==================

Medians are a measure of central tendency for a set of numbers.  In
distributions skewed by outliers, the median provides a more stable
estimate than an average (arithmetic mean).  A running median is an
online algorithm that updates continuously as new data arrives.

A running median can be efficiently implemented by balancing two
heaps, a max-heap for values at or below the midpoint and a min-heap
for values above the midpoint.  When the two heaps have the same size,
the new median is the average of the tops of the two heaps; otherwise,
the median is at the top of the larger heap:

   def running_median(iterable):
       "Yields the cumulative median of values seen so far."

       lo = []  # max-heap
       hi = []  # min-heap (same size as or one smaller than lo)

       for x in iterable:
           if len(lo) == len(hi):
               heappush_max(lo, heappushpop(hi, x))
               yield lo[0]
           else:
               heappush(hi, heappushpop_max(lo, x))
               yield (lo[0] + hi[0]) / 2

For example:

   >>> list(running_median([5.0, 9.0, 4.0, 12.0, 8.0, 9.0]))
   [5.0, 7.0, 5.0, 7.0, 8.0, 8.5]


Priority Queue Implementation Notes
===================================

A 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.

Another solution to the problem of non-comparable tasks is to create a
wrapper class that ignores the task item and only compares the
priority field:

   from dataclasses import dataclass, field
   from typing import Any

   @dataclass(order=True)
   class PrioritizedItem:
       priority: int
       item: Any=field(compare=False)

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 the entry as removed and add a new entry with the revised
priority:

   pq = []                         # list of entries arranged in a heap
   entry_finder = {}               # mapping of tasks to entries
   REMOVED = '<removed-task>'      # placeholder for a removed task
   counter = itertools.count()     # unique sequence count

   def add_task(task, priority=0):
       'Add a new task or update the priority of an existing task'
       if task in entry_finder:
           remove_task(task)
       count = next(counter)
       entry = [priority, count, task]
       entry_finder[task] = entry
       heappush(pq, entry)

   def remove_task(task):
       'Mark an existing task as REMOVED.  Raise KeyError if not found.'
       entry = entry_finder.pop(task)
       entry[-1] = REMOVED

   def pop_task():
       'Remove and return the lowest priority task. Raise KeyError if empty.'
       while pq:
           priority, count, task = heappop(pq)
           if task is not REMOVED:
               del entry_finder[task]
               return task
       raise KeyError('pop from an empty priority queue')


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]":

   [image: Example (min-heap) binary tree.][image]

In the tree above, each cell *k* is topping "2*k+1" and "2*k+2". In a
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 schedules 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, whose 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 [1]. It is very important that the initial
sort produces the longest runs possible.  Tournaments are a good way
to achieve 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. :-)

-[ Footnotes ]-

[1] 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! :-)
