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+\section{\module{heapq} ---
+         Heap queue algorithm}
+
+\declaremodule{standard}{heapq}
+\modulesynopsis{Heap queue algorithm (a.k.a. priority queue).}
+\moduleauthor{Kevin O'Connor}{}
+\sectionauthor{Guido van Rossum}{guido@python.org}
+% Theoretical explanation:
+\sectionauthor{Fran\c cois Pinard}{}
+\versionadded{2.3}
+
+
+This module provides an implementation of the heap queue algorithm,
+also known as the priority queue algorithm.
+
+Heaps are arrays for which
+\code{\var{heap}[\var{k}] <= \var{heap}[2*\var{k}+1]} and
+\code{\var{heap}[\var{k}] <= \var{heap}[2*\var{k}+2]}
+for all \var{k}, counting elements from zero.  For the sake of
+comparison, non-existing elements are considered to be infinite.  The
+interesting property of a heap is that \code{\var{heap}[0]} is always
+its smallest element.
+
+The API below 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) Our pop method returns the smallest item, not the largest (called a
+"min heap" in textbooks; a "max heap" is more common in texts because
+of its suitability for in-place sorting).
+
+These two make it possible to view the heap as a regular Python list
+without surprises: \code{\var{heap}[0]} is the smallest item, and
+\code{\var{heap}.sort()} maintains the heap invariant!
+
+To create a heap, use a list initialized to \code{[]}, or you can
+transform a populated list into a heap via function \function{heapify()}.
+
+The following functions are provided:
+
+\begin{funcdesc}{heappush}{heap, item}
+Push the value \var{item} onto the \var{heap}, maintaining the
+heap invariant.
+\end{funcdesc}
+
+\begin{funcdesc}{heappop}{heap}
+Pop and return the smallest item from the \var{heap}, maintaining the
+heap invariant.  If the heap is empty, \exception{IndexError} is raised.
+\end{funcdesc}
+
+\begin{funcdesc}{heapify}{x}
+Transform list \var{x} into a heap, in-place, in linear time.
+\end{funcdesc}
+
+\begin{funcdesc}{heapreplace}{heap, item}
+Pop and return the smallest item from the \var{heap}, and also push
+the new \var{item}.  The heap size doesn't change.
+If the heap is empty, \exception{IndexError} is raised.
+This is more efficient than \function{heappop()} followed
+by  \function{heappush()}, and can be more appropriate when using
+a fixed-size heap.  Note that the value returned may be larger
+than \var{item}!  That constrains reasonable uses of this routine
+unless written as part of a conditional replacement:
+\begin{verbatim}
+        if item > heap[0]:
+            item = heapreplace(heap, item)
+\end{verbatim}
+\end{funcdesc}
+
+Example of use:
+
+\begin{verbatim}
+>>> from heapq import heappush, heappop
+>>> heap = []
+>>> data = [1, 3, 5, 7, 9, 2, 4, 6, 8, 0]
+>>> for item in data:
+...     heappush(heap, item)
+...
+>>> ordered = []
+>>> while heap:
+...     ordered.append(heappop(heap))
+...
+>>> print ordered
+[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
+>>> data.sort()
+>>> print data == ordered
+True
+>>>
+\end{verbatim}
+
+The module also offers two general purpose functions based on heaps.
+
+\begin{funcdesc}{nlargest}{n, iterable\optional{, key}}
+Return a list with the \var{n} largest elements from the dataset defined
+by \var{iterable}.  \var{key}, if provided, specifies a function of one
+argument that is used to extract a comparison key from each element
+in the iterable:  \samp{\var{key}=\function{str.lower}}
+Equivalent to:  \samp{sorted(iterable, key=key, reverse=True)[:n]}
+\versionadded{2.4}
+\versionchanged[Added the optional \var{key} argument]{2.5}
+\end{funcdesc}
+
+\begin{funcdesc}{nsmallest}{n, iterable\optional{, key}}
+Return a list with the \var{n} smallest elements from the dataset defined
+by \var{iterable}.  \var{key}, if provided, specifies a function of one
+argument that is used to extract a comparison key from each element
+in the iterable:  \samp{\var{key}=\function{str.lower}}
+Equivalent to:  \samp{sorted(iterable, key=key)[:n]}
+\versionadded{2.4}
+\versionchanged[Added the optional \var{key} argument]{2.5}
+\end{funcdesc}
+
+Both functions perform best for smaller values of \var{n}.  For larger
+values, it is more efficient to use the \function{sorted()} function.  Also,
+when \code{n==1}, it is more efficient to use the builtin \function{min()}
+and \function{max()} functions.
+
+
+\subsection{Theory}
+
+(This explanation is due to François Pinard.  The Python
+code for this module was contributed by Kevin O'Connor.)
+
+Heaps are arrays for which \code{a[\var{k}] <= a[2*\var{k}+1]} and
+\code{a[\var{k}] <= a[2*\var{k}+2]}
+for all \var{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 \code{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 \var{k}, not
+\code{a[\var{k}]}:
+
+\begin{verbatim}
+                                   0
+
+                  1                                 2
+
+          3               4                5               6
+
+      7       8       9       10      11      12      13      14
+
+    15 16   17 18   19 20   21 22   23 24   25 26   27 28   29 30
+\end{verbatim}
+
+In the tree above, each cell \var{k} is topping \code{2*\var{k}+1} and
+\code{2*\var{k}+2}.
+In an 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 schedule 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, which 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\footnote{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! :-)}.
+It is very important that the initial
+sort produces the longest runs possible.  Tournaments are a good way
+to 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. :-)