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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 three general purpose functions based on heaps.
-
-\begin{funcdesc}{merge}{*iterables}
-Merge multiple sorted inputs into a single sorted output (for example, merge
-timestamped entries from multiple log files).  Returns an iterator over
-over the sorted values.
-
-Similar to \code{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).            
-\versionadded{2.6}
-\end{funcdesc}
-
-\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}
-
-The latter two 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. :-)