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\section{\module{heapq} --- | ||
Heap queue algorithm} | ||
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\declaremodule{standard}{heapq} | ||
\modulesynopsis{Heap queue algorithm (a.k.a. priority queue).} | ||
\sectionauthor{Guido van Rossum}{[email protected]} | ||
% Implementation contributed by Kevin O'Connor | ||
% Theoretical explanation by François Pinard | ||
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This module provides an implementation of the heap queue algorithm, | ||
also known as the priority queue algorithm. | ||
\versionadded{2.3} | ||
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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. | ||
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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. | ||
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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! | ||
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To create a heap, use a list initialized to \code{[]}. | ||
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The following functions are provided: | ||
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\begin{funcdesc}{heappush}{heap, item} | ||
Push the value \var{item} onto the \var{heap}, maintaining the | ||
heap invariant. | ||
\end{funcdesc} | ||
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\begin{funcdesc}{heappop}{heap} | ||
Pop and return the smallest item from the \var{heap}, maintaining the | ||
heap invariant. | ||
\end{funcdesc} | ||
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Example of use: | ||
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\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) | ||
... | ||
>>> sorted = [] | ||
>>> while heap: | ||
... sorted.append(heappop(heap)) | ||
... | ||
>>> print sorted | ||
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] | ||
>>> data.sort() | ||
>>> print data == sorted | ||
True | ||
>>> | ||
\end{verbatim} | ||
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\subsection{Theory} | ||
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(This explanation is due to François Pinard. The Python | ||
code for this module was contributed by Kevin O'Connor.) | ||
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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. | ||
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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}]}: | ||
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\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} | ||
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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. | ||
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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. | ||
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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 :-). | ||
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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. | ||
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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. | ||
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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! | ||
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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. :-) |