207 lines
8.6 KiB
TeX
207 lines
8.6 KiB
TeX
\section{\module{heapq} ---
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Heap queue algorithm}
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\declaremodule{standard}{heapq}
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\modulesynopsis{Heap queue algorithm (a.k.a. priority queue).}
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\moduleauthor{Kevin O'Connor}{}
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\sectionauthor{Guido van Rossum}{guido@python.org}
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% Theoretical explanation:
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\sectionauthor{Fran\c cois Pinard}{}
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\versionadded{2.3}
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This module provides an implementation of the heap queue algorithm,
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also known as the priority queue algorithm.
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Heaps are arrays for which
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\code{\var{heap}[\var{k}] <= \var{heap}[2*\var{k}+1]} and
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\code{\var{heap}[\var{k}] <= \var{heap}[2*\var{k}+2]}
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for all \var{k}, counting elements from zero. For the sake of
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comparison, non-existing elements are considered to be infinite. The
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interesting property of a heap is that \code{\var{heap}[0]} is always
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its smallest element.
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The API below differs from textbook heap algorithms in two aspects:
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(a) We use zero-based indexing. This makes the relationship between the
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index for a node and the indexes for its children slightly less
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obvious, but is more suitable since Python uses zero-based indexing.
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(b) Our pop method returns the smallest item, not the largest (called a
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"min heap" in textbooks; a "max heap" is more common in texts because
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of its suitability for in-place sorting).
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These two make it possible to view the heap as a regular Python list
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without surprises: \code{\var{heap}[0]} is the smallest item, and
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\code{\var{heap}.sort()} maintains the heap invariant!
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To create a heap, use a list initialized to \code{[]}, or you can
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transform a populated list into a heap via function \function{heapify()}.
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The following functions are provided:
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\begin{funcdesc}{heappush}{heap, item}
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Push the value \var{item} onto the \var{heap}, maintaining the
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heap invariant.
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\end{funcdesc}
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\begin{funcdesc}{heappop}{heap}
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Pop and return the smallest item from the \var{heap}, maintaining the
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heap invariant. If the heap is empty, \exception{IndexError} is raised.
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\end{funcdesc}
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\begin{funcdesc}{heapify}{x}
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Transform list \var{x} into a heap, in-place, in linear time.
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\end{funcdesc}
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\begin{funcdesc}{heapreplace}{heap, item}
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Pop and return the smallest item from the \var{heap}, and also push
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the new \var{item}. The heap size doesn't change.
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If the heap is empty, \exception{IndexError} is raised.
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This is more efficient than \function{heappop()} followed
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by \function{heappush()}, and can be more appropriate when using
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a fixed-size heap. Note that the value returned may be larger
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than \var{item}! That constrains reasonable uses of this routine
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unless written as part of a conditional replacement:
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\begin{verbatim}
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if item > heap[0]:
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item = heapreplace(heap, item)
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\end{verbatim}
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\end{funcdesc}
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Example of use:
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\begin{verbatim}
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>>> from heapq import heappush, heappop
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>>> heap = []
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>>> data = [1, 3, 5, 7, 9, 2, 4, 6, 8, 0]
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>>> for item in data:
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... heappush(heap, item)
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...
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>>> sorted = []
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>>> while heap:
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... sorted.append(heappop(heap))
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...
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>>> print sorted
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[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
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>>> data.sort()
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>>> print data == sorted
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True
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>>>
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\end{verbatim}
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The module also offers two general purpose functions based on heaps.
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\begin{funcdesc}{nlargest}{n, iterable}
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Return a list with the \var{n} largest elements from the dataset defined
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by \var{iterable}. Equivalent to: \code{sorted(iterable, reverse=True)[:n]}
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\versionadded{2.4}
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\end{funcdesc}
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\begin{funcdesc}{nsmallest}{n, iterable}
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Return a list with the \var{n} smallest elements from the dataset defined
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by \var{iterable}. Equivalent to: \code{sorted(iterable)[:n]}
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\versionadded{2.4}
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\end{funcdesc}
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Both functions perform best for smaller values of \var{n}. For larger
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values, it is more efficient to use the \function{sorted()} function. Also,
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when \code{n==1}, it is more efficient to use the builtin \function{min()}
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and \function{max()} functions.
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\subsection{Theory}
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(This explanation is due to Fran<61>ois Pinard. The Python
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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
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\code{a[\var{k}] <= a[2*\var{k}+2]}
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for all \var{k}, counting elements from 0. For the sake of comparison,
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non-existing elements are considered to be infinite. The interesting
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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
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representation for a tournament. The numbers below are \var{k}, not
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\code{a[\var{k}]}:
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\begin{verbatim}
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0
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1 2
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3 4 5 6
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7 8 9 10 11 12 13 14
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15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
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\end{verbatim}
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In the tree above, each cell \var{k} is topping \code{2*\var{k}+1} and
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\code{2*\var{k}+2}.
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In an usual binary tournament we see in sports, each cell is the winner
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over the two cells it tops, and we can trace the winner down the tree
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to see all opponents s/he had. However, in many computer applications
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of such tournaments, we do not need to trace the history of a winner.
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To be more memory efficient, when a winner is promoted, we try to
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replace it by something else at a lower level, and the rule becomes
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that a cell and the two cells it tops contain three different items,
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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
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the overall winner. The simplest algorithmic way to remove it and
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find the "next" winner is to move some loser (let's say cell 30 in the
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diagram above) into the 0 position, and then percolate this new 0 down
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the tree, exchanging values, until the invariant is re-established.
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This is clearly logarithmic on the total number of items in the tree.
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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
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items while the sort is going on, provided that the inserted items are
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not "better" than the last 0'th element you extracted. This is
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especially useful in simulation contexts, where the tree holds all
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incoming events, and the "win" condition means the smallest scheduled
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time. When an event schedule other events for execution, they are
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scheduled into the future, so they can easily go into the heap. So, a
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heap is a good structure for implementing schedulers (this is what I
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used for my MIDI sequencer :-).
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Various structures for implementing schedulers have been extensively
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studied, and heaps are good for this, as they are reasonably speedy,
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the speed is almost constant, and the worst case is not much different
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than the average case. However, there are other representations which
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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
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know that a big sort implies producing "runs" (which are pre-sorted
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sequences, which size is usually related to the amount of CPU memory),
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followed by a merging passes for these runs, which merging is often
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very cleverly organised\footnote{The disk balancing algorithms which
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are current, nowadays, are
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more annoying than clever, and this is a consequence of the seeking
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capabilities of the disks. On devices which cannot seek, like big
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tape drives, the story was quite different, and one had to be very
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clever to ensure (far in advance) that each tape movement will be the
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most effective possible (that is, will best participate at
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"progressing" the merge). Some tapes were even able to read
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backwards, and this was also used to avoid the rewinding time.
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Believe me, real good tape sorts were quite spectacular to watch!
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From all times, sorting has always been a Great Art! :-)}.
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It is very important that the initial
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sort produces the longest runs possible. Tournaments are a good way
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to that. If, using all the memory available to hold a tournament, you
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replace and percolate items that happen to fit the current run, you'll
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produce runs which are twice the size of the memory for random input,
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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
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may not fit in the current tournament (because the value "wins" over
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the last output value), it cannot fit in the heap, so the size of the
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heap decreases. The freed memory could be cleverly reused immediately
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for progressively building a second heap, which grows at exactly the
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same rate the first heap is melting. When the first heap completely
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vanishes, you switch heaps and start a new run. Clever and quite
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effective!
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In a word, heaps are useful memory structures to know. I use them in
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a few applications, and I think it is good to keep a `heap' module
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around. :-)
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