Adding the heap queue algorithm, per discussion in python-dev last
week.
diff --git a/Lib/heapq.py b/Lib/heapq.py
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+++ b/Lib/heapq.py
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+"""Heap queue algorithm (a.k.a. priority queue).
+
+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.
+
+Usage:
+
+heap = [] # creates an empty heap
+heappush(heap, item) # pushes a new item on the heap
+item = heappop(heap) # pops the smallest item from the heap
+item = heap[0] # smallest item on the heap without popping it
+
+Our API differs from textbook heap algorithms as follows:
+
+- We use 0-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 0-based indexing.
+
+- Our heappop() method returns the smallest item, not the largest.
+
+These two 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!
+"""
+
+__about__ = """Heap queues
+
+[explanation by François Pinard]
+
+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]:
+
+ 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
+
+
+In the tree above, each cell `k' is topping `2*k+1' and `2*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 ln 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[1]. 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. :-)
+
+--------------------
+[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! :-)
+"""
+
+def heappush(heap, item):
+ """Push item onto heap, maintaining the heap invariant."""
+ pos = len(heap)
+ heap.append(None)
+ while pos:
+ parentpos = (pos - 1) / 2
+ parent = heap[parentpos]
+ if item >= parent:
+ break
+ heap[pos] = parent
+ pos = parentpos
+ heap[pos] = item
+
+def heappop(heap):
+ """Pop the smallest item off the heap, maintaining the heap invariant."""
+ endpos = len(heap) - 1
+ if endpos <= 0:
+ return heap.pop()
+ returnitem = heap[0]
+ item = heap.pop()
+ pos = 0
+ while 1:
+ child2pos = (pos + 1) * 2
+ child1pos = child2pos - 1
+ if child2pos < endpos:
+ child1 = heap[child1pos]
+ child2 = heap[child2pos]
+ if item <= child1 and item <= child2:
+ break
+ if child1 < child2:
+ heap[pos] = child1
+ pos = child1pos
+ continue
+ heap[pos] = child2
+ pos = child2pos
+ continue
+ if child1pos < endpos:
+ child1 = heap[child1pos]
+ if child1 < item:
+ heap[pos] = child1
+ pos = child1pos
+ break
+ heap[pos] = item
+ return returnitem
+
+if __name__ == "__main__":
+ # Simple sanity test
+ heap = []
+ data = [1, 3, 5, 7, 9, 2, 4, 6, 8, 0]
+ for item in data:
+ heappush(heap, item)
+ sort = []
+ while heap:
+ sort.append(heappop(heap))
+ print sort