From c25faa844a23ec6a4e1654d02c71ac07458dde96 Mon Sep 17 00:00:00 2001 From: Guido van Rossum Date: Fri, 2 Aug 2002 16:44:32 +0000 Subject: Adding the heap queue algorithm, per discussion in python-dev last week. --- Lib/heapq.py | 176 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 176 insertions(+) create mode 100644 Lib/heapq.py (limited to 'Lib/heapq.py') diff --git a/Lib/heapq.py b/Lib/heapq.py new file mode 100644 index 0000000000..4654f5ebe2 --- /dev/null +++ b/Lib/heapq.py @@ -0,0 +1,176 @@ +"""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 -- cgit v1.2.1