# -*- coding: latin-1 -*- | |
"""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 | |
heapify(x) # transforms list into a heap, in-place, in linear time | |
item = heapreplace(heap, item) # pops and returns smallest item, and adds | |
# new item; the heap size is unchanged | |
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! | |
""" | |
# Original code by Kevin O'Connor, augmented by Tim Peters and Raymond Hettinger | |
__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! :-) | |
""" | |
__all__ = ['heappush', 'heappop', 'heapify', 'heapreplace', 'merge', | |
'nlargest', 'nsmallest', 'heappushpop'] | |
from itertools import islice, repeat, count, imap, izip, tee, chain | |
from operator import itemgetter | |
import bisect | |
def cmp_lt(x, y): | |
# Use __lt__ if available; otherwise, try __le__. | |
# In Py3.x, only __lt__ will be called. | |
return (x < y) if hasattr(x, '__lt__') else (not y <= x) | |
def heappush(heap, item): | |
"""Push item onto heap, maintaining the heap invariant.""" | |
heap.append(item) | |
_siftdown(heap, 0, len(heap)-1) | |
def heappop(heap): | |
"""Pop the smallest item off the heap, maintaining the heap invariant.""" | |
lastelt = heap.pop() # raises appropriate IndexError if heap is empty | |
if heap: | |
returnitem = heap[0] | |
heap[0] = lastelt | |
_siftup(heap, 0) | |
else: | |
returnitem = lastelt | |
return returnitem | |
def heapreplace(heap, item): | |
"""Pop and return the current smallest value, and add the new item. | |
This is more efficient than heappop() followed by heappush(), and can be | |
more appropriate when using a fixed-size heap. Note that the value | |
returned may be larger than item! That constrains reasonable uses of | |
this routine unless written as part of a conditional replacement: | |
if item > heap[0]: | |
item = heapreplace(heap, item) | |
""" | |
returnitem = heap[0] # raises appropriate IndexError if heap is empty | |
heap[0] = item | |
_siftup(heap, 0) | |
return returnitem | |
def heappushpop(heap, item): | |
"""Fast version of a heappush followed by a heappop.""" | |
if heap and cmp_lt(heap[0], item): | |
item, heap[0] = heap[0], item | |
_siftup(heap, 0) | |
return item | |
def heapify(x): | |
"""Transform list into a heap, in-place, in O(len(x)) time.""" | |
n = len(x) | |
# Transform bottom-up. The largest index there's any point to looking at | |
# is the largest with a child index in-range, so must have 2*i + 1 < n, | |
# or i < (n-1)/2. If n is even = 2*j, this is (2*j-1)/2 = j-1/2 so | |
# j-1 is the largest, which is n//2 - 1. If n is odd = 2*j+1, this is | |
# (2*j+1-1)/2 = j so j-1 is the largest, and that's again n//2-1. | |
for i in reversed(xrange(n//2)): | |
_siftup(x, i) | |
def nlargest(n, iterable): | |
"""Find the n largest elements in a dataset. | |
Equivalent to: sorted(iterable, reverse=True)[:n] | |
""" | |
it = iter(iterable) | |
result = list(islice(it, n)) | |
if not result: | |
return result | |
heapify(result) | |
_heappushpop = heappushpop | |
for elem in it: | |
_heappushpop(result, elem) | |
result.sort(reverse=True) | |
return result | |
def nsmallest(n, iterable): | |
"""Find the n smallest elements in a dataset. | |
Equivalent to: sorted(iterable)[:n] | |
""" | |
if hasattr(iterable, '__len__') and n * 10 <= len(iterable): | |
# For smaller values of n, the bisect method is faster than a minheap. | |
# It is also memory efficient, consuming only n elements of space. | |
it = iter(iterable) | |
result = sorted(islice(it, 0, n)) | |
if not result: | |
return result | |
insort = bisect.insort | |
pop = result.pop | |
los = result[-1] # los --> Largest of the nsmallest | |
for elem in it: | |
if cmp_lt(elem, los): | |
insort(result, elem) | |
pop() | |
los = result[-1] | |
return result | |
# An alternative approach manifests the whole iterable in memory but | |
# saves comparisons by heapifying all at once. Also, saves time | |
# over bisect.insort() which has O(n) data movement time for every | |
# insertion. Finding the n smallest of an m length iterable requires | |
# O(m) + O(n log m) comparisons. | |
h = list(iterable) | |
heapify(h) | |
return map(heappop, repeat(h, min(n, len(h)))) | |
# 'heap' is a heap at all indices >= startpos, except possibly for pos. pos | |
# is the index of a leaf with a possibly out-of-order value. Restore the | |
# heap invariant. | |
def _siftdown(heap, startpos, pos): | |
newitem = heap[pos] | |
# Follow the path to the root, moving parents down until finding a place | |
# newitem fits. | |
while pos > startpos: | |
parentpos = (pos - 1) >> 1 | |
parent = heap[parentpos] | |
if cmp_lt(newitem, parent): | |
heap[pos] = parent | |
pos = parentpos | |
continue | |
break | |
heap[pos] = newitem | |
# The child indices of heap index pos are already heaps, and we want to make | |
# a heap at index pos too. We do this by bubbling the smaller child of | |
# pos up (and so on with that child's children, etc) until hitting a leaf, | |
# then using _siftdown to move the oddball originally at index pos into place. | |
# | |
# We *could* break out of the loop as soon as we find a pos where newitem <= | |
# both its children, but turns out that's not a good idea, and despite that | |
# many books write the algorithm that way. During a heap pop, the last array | |
# element is sifted in, and that tends to be large, so that comparing it | |
# against values starting from the root usually doesn't pay (= usually doesn't | |
# get us out of the loop early). See Knuth, Volume 3, where this is | |
# explained and quantified in an exercise. | |
# | |
# Cutting the # of comparisons is important, since these routines have no | |
# way to extract "the priority" from an array element, so that intelligence | |
# is likely to be hiding in custom __cmp__ methods, or in array elements | |
# storing (priority, record) tuples. Comparisons are thus potentially | |
# expensive. | |
# | |
# On random arrays of length 1000, making this change cut the number of | |
# comparisons made by heapify() a little, and those made by exhaustive | |
# heappop() a lot, in accord with theory. Here are typical results from 3 | |
# runs (3 just to demonstrate how small the variance is): | |
# | |
# Compares needed by heapify Compares needed by 1000 heappops | |
# -------------------------- -------------------------------- | |
# 1837 cut to 1663 14996 cut to 8680 | |
# 1855 cut to 1659 14966 cut to 8678 | |
# 1847 cut to 1660 15024 cut to 8703 | |
# | |
# Building the heap by using heappush() 1000 times instead required | |
# 2198, 2148, and 2219 compares: heapify() is more efficient, when | |
# you can use it. | |
# | |
# The total compares needed by list.sort() on the same lists were 8627, | |
# 8627, and 8632 (this should be compared to the sum of heapify() and | |
# heappop() compares): list.sort() is (unsurprisingly!) more efficient | |
# for sorting. | |
def _siftup(heap, pos): | |
endpos = len(heap) | |
startpos = pos | |
newitem = heap[pos] | |
# Bubble up the smaller child until hitting a leaf. | |
childpos = 2*pos + 1 # leftmost child position | |
while childpos < endpos: | |
# Set childpos to index of smaller child. | |
rightpos = childpos + 1 | |
if rightpos < endpos and not cmp_lt(heap[childpos], heap[rightpos]): | |
childpos = rightpos | |
# Move the smaller child up. | |
heap[pos] = heap[childpos] | |
pos = childpos | |
childpos = 2*pos + 1 | |
# The leaf at pos is empty now. Put newitem there, and bubble it up | |
# to its final resting place (by sifting its parents down). | |
heap[pos] = newitem | |
_siftdown(heap, startpos, pos) | |
# If available, use C implementation | |
try: | |
from _heapq import * | |
except ImportError: | |
pass | |
def merge(*iterables): | |
'''Merge multiple sorted inputs into a single sorted output. | |
Similar to sorted(itertools.chain(*iterables)) but returns a generator, | |
does not pull the data into memory all at once, and assumes that each of | |
the input streams is already sorted (smallest to largest). | |
>>> list(merge([1,3,5,7], [0,2,4,8], [5,10,15,20], [], [25])) | |
[0, 1, 2, 3, 4, 5, 5, 7, 8, 10, 15, 20, 25] | |
''' | |
_heappop, _heapreplace, _StopIteration = heappop, heapreplace, StopIteration | |
h = [] | |
h_append = h.append | |
for itnum, it in enumerate(map(iter, iterables)): | |
try: | |
next = it.next | |
h_append([next(), itnum, next]) | |
except _StopIteration: | |
pass | |
heapify(h) | |
while 1: | |
try: | |
while 1: | |
v, itnum, next = s = h[0] # raises IndexError when h is empty | |
yield v | |
s[0] = next() # raises StopIteration when exhausted | |
_heapreplace(h, s) # restore heap condition | |
except _StopIteration: | |
_heappop(h) # remove empty iterator | |
except IndexError: | |
return | |
# Extend the implementations of nsmallest and nlargest to use a key= argument | |
_nsmallest = nsmallest | |
def nsmallest(n, iterable, key=None): | |
"""Find the n smallest elements in a dataset. | |
Equivalent to: sorted(iterable, key=key)[:n] | |
""" | |
# Short-cut for n==1 is to use min() when len(iterable)>0 | |
if n == 1: | |
it = iter(iterable) | |
head = list(islice(it, 1)) | |
if not head: | |
return [] | |
if key is None: | |
return [min(chain(head, it))] | |
return [min(chain(head, it), key=key)] | |
# When n>=size, it's faster to use sorted() | |
try: | |
size = len(iterable) | |
except (TypeError, AttributeError): | |
pass | |
else: | |
if n >= size: | |
return sorted(iterable, key=key)[:n] | |
# When key is none, use simpler decoration | |
if key is None: | |
it = izip(iterable, count()) # decorate | |
result = _nsmallest(n, it) | |
return map(itemgetter(0), result) # undecorate | |
# General case, slowest method | |
in1, in2 = tee(iterable) | |
it = izip(imap(key, in1), count(), in2) # decorate | |
result = _nsmallest(n, it) | |
return map(itemgetter(2), result) # undecorate | |
_nlargest = nlargest | |
def nlargest(n, iterable, key=None): | |
"""Find the n largest elements in a dataset. | |
Equivalent to: sorted(iterable, key=key, reverse=True)[:n] | |
""" | |
# Short-cut for n==1 is to use max() when len(iterable)>0 | |
if n == 1: | |
it = iter(iterable) | |
head = list(islice(it, 1)) | |
if not head: | |
return [] | |
if key is None: | |
return [max(chain(head, it))] | |
return [max(chain(head, it), key=key)] | |
# When n>=size, it's faster to use sorted() | |
try: | |
size = len(iterable) | |
except (TypeError, AttributeError): | |
pass | |
else: | |
if n >= size: | |
return sorted(iterable, key=key, reverse=True)[:n] | |
# When key is none, use simpler decoration | |
if key is None: | |
it = izip(iterable, count(0,-1)) # decorate | |
result = _nlargest(n, it) | |
return map(itemgetter(0), result) # undecorate | |
# General case, slowest method | |
in1, in2 = tee(iterable) | |
it = izip(imap(key, in1), count(0,-1), in2) # decorate | |
result = _nlargest(n, it) | |
return map(itemgetter(2), result) # undecorate | |
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 | |
import doctest | |
doctest.testmod() |