android / platform / prebuilts / python / darwin-x86 / 2.7.5 / refs/tags/android-n-mr1-preview-2 / . / lib / python2.7 / heapq.py

# -*- 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, count, imap, izip, tee, chain | |

from operator import itemgetter | |

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 _heappushpop_max(heap, item): | |

"""Maxheap version of a heappush followed by a heappop.""" | |

if heap and cmp_lt(item, heap[0]): | |

item, heap[0] = heap[0], item | |

_siftup_max(heap, 0) | |

return item | |

def _heapify_max(x): | |

"""Transform list into a maxheap, in-place, in O(len(x)) time.""" | |

n = len(x) | |

for i in reversed(range(n//2)): | |

_siftup_max(x, i) | |

def nlargest(n, iterable): | |

"""Find the n largest elements in a dataset. | |

Equivalent to: sorted(iterable, reverse=True)[:n] | |

""" | |

if n < 0: | |

return [] | |

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 n < 0: | |

return [] | |

it = iter(iterable) | |

result = list(islice(it, n)) | |

if not result: | |

return result | |

_heapify_max(result) | |

_heappushpop = _heappushpop_max | |

for elem in it: | |

_heappushpop(result, elem) | |

result.sort() | |

return result | |

# '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) | |

def _siftdown_max(heap, startpos, pos): | |

'Maxheap variant of _siftdown' | |

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(parent, newitem): | |

heap[pos] = parent | |

pos = parentpos | |

continue | |

break | |

heap[pos] = newitem | |

def _siftup_max(heap, pos): | |

'Maxheap variant of _siftup' | |

endpos = len(heap) | |

startpos = pos | |

newitem = heap[pos] | |

# Bubble up the larger child until hitting a leaf. | |

childpos = 2*pos + 1 # leftmost child position | |

while childpos < endpos: | |

# Set childpos to index of larger child. | |

rightpos = childpos + 1 | |

if rightpos < endpos and not cmp_lt(heap[rightpos], heap[childpos]): | |

childpos = rightpos | |

# Move the larger 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_max(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() |