"""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

a 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']

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)

return returnitem

return lastelt

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 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(range(n//2)):

_siftup(x, i)

def _heappop_max(heap):

"""Maxheap version of a heappop."""

lastelt = heap.pop() # raises appropriate IndexError if heap is empty

if heap:

returnitem = heap[0]

heap[0] = lastelt

_siftup_max(heap, 0)

return returnitem

return lastelt

def _heapreplace_max(heap, item):

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

returnitem = heap[0] # raises appropriate IndexError if heap is empty

heap[0] = item

_siftup_max(heap, 0)

return returnitem

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)

# '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 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 comparison 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 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 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 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)

def merge(*iterables, key=None, reverse=False):

'''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]

If *key* is not None, applies a key function to each element to determine

its sort order.

>>> list(merge(['dog', 'horse'], ['cat', 'fish', 'kangaroo'], key=len))

['dog', 'cat', 'fish', 'horse', 'kangaroo']

'''

h = []

h_append = h.append

if reverse:

_heapify = _heapify_max

_heappop = _heappop_max

_heapreplace = _heapreplace_max

direction = -1

else:

_heapify = heapify

_heappop = heappop

_heapreplace = heapreplace

direction = 1

if key is None:

for order, it in enumerate(map(iter, iterables)):

try:

next = it.__next__

h_append([next(), order * direction, next])

except StopIteration:

pass

_heapify(h)

while len(h) > 1:

try:

while True:

value, order, next = s = h[0]

yield value

s[0] = next() # raises StopIteration when exhausted

_heapreplace(h, s) # restore heap condition

except StopIteration:

_heappop(h) # remove empty iterator

if h:

# fast case when only a single iterator remains

value, order, next = h[0]

yield value

yield from next.__self__

return

for order, it in enumerate(map(iter, iterables)):

try:

next = it.__next__

value = next()

h_append([key(value), order * direction, value, next])

except StopIteration:

pass

_heapify(h)

while len(h) > 1:

try:

while True:

key_value, order, value, next = s = h[0]

yield value

value = next()

s[0] = key(value)

s[2] = value

_heapreplace(h, s)

except StopIteration:

_heappop(h)

if h:

key_value, order, value, next = h[0]

yield value

yield from next.__self__

# Algorithm notes for nlargest() and nsmallest()

# ==============================================

#

# Make a single pass over the data while keeping the k most extreme values

# in a heap. Memory consumption is limited to keeping k values in a list.

#

# Measured performance for random inputs:

#

# number of comparisons

# n inputs k-extreme values (average of 5 trials) % more than min()

# ------------- ---------------- --------------------- -----------------

# 1,000 100 3,317 231.7%

# 10,000 100 14,046 40.5%

# 100,000 100 105,749 5.7%

# 1,000,000 100 1,007,751 0.8%

# 10,000,000 100 10,009,401 0.1%

#

# Theoretical number of comparisons for k smallest of n random inputs:

#

# Step Comparisons Action

# ---- -------------------------- ---------------------------

# 1 1.66 * k heapify the first k-inputs

# 2 n - k compare remaining elements to top of heap

# 3 k * (1 + lg2(k)) * ln(n/k) replace the topmost value on the heap

# 4 k * lg2(k) - (k/2) final sort of the k most extreme values

#

# Combining and simplifying for a rough estimate gives:

#

# comparisons = n + k * (log(k, 2) * log(n/k) + log(k, 2) + log(n/k))

#

# Computing the number of comparisons for step 3:

# -----------------------------------------------

# * For the i-th new value from the iterable, the probability of being in the

# k most extreme values is k/i. For example, the probability of the 101st

# value seen being in the 100 most extreme values is 100/101.

# * If the value is a new extreme value, the cost of inserting it into the

# heap is 1 + log(k, 2).

# * The probability times the cost gives:

# (k/i) * (1 + log(k, 2))

# * Summing across the remaining n-k elements gives:

# sum((k/i) * (1 + log(k, 2)) for i in range(k+1, n+1))

# * This reduces to:

# (H(n) - H(k)) * k * (1 + log(k, 2))

# * Where H(n) is the n-th harmonic number estimated by:

# gamma = 0.5772156649

# H(n) = log(n, e) + gamma + 1 / (2 * n)

# http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Rate_of_divergence

# * Substituting the H(n) formula:

# comparisons = k * (1 + log(k, 2)) * (log(n/k, e) + (1/n - 1/k) / 2)

#

# Worst-case for step 3:

# ----------------------

# In the worst case, the input data is reversed sorted so that every new element

# must be inserted in the heap:

#

# comparisons = 1.66 * k + log(k, 2) * (n - k)

#

# Alternative Algorithms

# ----------------------

# Other algorithms were not used because they:

# 1) Took much more auxiliary memory,

# 2) Made multiple passes over the data.

# 3) Made more comparisons in common cases (small k, large n, semi-random input).

# See the more detailed comparison of approach at:

# http://code.activestate.com/recipes/577573-compare-algorithms-for-heapqsmallest

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()

if n == 1:

it = iter(iterable)

sentinel = object()

if key is None:

result = min(it, default=sentinel)

else:

result = min(it, default=sentinel, key=key)

return [] if result is sentinel else [result]

# 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 = iter(iterable)

# put the range(n) first so that zip() doesn't

# consume one too many elements from the iterator

result = [(elem, i) for i, elem in zip(range(n), it)]

if not result:

return result

_heapify_max(result)

top = result[0][0]

order = n

_heapreplace = _heapreplace_max

for elem in it:

if elem < top:

_heapreplace(result, (elem, order))

top, _order = result[0]

order += 1

result.sort()

return [elem for (elem, order) in result]

# General case, slowest method

it = iter(iterable)

result = [(key(elem), i, elem) for i, elem in zip(range(n), it)]

if not result:

return result

_heapify_max(result)

top = result[0][0]

order = n

_heapreplace = _heapreplace_max

for elem in it:

k = key(elem)

if k < top:

_heapreplace(result, (k, order, elem))

top, _order, _elem = result[0]

order += 1

result.sort()

return [elem for (k, order, elem) in result]

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()

if n == 1:

it = iter(iterable)

sentinel = object()

if key is None:

result = max(it, default=sentinel)

else:

result = max(it, default=sentinel, key=key)

return [] if result is sentinel else [result]

# 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 = iter(iterable)

result = [(elem, i) for i, elem in zip(range(0, -n, -1), it)]

if not result:

return result

heapify(result)

top = result[0][0]

order = -n

_heapreplace = heapreplace

for elem in it:

if top < elem:

_heapreplace(result, (elem, order))

top, _order = result[0]

order -= 1

result.sort(reverse=True)

return [elem for (elem, order) in result]

# General case, slowest method

it = iter(iterable)

result = [(key(elem), i, elem) for i, elem in zip(range(0, -n, -1), it)]

if not result:

return result

heapify(result)

top = result[0][0]

order = -n

_heapreplace = heapreplace

for elem in it:

k = key(elem)

if top < k:

_heapreplace(result, (k, order, elem))

top, _order, _elem = result[0]

order -= 1

result.sort(reverse=True)

return [elem for (k, order, elem) in result]

# If available, use C implementation

try:

from _heapq import *

except ImportError:

pass

try:

from _heapq import _heapreplace_max

except ImportError:

pass

try:

from _heapq import _heapify_max

except ImportError:

pass

try:

from _heapq import _heappop_max

except ImportError:

pass

if __name__ == "__main__":

import doctest

print(doctest.testmod())