+++ /dev/null
-"""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 = result[0][0]
- order += 1
- result.sort()
- return [r[0] for r 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 = result[0][0]
- order += 1
- result.sort()
- return [r[2] for r 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 = result[0][0]
- order -= 1
- result.sort(reverse=True)
- return [r[0] for r 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 = result[0][0]
- order -= 1
- result.sort(reverse=True)
- return [r[2] for r 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())
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