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| """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()) |