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New in version 2.3.
Source code: Lib/heapq.py
本模块提供heap queue algorithm 的实现，正如已经知道的priority queue algorithm
heaps 是一个任一父节点值小于或等于子节点值的二叉树This implementation uses arrays for which heap[k] <= heap[2*k+1] and heap[k] <= heap[2*k+2] for all k, counting elements from zero. 为了比较，不存在的元素被认为是无限大。堆的有趣的属性是它的最小元素总是根， heap[0].
下面的API与教科书堆算法在两个方面不同：（a）我们使用基于零的索引。这使得节点的索引和其子节点的索引之间的关系稍微不那么明显，但是更合适，因为Python使用基于零的索引。（b）我们的pop方法返回最小的项而不是最大的（在教科书中称为“最小堆”；但是“最大堆”适合于就地排序，所以“最大堆”在教材中更常见）。
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!
要创建堆，请使用初始化为[]的列表，或者您可以通过函数heapify()将填充列表转换为堆。
提供以下功能
将值 item push到 heap 中, 然后调用 heap 的调序算法。
弹出并返回堆的最小值 heap, 调整堆。If the heap is empty, IndexError is raised.
先 push item 到堆中, 然后弹出并返回这个堆 heap 中的最小值。这个组合操作比调用heappush() 再调用heappop() 更加的有效率.
New in version 2.6.
将一个 list x 就地转换成 heap, 只需要线性的时间。
先弹出并返回堆顶 heap, 再 push 新的 item.。堆的大小不会变动。If the heap is empty, IndexError is raised.
这个方法比先调用 heappop() 再调用 heappush() 更加有效率，而且在固定大小的堆中，这个方法更加的合适。The pop/push combination always returns an element from the heap and replaces it with item.
The value returned may be larger than the item added. If that isn’t desired, consider using heappushpop() instead. Its push/pop combination returns the smaller of the two values, leaving the larger value on the heap.
The module also offers three general purpose functions based on heaps.
合并多个已排序的输入到一个单一的排序输出（例如，合并来自多个日志文件的时间戳的条目） Returns an iterator over the sorted values.
Similar to sorted(itertools.chain(*iterables)) but returns an iterable, does not pull the data into memory all at once, and assumes that each of the input streams is already sorted (smallest to largest).
New in version 2.6.
Return a list with the n largest elements from the dataset defined by iterable. key, if provided, specifies a function of one argument that is used to extract a comparison key from each element in the iterable: key=str.lower Equivalent to: sorted(iterable, key=key, reverse=True)[:n]
New in version 2.4.
Changed in version 2.5: Added the optional key argument.
Return a list with the n smallest elements from the dataset defined by iterable. key, if provided, specifies a function of one argument that is used to extract a comparison key from each element in the iterable: key=str.lower Equivalent to: sorted(iterable, key=key)[:n]
New in version 2.4.
Changed in version 2.5: Added the optional key argument.
The latter two functions perform best for smaller values of n. For larger values, it is more efficient to use the sorted() function. Also, when n==1, it is more efficient to use the built-in min() and max() functions.
A heapsort can be implemented by pushing all values onto a heap and then popping off the smallest values one at a time:
>>> def heapsort(iterable):
... 'Equivalent to sorted(iterable)'
... h = []
... for value in iterable:
... heappush(h, value)
... return [heappop(h) for i in range(len(h))]
...
>>> heapsort([1, 3, 5, 7, 9, 2, 4, 6, 8, 0])
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
Heap elements can be tuples. This is useful for assigning comparison values (such as task priorities) alongside the main record being tracked:
>>> h = []
>>> heappush(h, (5, 'write code'))
>>> heappush(h, (7, 'release product'))
>>> heappush(h, (1, 'write spec'))
>>> heappush(h, (3, 'create tests'))
>>> heappop(h)
(1, 'write spec')
A priority queue is common use for a heap, and it presents several implementation challenges:
前两个挑战的解决方案是：存3个元素到堆中，包括 优先级，条目计数和任务的3元素列表。The entry count serves as a tie-breaker so that two tasks with the same priority are returned in the order they were added. And since no two entry counts are the same, the tuple comparison will never attempt to directly compare two tasks.
剩下的问题就是 如何找到一个已经存在的 task，修改它的优先级，或者移除它。找到 task 可以通过一个字典来指向 queue 中对应的 entry。
删除 queue中的 entry，或者修改 entry的 优先级就更加的复杂了，因为这将破坏 queue的结构。所以，一个可能的方案就是，我们将对应的 entry 添加一个标志为，用以表示删除。再新加一个 更新后的 entry。
pq = [] # list of entries arranged in a heap
entry_finder = {} # mapping of tasks to entries
REMOVED = '<removed-task>' # placeholder for a removed task
counter = itertools.count() # unique sequence count
def add_task(task, priority=0):
'Add a new task or update the priority of an existing task'
if task in entry_finder:
remove_task(task)
count = next(counter)
entry = [priority, count, task]
entry_finder[task] = entry
heappush(pq, entry)
def remove_task(task):
'Mark an existing task as REMOVED. Raise KeyError if not found.'
entry = entry_finder.pop(task)
entry[-1] = REMOVED
def pop_task():
'Remove and return the lowest priority task. Raise KeyError if empty.'
while pq:
priority, count, task = heappop(pq)
if task is not REMOVED:
del entry_finder[task]
return task
raise KeyError('pop from an empty 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.
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 log 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 schedules 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. :-)
Footnotes
| [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! :-) |