8.4.1. Basic Examples

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')

8.4.2. Priority Queue Implementation Notes

A priority queue is common use for a heap, and it presents several implementation challenges:

• Sort stability: how do you get two tasks with equal priorities to be returned in the order they were originally added?
• In the future with Python 3, tuple comparison breaks for (priority, task) pairs if the priorities are equal and the tasks do not have a default comparison order.
• If the priority of a task changes, how do you move it to a new position in the heap?
• Or if a pending task needs to be deleted, how do you find it and remove it from the queue?

A solution to the first two challenges is to store entries as 3-element list including the priority, an entry count, and the task. 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.

The remaining challenges revolve around finding a pending task and making changes to its priority or removing it entirely. Finding a task can be done with a dictionary pointing to an entry in the queue.

Removing the entry or changing its priority is more difficult because it would break the heap structure invariants. So, a possible solution is to mark an entry as invalid and optionally add a new entry with the revised priority:

pq = []                         # the priority queue list
counter = itertools.count(1)    # unique sequence count
task_finder = {}                # mapping of tasks to entries
INVALID = 0                     # mark an entry as deleted

def add_task(priority, task, count=None):
    if count is None:
        count = next(counter)
    entry = [priority, count, task]
    task_finder[task] = entry
    heappush(pq, entry)

def get_top_priority():
    while True:
        priority, count, task = heappop(pq)
        del task_finder[task]
        if count is not INVALID:
            return task

def delete_task(task):
    entry = task_finder[task]
    entry[1] = INVALID

def reprioritize(priority, task):
    entry = task_finder[task]
    add_task(priority, task, entry[1])
    entry[1] = INVALID

8.4.3. Theory

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 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. :-)

Footnotes