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There is no such thing as a "most efficient priority queue implementation" in any language.
A priority queue is all about trade-offs. See http://en.wikipedia.org/wiki/Priority_queue
You should choose one of these two, based on how you plan to use it:
O(log(N))insertion time andO(1)(findMin+deleteMin)* time, orO(1)insertion time andO(log(N))(findMin+deleteMin)* time
(* sidenote: the findMin time of most queues is almost always O(1), so here I mostly mean the deleteMin time can either be O(1) quick if the insertion time is O(log(N)) slow, or the deleteMin time must be O(log(N)) slow if the insertion time is O(1) fast. One should note that both may also be unnecessarily slow like with binary-tree based priority queues.)
In the latter case, you can choose to implement a priority queue with a Fibonacci heap: http://en.wikipedia.org/wiki/Heap_(data_structure)#Comparison_of_theoretic_bounds_for_variants (as you can see, heapq which is basically a binary tree, must necessarily have O(log(N)) for both insertion and findMin+deleteMin)
If you are dealing with data with special properties (such as bounded data), then you can achieve O(1) insertion and O(1) findMin+deleteMin time. You can only do this with certain kinds of data because otherwise you could abuse your priority queue to violate the O(N log(N)) bound on sorting. vEB trees kind of fall under a similar category, since you have a maximum set size (O(log(log(M)) is not referring to the number of elements, but the maximum number of elements) and thus you cannot circumvent the theoretical O(N log(N)) general-purpose comparison-sorting bound.
To implement any queue in any language, all you need is to define the insert(value) and extractMin() -> value operations. This generally just involves a minimal wrapping of the underlying heap; see http://en.wikipedia.org/wiki/Fibonacci_heap to implement your own, or use an off-the-shelf library of a similar heap like a Pairing Heap (a Google search revealed http://svn.python.org/projects/sandbox/trunk/collections/pairing_heap.py )
If you only care about which of the two you referenced are more efficient (the heapq-based code from http://docs.python.org/library/heapq.html#priority-queue-implementation-notes which you included above, versus Queue.PriorityQueue), then:
There doesn't seem to be any easily-findable discussion on the web as to what Queue.PriorityQueue is actually doing; you would have to source dive into the code, which is linked to from the help documentation: http://hg.python.org/cpython/file/2.7/Lib/Queue.py
Copy 224 def _put(self, item, heappush=heapq.heappush):
225 heappush(self.queue, item)
226
227 def _get(self, heappop=heapq.heappop):
228 return heappop(self.queue)
As we can see, Queue.PriorityQueue is also using heapq as an underlying mechanism. Therefore they are equally bad (asymptotically speaking). Queue.PriorityQueue may allow for parallel queries, so I would wager that it might have a very slightly constant-factor more of overhead. But because you know the underlying implementation (and asymptotic behavior) must be the same, the simplest way would simply be to run them on the same large dataset.
(Do note that Queue.PriorityQueue does not seem to have a way to remove entries, while heapq does. However this is a double-edged sword: Good priority queue implementations might possibly allow you to delete elements in O(1) or O(log(N)) time, but if you use the remove_task function you mention, and let those zombie tasks accumulate in your queue because you aren't extracting them off the min, then you will see asymptotic slowdown which you wouldn't otherwise see. Of course, you couldn't do this with Queue.PriorityQueue in the first place, so no comparison can be made here.)
The version in the Queue module is implemented using the heapq module, so they have equal efficiency for the underlying heap operations.
That said, the Queue version is slower because it adds locks, encapsulation, and a nice object oriented API.
The priority queue suggestions shown in the heapq docs are meant to show how to add additional capabilities to a priority queue (such as sort stability and the ability to change the priority of a previously enqueued task). If you don't need those capabilities, then the basic heappush and heappop functions will give you the fastest performance.
You can use Queue.PriorityQueue.
Recall that Python isn't strongly typed, so you can save anything you like: just make a tuple of (priority, thing) and you're set.
When using a priority queue, decrease-key is a must-have operation for many algorithms (Dijkstra's Algorithm, A*, OPTICS), I wonder why Python's built-in priority queue does not support it. None of the other answers supply a solution that supports this functionality.
A priority queue which also supports decrease-key operation is this implementation by Daniel Stutzbach worked perfectly for me with Python 3.5.
from heapdict import heapdict
hd = heapdict()
hd["two"] = 2
hd["one"] = 1
obj = hd.popitem()
print("object:",obj[0])
print("priority:",obj[1])
# object: one
# priority: 1