1:mod:`!heapq` --- Heap queue algorithm 2====================================== 3 4.. module:: heapq 5 :synopsis: Heap queue algorithm (a.k.a. priority queue). 6 7.. moduleauthor:: Kevin O'Connor 8.. sectionauthor:: Guido van Rossum <guido@python.org> 9.. sectionauthor:: François Pinard 10.. sectionauthor:: Raymond Hettinger 11 12**Source code:** :source:`Lib/heapq.py` 13 14-------------- 15 16This module provides an implementation of the heap queue algorithm, also known 17as the priority queue algorithm. 18 19Heaps are binary trees for which every parent node has a value less than or 20equal to any of its children. We refer to this condition as the heap invariant. 21 22This implementation uses arrays for which 23``heap[k] <= heap[2*k+1]`` and ``heap[k] <= heap[2*k+2]`` for all *k*, counting 24elements from zero. For the sake of comparison, non-existing elements are 25considered to be infinite. The interesting property of a heap is that its 26smallest element is always the root, ``heap[0]``. 27 28The API below differs from textbook heap algorithms in two aspects: (a) We use 29zero-based indexing. This makes the relationship between the index for a node 30and the indexes for its children slightly less obvious, but is more suitable 31since Python uses zero-based indexing. (b) Our pop method returns the smallest 32item, not the largest (called a "min heap" in textbooks; a "max heap" is more 33common in texts because of its suitability for in-place sorting). 34 35These two make it possible to view the heap as a regular Python list without 36surprises: ``heap[0]`` is the smallest item, and ``heap.sort()`` maintains the 37heap invariant! 38 39To create a heap, use a list initialized to ``[]``, or you can transform a 40populated list into a heap via function :func:`heapify`. 41 42The following functions are provided: 43 44 45.. function:: heappush(heap, item) 46 47 Push the value *item* onto the *heap*, maintaining the heap invariant. 48 49 50.. function:: heappop(heap) 51 52 Pop and return the smallest item from the *heap*, maintaining the heap 53 invariant. If the heap is empty, :exc:`IndexError` is raised. To access the 54 smallest item without popping it, use ``heap[0]``. 55 56 57.. function:: heappushpop(heap, item) 58 59 Push *item* on the heap, then pop and return the smallest item from the 60 *heap*. The combined action runs more efficiently than :func:`heappush` 61 followed by a separate call to :func:`heappop`. 62 63 64.. function:: heapify(x) 65 66 Transform list *x* into a heap, in-place, in linear time. 67 68 69.. function:: heapreplace(heap, item) 70 71 Pop and return the smallest item from the *heap*, and also push the new *item*. 72 The heap size doesn't change. If the heap is empty, :exc:`IndexError` is raised. 73 74 This one step operation is more efficient than a :func:`heappop` followed by 75 :func:`heappush` and can be more appropriate when using a fixed-size heap. 76 The pop/push combination always returns an element from the heap and replaces 77 it with *item*. 78 79 The value returned may be larger than the *item* added. If that isn't 80 desired, consider using :func:`heappushpop` instead. Its push/pop 81 combination returns the smaller of the two values, leaving the larger value 82 on the heap. 83 84 85The module also offers three general purpose functions based on heaps. 86 87 88.. function:: merge(*iterables, key=None, reverse=False) 89 90 Merge multiple sorted inputs into a single sorted output (for example, merge 91 timestamped entries from multiple log files). Returns an :term:`iterator` 92 over the sorted values. 93 94 Similar to ``sorted(itertools.chain(*iterables))`` but returns an iterable, does 95 not pull the data into memory all at once, and assumes that each of the input 96 streams is already sorted (smallest to largest). 97 98 Has two optional arguments which must be specified as keyword arguments. 99 100 *key* specifies a :term:`key function` of one argument that is used to 101 extract a comparison key from each input element. The default value is 102 ``None`` (compare the elements directly). 103 104 *reverse* is a boolean value. If set to ``True``, then the input elements 105 are merged as if each comparison were reversed. To achieve behavior similar 106 to ``sorted(itertools.chain(*iterables), reverse=True)``, all iterables must 107 be sorted from largest to smallest. 108 109 .. versionchanged:: 3.5 110 Added the optional *key* and *reverse* parameters. 111 112 113.. function:: nlargest(n, iterable, key=None) 114 115 Return a list with the *n* largest elements from the dataset defined by 116 *iterable*. *key*, if provided, specifies a function of one argument that is 117 used to extract a comparison key from each element in *iterable* (for example, 118 ``key=str.lower``). Equivalent to: ``sorted(iterable, key=key, 119 reverse=True)[:n]``. 120 121 122.. function:: nsmallest(n, iterable, key=None) 123 124 Return a list with the *n* smallest elements from the dataset defined by 125 *iterable*. *key*, if provided, specifies a function of one argument that is 126 used to extract a comparison key from each element in *iterable* (for example, 127 ``key=str.lower``). Equivalent to: ``sorted(iterable, key=key)[:n]``. 128 129 130The latter two functions perform best for smaller values of *n*. For larger 131values, it is more efficient to use the :func:`sorted` function. Also, when 132``n==1``, it is more efficient to use the built-in :func:`min` and :func:`max` 133functions. If repeated usage of these functions is required, consider turning 134the iterable into an actual heap. 135 136 137Basic Examples 138-------------- 139 140A `heapsort <https://en.wikipedia.org/wiki/Heapsort>`_ can be implemented by 141pushing all values onto a heap and then popping off the smallest values one at a 142time:: 143 144 >>> def heapsort(iterable): 145 ... h = [] 146 ... for value in iterable: 147 ... heappush(h, value) 148 ... return [heappop(h) for i in range(len(h))] 149 ... 150 >>> heapsort([1, 3, 5, 7, 9, 2, 4, 6, 8, 0]) 151 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] 152 153This is similar to ``sorted(iterable)``, but unlike :func:`sorted`, this 154implementation is not stable. 155 156Heap elements can be tuples. This is useful for assigning comparison values 157(such as task priorities) alongside the main record being tracked:: 158 159 >>> h = [] 160 >>> heappush(h, (5, 'write code')) 161 >>> heappush(h, (7, 'release product')) 162 >>> heappush(h, (1, 'write spec')) 163 >>> heappush(h, (3, 'create tests')) 164 >>> heappop(h) 165 (1, 'write spec') 166 167 168Priority Queue Implementation Notes 169----------------------------------- 170 171A `priority queue <https://en.wikipedia.org/wiki/Priority_queue>`_ is common use 172for a heap, and it presents several implementation challenges: 173 174* Sort stability: how do you get two tasks with equal priorities to be returned 175 in the order they were originally added? 176 177* Tuple comparison breaks for (priority, task) pairs if the priorities are equal 178 and the tasks do not have a default comparison order. 179 180* If the priority of a task changes, how do you move it to a new position in 181 the heap? 182 183* Or if a pending task needs to be deleted, how do you find it and remove it 184 from the queue? 185 186A solution to the first two challenges is to store entries as 3-element list 187including the priority, an entry count, and the task. The entry count serves as 188a tie-breaker so that two tasks with the same priority are returned in the order 189they were added. And since no two entry counts are the same, the tuple 190comparison will never attempt to directly compare two tasks. 191 192Another solution to the problem of non-comparable tasks is to create a wrapper 193class that ignores the task item and only compares the priority field:: 194 195 from dataclasses import dataclass, field 196 from typing import Any 197 198 @dataclass(order=True) 199 class PrioritizedItem: 200 priority: int 201 item: Any=field(compare=False) 202 203The remaining challenges revolve around finding a pending task and making 204changes to its priority or removing it entirely. Finding a task can be done 205with a dictionary pointing to an entry in the queue. 206 207Removing the entry or changing its priority is more difficult because it would 208break the heap structure invariants. So, a possible solution is to mark the 209entry as removed and add a new entry with the revised priority:: 210 211 pq = [] # list of entries arranged in a heap 212 entry_finder = {} # mapping of tasks to entries 213 REMOVED = '<removed-task>' # placeholder for a removed task 214 counter = itertools.count() # unique sequence count 215 216 def add_task(task, priority=0): 217 'Add a new task or update the priority of an existing task' 218 if task in entry_finder: 219 remove_task(task) 220 count = next(counter) 221 entry = [priority, count, task] 222 entry_finder[task] = entry 223 heappush(pq, entry) 224 225 def remove_task(task): 226 'Mark an existing task as REMOVED. Raise KeyError if not found.' 227 entry = entry_finder.pop(task) 228 entry[-1] = REMOVED 229 230 def pop_task(): 231 'Remove and return the lowest priority task. Raise KeyError if empty.' 232 while pq: 233 priority, count, task = heappop(pq) 234 if task is not REMOVED: 235 del entry_finder[task] 236 return task 237 raise KeyError('pop from an empty priority queue') 238 239 240Theory 241------ 242 243Heaps are arrays for which ``a[k] <= a[2*k+1]`` and ``a[k] <= a[2*k+2]`` for all 244*k*, counting elements from 0. For the sake of comparison, non-existing 245elements are considered to be infinite. The interesting property of a heap is 246that ``a[0]`` is always its smallest element. 247 248The strange invariant above is meant to be an efficient memory representation 249for a tournament. The numbers below are *k*, not ``a[k]``:: 250 251 0 252 253 1 2 254 255 3 4 5 6 256 257 7 8 9 10 11 12 13 14 258 259 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 260 261In the tree above, each cell *k* is topping ``2*k+1`` and ``2*k+2``. In a usual 262binary tournament we see in sports, each cell is the winner over the two cells 263it tops, and we can trace the winner down the tree to see all opponents s/he 264had. However, in many computer applications of such tournaments, we do not need 265to trace the history of a winner. To be more memory efficient, when a winner is 266promoted, we try to replace it by something else at a lower level, and the rule 267becomes that a cell and the two cells it tops contain three different items, but 268the top cell "wins" over the two topped cells. 269 270If this heap invariant is protected at all time, index 0 is clearly the overall 271winner. The simplest algorithmic way to remove it and find the "next" winner is 272to move some loser (let's say cell 30 in the diagram above) into the 0 position, 273and then percolate this new 0 down the tree, exchanging values, until the 274invariant is re-established. This is clearly logarithmic on the total number of 275items in the tree. By iterating over all items, you get an *O*\ (*n* log *n*) sort. 276 277A nice feature of this sort is that you can efficiently insert new items while 278the sort is going on, provided that the inserted items are not "better" than the 279last 0'th element you extracted. This is especially useful in simulation 280contexts, where the tree holds all incoming events, and the "win" condition 281means the smallest scheduled time. When an event schedules other events for 282execution, they are scheduled into the future, so they can easily go into the 283heap. So, a heap is a good structure for implementing schedulers (this is what 284I used for my MIDI sequencer :-). 285 286Various structures for implementing schedulers have been extensively studied, 287and heaps are good for this, as they are reasonably speedy, the speed is almost 288constant, and the worst case is not much different than the average case. 289However, there are other representations which are more efficient overall, yet 290the worst cases might be terrible. 291 292Heaps are also very useful in big disk sorts. You most probably all know that a 293big sort implies producing "runs" (which are pre-sorted sequences, whose size is 294usually related to the amount of CPU memory), followed by a merging passes for 295these runs, which merging is often very cleverly organised [#]_. It is very 296important that the initial sort produces the longest runs possible. Tournaments 297are a good way to achieve that. If, using all the memory available to hold a 298tournament, you replace and percolate items that happen to fit the current run, 299you'll produce runs which are twice the size of the memory for random input, and 300much better for input fuzzily ordered. 301 302Moreover, if you output the 0'th item on disk and get an input which may not fit 303in the current tournament (because the value "wins" over the last output value), 304it cannot fit in the heap, so the size of the heap decreases. The freed memory 305could be cleverly reused immediately for progressively building a second heap, 306which grows at exactly the same rate the first heap is melting. When the first 307heap completely vanishes, you switch heaps and start a new run. Clever and 308quite effective! 309 310In a word, heaps are useful memory structures to know. I use them in a few 311applications, and I think it is good to keep a 'heap' module around. :-) 312 313.. rubric:: Footnotes 314 315.. [#] The disk balancing algorithms which are current, nowadays, are more annoying 316 than clever, and this is a consequence of the seeking capabilities of the disks. 317 On devices which cannot seek, like big tape drives, the story was quite 318 different, and one had to be very clever to ensure (far in advance) that each 319 tape movement will be the most effective possible (that is, will best 320 participate at "progressing" the merge). Some tapes were even able to read 321 backwards, and this was also used to avoid the rewinding time. Believe me, real 322 good tape sorts were quite spectacular to watch! From all times, sorting has 323 always been a Great Art! :-) 324