1 :mod:`heapq` --- Heap queue algorithm 2 ===================================== 3 4 .. module:: heapq 5 :synopsis: Heap queue algorithm (a.k.a. priority queue). 6 .. moduleauthor:: Kevin O'Connor 7 .. sectionauthor:: Guido van Rossum <guido (a] python.org> 8 .. sectionauthor:: Franois Pinard 9 .. sectionauthor:: Raymond Hettinger 10 11 .. versionadded:: 2.3 12 13 **Source code:** :source:`Lib/heapq.py` 14 15 -------------- 16 17 This module provides an implementation of the heap queue algorithm, also known 18 as the priority queue algorithm. 19 20 Heaps are binary trees for which every parent node has a value less than or 21 equal to any of its children. This implementation uses arrays for which 22 ``heap[k] <= heap[2*k+1]`` and ``heap[k] <= heap[2*k+2]`` for all *k*, counting 23 elements from zero. For the sake of comparison, non-existing elements are 24 considered to be infinite. The interesting property of a heap is that its 25 smallest element is always the root, ``heap[0]``. 26 27 The API below differs from textbook heap algorithms in two aspects: (a) We use 28 zero-based indexing. This makes the relationship between the index for a node 29 and the indexes for its children slightly less obvious, but is more suitable 30 since Python uses zero-based indexing. (b) Our pop method returns the smallest 31 item, not the largest (called a "min heap" in textbooks; a "max heap" is more 32 common in texts because of its suitability for in-place sorting). 33 34 These two make it possible to view the heap as a regular Python list without 35 surprises: ``heap[0]`` is the smallest item, and ``heap.sort()`` maintains the 36 heap invariant! 37 38 To create a heap, use a list initialized to ``[]``, or you can transform a 39 populated list into a heap via function :func:`heapify`. 40 41 The following functions are provided: 42 43 44 .. function:: heappush(heap, item) 45 46 Push the value *item* onto the *heap*, maintaining the heap invariant. 47 48 49 .. function:: heappop(heap) 50 51 Pop and return the smallest item from the *heap*, maintaining the heap 52 invariant. If the heap is empty, :exc:`IndexError` is raised. To access the 53 smallest item without popping it, use ``heap[0]``. 54 55 .. function:: heappushpop(heap, item) 56 57 Push *item* on the heap, then pop and return the smallest item from the 58 *heap*. The combined action runs more efficiently than :func:`heappush` 59 followed by a separate call to :func:`heappop`. 60 61 .. versionadded:: 2.6 62 63 .. function:: heapify(x) 64 65 Transform list *x* into a heap, in-place, in linear time. 66 67 68 .. function:: heapreplace(heap, item) 69 70 Pop and return the smallest item from the *heap*, and also push the new *item*. 71 The heap size doesn't change. If the heap is empty, :exc:`IndexError` is raised. 72 73 This one step operation is more efficient than a :func:`heappop` followed by 74 :func:`heappush` and can be more appropriate when using a fixed-size heap. 75 The pop/push combination always returns an element from the heap and replaces 76 it with *item*. 77 78 The value returned may be larger than the *item* added. If that isn't 79 desired, consider using :func:`heappushpop` instead. Its push/pop 80 combination returns the smaller of the two values, leaving the larger value 81 on the heap. 82 83 84 The module also offers three general purpose functions based on heaps. 85 86 87 .. function:: merge(*iterables) 88 89 Merge multiple sorted inputs into a single sorted output (for example, merge 90 timestamped entries from multiple log files). Returns an :term:`iterator` 91 over the sorted values. 92 93 Similar to ``sorted(itertools.chain(*iterables))`` but returns an iterable, does 94 not pull the data into memory all at once, and assumes that each of the input 95 streams is already sorted (smallest to largest). 96 97 .. versionadded:: 2.6 98 99 100 .. function:: nlargest(n, iterable[, key]) 101 102 Return a list with the *n* largest elements from the dataset defined by 103 *iterable*. *key*, if provided, specifies a function of one argument that is 104 used to extract a comparison key from each element in the iterable: 105 ``key=str.lower`` Equivalent to: ``sorted(iterable, key=key, 106 reverse=True)[:n]`` 107 108 .. versionadded:: 2.4 109 110 .. versionchanged:: 2.5 111 Added the optional *key* argument. 112 113 114 .. function:: nsmallest(n, iterable[, key]) 115 116 Return a list with the *n* smallest elements from the dataset defined by 117 *iterable*. *key*, if provided, specifies a function of one argument that is 118 used to extract a comparison key from each element in the iterable: 119 ``key=str.lower`` Equivalent to: ``sorted(iterable, key=key)[:n]`` 120 121 .. versionadded:: 2.4 122 123 .. versionchanged:: 2.5 124 Added the optional *key* argument. 125 126 The latter two functions perform best for smaller values of *n*. For larger 127 values, it is more efficient to use the :func:`sorted` function. Also, when 128 ``n==1``, it is more efficient to use the built-in :func:`min` and :func:`max` 129 functions. If repeated usage of these functions is required, consider turning 130 the iterable into an actual heap. 131 132 133 Basic Examples 134 -------------- 135 136 A `heapsort <https://en.wikipedia.org/wiki/Heapsort>`_ can be implemented by 137 pushing all values onto a heap and then popping off the smallest values one at a 138 time:: 139 140 >>> def heapsort(iterable): 141 ... h = [] 142 ... for value in iterable: 143 ... heappush(h, value) 144 ... return [heappop(h) for i in range(len(h))] 145 ... 146 >>> heapsort([1, 3, 5, 7, 9, 2, 4, 6, 8, 0]) 147 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] 148 149 This is similar to ``sorted(iterable)``, but unlike :func:`sorted`, this 150 implementation is not stable. 151 152 Heap elements can be tuples. This is useful for assigning comparison values 153 (such as task priorities) alongside the main record being tracked:: 154 155 >>> h = [] 156 >>> heappush(h, (5, 'write code')) 157 >>> heappush(h, (7, 'release product')) 158 >>> heappush(h, (1, 'write spec')) 159 >>> heappush(h, (3, 'create tests')) 160 >>> heappop(h) 161 (1, 'write spec') 162 163 164 Priority Queue Implementation Notes 165 ----------------------------------- 166 167 A `priority queue <https://en.wikipedia.org/wiki/Priority_queue>`_ is common use 168 for a heap, and it presents several implementation challenges: 169 170 * Sort stability: how do you get two tasks with equal priorities to be returned 171 in the order they were originally added? 172 173 * In the future with Python 3, tuple comparison breaks for (priority, task) 174 pairs if the priorities are equal and the tasks do not have a default 175 comparison order. 176 177 * If the priority of a task changes, how do you move it to a new position in 178 the heap? 179 180 * Or if a pending task needs to be deleted, how do you find it and remove it 181 from the queue? 182 183 A solution to the first two challenges is to store entries as 3-element list 184 including the priority, an entry count, and the task. The entry count serves as 185 a tie-breaker so that two tasks with the same priority are returned in the order 186 they were added. And since no two entry counts are the same, the tuple 187 comparison will never attempt to directly compare two tasks. 188 189 The remaining challenges revolve around finding a pending task and making 190 changes to its priority or removing it entirely. Finding a task can be done 191 with a dictionary pointing to an entry in the queue. 192 193 Removing the entry or changing its priority is more difficult because it would 194 break the heap structure invariants. So, a possible solution is to mark the 195 existing entry as removed and add a new entry with the revised priority:: 196 197 pq = [] # list of entries arranged in a heap 198 entry_finder = {} # mapping of tasks to entries 199 REMOVED = '<removed-task>' # placeholder for a removed task 200 counter = itertools.count() # unique sequence count 201 202 def add_task(task, priority=0): 203 'Add a new task or update the priority of an existing task' 204 if task in entry_finder: 205 remove_task(task) 206 count = next(counter) 207 entry = [priority, count, task] 208 entry_finder[task] = entry 209 heappush(pq, entry) 210 211 def remove_task(task): 212 'Mark an existing task as REMOVED. Raise KeyError if not found.' 213 entry = entry_finder.pop(task) 214 entry[-1] = REMOVED 215 216 def pop_task(): 217 'Remove and return the lowest priority task. Raise KeyError if empty.' 218 while pq: 219 priority, count, task = heappop(pq) 220 if task is not REMOVED: 221 del entry_finder[task] 222 return task 223 raise KeyError('pop from an empty priority queue') 224 225 226 Theory 227 ------ 228 229 Heaps are arrays for which ``a[k] <= a[2*k+1]`` and ``a[k] <= a[2*k+2]`` for all 230 *k*, counting elements from 0. For the sake of comparison, non-existing 231 elements are considered to be infinite. The interesting property of a heap is 232 that ``a[0]`` is always its smallest element. 233 234 The strange invariant above is meant to be an efficient memory representation 235 for a tournament. The numbers below are *k*, not ``a[k]``:: 236 237 0 238 239 1 2 240 241 3 4 5 6 242 243 7 8 9 10 11 12 13 14 244 245 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 246 247 In the tree above, each cell *k* is topping ``2*k+1`` and ``2*k+2``. In a usual 248 binary tournament we see in sports, each cell is the winner over the two cells 249 it tops, and we can trace the winner down the tree to see all opponents s/he 250 had. However, in many computer applications of such tournaments, we do not need 251 to trace the history of a winner. To be more memory efficient, when a winner is 252 promoted, we try to replace it by something else at a lower level, and the rule 253 becomes that a cell and the two cells it tops contain three different items, but 254 the top cell "wins" over the two topped cells. 255 256 If this heap invariant is protected at all time, index 0 is clearly the overall 257 winner. The simplest algorithmic way to remove it and find the "next" winner is 258 to move some loser (let's say cell 30 in the diagram above) into the 0 position, 259 and then percolate this new 0 down the tree, exchanging values, until the 260 invariant is re-established. This is clearly logarithmic on the total number of 261 items in the tree. By iterating over all items, you get an O(n log n) sort. 262 263 A nice feature of this sort is that you can efficiently insert new items while 264 the sort is going on, provided that the inserted items are not "better" than the 265 last 0'th element you extracted. This is especially useful in simulation 266 contexts, where the tree holds all incoming events, and the "win" condition 267 means the smallest scheduled time. When an event schedules other events for 268 execution, they are scheduled into the future, so they can easily go into the 269 heap. So, a heap is a good structure for implementing schedulers (this is what 270 I used for my MIDI sequencer :-). 271 272 Various structures for implementing schedulers have been extensively studied, 273 and heaps are good for this, as they are reasonably speedy, the speed is almost 274 constant, and the worst case is not much different than the average case. 275 However, there are other representations which are more efficient overall, yet 276 the worst cases might be terrible. 277 278 Heaps are also very useful in big disk sorts. You most probably all know that a 279 big sort implies producing "runs" (which are pre-sorted sequences, whose size is 280 usually related to the amount of CPU memory), followed by a merging passes for 281 these runs, which merging is often very cleverly organised [#]_. It is very 282 important that the initial sort produces the longest runs possible. Tournaments 283 are a good way to achieve that. If, using all the memory available to hold a 284 tournament, you replace and percolate items that happen to fit the current run, 285 you'll produce runs which are twice the size of the memory for random input, and 286 much better for input fuzzily ordered. 287 288 Moreover, if you output the 0'th item on disk and get an input which may not fit 289 in the current tournament (because the value "wins" over the last output value), 290 it cannot fit in the heap, so the size of the heap decreases. The freed memory 291 could be cleverly reused immediately for progressively building a second heap, 292 which grows at exactly the same rate the first heap is melting. When the first 293 heap completely vanishes, you switch heaps and start a new run. Clever and 294 quite effective! 295 296 In a word, heaps are useful memory structures to know. I use them in a few 297 applications, and I think it is good to keep a 'heap' module around. :-) 298 299 .. rubric:: Footnotes 300 301 .. [#] The disk balancing algorithms which are current, nowadays, are more annoying 302 than clever, and this is a consequence of the seeking capabilities of the disks. 303 On devices which cannot seek, like big tape drives, the story was quite 304 different, and one had to be very clever to ensure (far in advance) that each 305 tape movement will be the most effective possible (that is, will best 306 participate at "progressing" the merge). Some tapes were even able to read 307 backwards, and this was also used to avoid the rewinding time. Believe me, real 308 good tape sorts were quite spectacular to watch! From all times, sorting has 309 always been a Great Art! :-) 310 311
