1 """Heap queue algorithm (a.k.a. priority queue).
2
3 Heaps are arrays for which a[k] <= a[2*k+1] and a[k] <= a[2*k+2] for
4 all k, counting elements from 0. For the sake of comparison,
5 non-existing elements are considered to be infinite. The interesting
6 property of a heap is that a[0] is always its smallest element.
7
8 Usage:
9
10 heap = [] # creates an empty heap
11 heappush(heap, item) # pushes a new item on the heap
12 item = heappop(heap) # pops the smallest item from the heap
13 item = heap[0] # smallest item on the heap without popping it
14 heapify(x) # transforms list into a heap, in-place, in linear time
15 item = heappushpop(heap, item) # pushes a new item and then returns
16 # the smallest item; the heap size is unchanged
17 item = heapreplace(heap, item) # pops and returns smallest item, and adds
18 # new item; the heap size is unchanged
19
20 Our API differs from textbook heap algorithms as follows:
21
22 - We use 0-based indexing. This makes the relationship between the
23 index for a node and the indexes for its children slightly less
24 obvious, but is more suitable since Python uses 0-based indexing.
25
26 - Our heappop() method returns the smallest item, not the largest.
27
28 These two make it possible to view the heap as a regular Python list
29 without surprises: heap[0] is the smallest item, and heap.sort()
30 maintains the heap invariant!
31 """
32
33 # Original code by Kevin O'Connor, augmented by Tim Peters and Raymond Hettinger
34
35 __about__ = """Heap queues
36
37 [explanation by François Pinard]
38
39 Heaps are arrays for which a[k] <= a[2*k+1] and a[k] <= a[2*k+2] for
40 all k, counting elements from 0. For the sake of comparison,
41 non-existing elements are considered to be infinite. The interesting
42 property of a heap is that a[0] is always its smallest element.
43
44 The strange invariant above is meant to be an efficient memory
45 representation for a tournament. The numbers below are `k', not a[k]:
46
47 0
48
49 1 2
50
51 3 4 5 6
52
53 7 8 9 10 11 12 13 14
54
55 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
56
57
58 In the tree above, each cell `k' is topping `2*k+1' and `2*k+2'. In
59 a usual binary tournament we see in sports, each cell is the winner
60 over the two cells it tops, and we can trace the winner down the tree
61 to see all opponents s/he had. However, in many computer applications
62 of such tournaments, we do not need to trace the history of a winner.
63 To be more memory efficient, when a winner is promoted, we try to
64 replace it by something else at a lower level, and the rule becomes
65 that a cell and the two cells it tops contain three different items,
66 but the top cell "wins" over the two topped cells.
67
68 If this heap invariant is protected at all time, index 0 is clearly
69 the overall winner. The simplest algorithmic way to remove it and
70 find the "next" winner is to move some loser (let's say cell 30 in the
71 diagram above) into the 0 position, and then percolate this new 0 down
72 the tree, exchanging values, until the invariant is re-established.
73 This is clearly logarithmic on the total number of items in the tree.
74 By iterating over all items, you get an O(n ln n) sort.
75
76 A nice feature of this sort is that you can efficiently insert new
77 items while the sort is going on, provided that the inserted items are
78 not "better" than the last 0'th element you extracted. This is
79 especially useful in simulation contexts, where the tree holds all
80 incoming events, and the "win" condition means the smallest scheduled
81 time. When an event schedule other events for execution, they are
82 scheduled into the future, so they can easily go into the heap. So, a
83 heap is a good structure for implementing schedulers (this is what I
84 used for my MIDI sequencer :-).
85
86 Various structures for implementing schedulers have been extensively
87 studied, and heaps are good for this, as they are reasonably speedy,
88 the speed is almost constant, and the worst case is not much different
89 than the average case. However, there are other representations which
90 are more efficient overall, yet the worst cases might be terrible.
91
92 Heaps are also very useful in big disk sorts. You most probably all
93 know that a big sort implies producing "runs" (which are pre-sorted
94 sequences, which size is usually related to the amount of CPU memory),
95 followed by a merging passes for these runs, which merging is often
96 very cleverly organised[1]. It is very important that the initial
97 sort produces the longest runs possible. Tournaments are a good way
98 to that. If, using all the memory available to hold a tournament, you
99 replace and percolate items that happen to fit the current run, you'll
100 produce runs which are twice the size of the memory for random input,
101 and much better for input fuzzily ordered.
102
103 Moreover, if you output the 0'th item on disk and get an input which
104 may not fit in the current tournament (because the value "wins" over
105 the last output value), it cannot fit in the heap, so the size of the
106 heap decreases. The freed memory could be cleverly reused immediately
107 for progressively building a second heap, which grows at exactly the
108 same rate the first heap is melting. When the first heap completely
109 vanishes, you switch heaps and start a new run. Clever and quite
110 effective!
111
112 In a word, heaps are useful memory structures to know. I use them in
113 a few applications, and I think it is good to keep a `heap' module
114 around. :-)
115
116 --------------------
117 [1] The disk balancing algorithms which are current, nowadays, are
118 more annoying than clever, and this is a consequence of the seeking
119 capabilities of the disks. On devices which cannot seek, like big
120 tape drives, the story was quite different, and one had to be very
121 clever to ensure (far in advance) that each tape movement will be the
122 most effective possible (that is, will best participate at
123 "progressing" the merge). Some tapes were even able to read
124 backwards, and this was also used to avoid the rewinding time.
125 Believe me, real good tape sorts were quite spectacular to watch!
126 From all times, sorting has always been a Great Art! :-)
127 """
128
129 __all__ = ['heappush', 'heappop', 'heapify', 'heapreplace', 'merge',
130 'nlargest', 'nsmallest', 'heappushpop']
131
132 def heappush(heap, item):
133 """Push item onto heap, maintaining the heap invariant."""
134 heap.append(item)
135 _siftdown(heap, 0, len(heap)-1)
136
137 def heappop(heap):
138 """Pop the smallest item off the heap, maintaining the heap invariant."""
139 lastelt = heap.pop() # raises appropriate IndexError if heap is empty
140 if heap:
141 returnitem = heap[0]
142 heap[0] = lastelt
143 _siftup(heap, 0)
144 return returnitem
145 return lastelt
146
147 def heapreplace(heap, item):
148 """Pop and return the current smallest value, and add the new item.
149
150 This is more efficient than heappop() followed by heappush(), and can be
151 more appropriate when using a fixed-size heap. Note that the value
152 returned may be larger than item! That constrains reasonable uses of
153 this routine unless written as part of a conditional replacement:
154
155 if item > heap[0]:
156 item = heapreplace(heap, item)
157 """
158 returnitem = heap[0] # raises appropriate IndexError if heap is empty
159 heap[0] = item
160 _siftup(heap, 0)
161 return returnitem
162
163 def heappushpop(heap, item):
164 """Fast version of a heappush followed by a heappop."""
165 if heap and heap[0] < item:
166 item, heap[0] = heap[0], item
167 _siftup(heap, 0)
168 return item
169
170 def heapify(x):
171 """Transform list into a heap, in-place, in O(len(x)) time."""
172 n = len(x)
173 # Transform bottom-up. The largest index there's any point to looking at
174 # is the largest with a child index in-range, so must have 2*i + 1 < n,
175 # or i < (n-1)/2. If n is even = 2*j, this is (2*j-1)/2 = j-1/2 so
176 # j-1 is the largest, which is n//2 - 1. If n is odd = 2*j+1, this is
177 # (2*j+1-1)/2 = j so j-1 is the largest, and that's again n//2-1.
178 for i in reversed(range(n//2)):
179 _siftup(x, i)
180
181 def _heappop_max(heap):
182 """Maxheap version of a heappop."""
183 lastelt = heap.pop() # raises appropriate IndexError if heap is empty
184 if heap:
185 returnitem = heap[0]
186 heap[0] = lastelt
187 _siftup_max(heap, 0)
188 return returnitem
189 return lastelt
190
191 def _heapreplace_max(heap, item):
192 """Maxheap version of a heappop followed by a heappush."""
193 returnitem = heap[0] # raises appropriate IndexError if heap is empty
194 heap[0] = item
195 _siftup_max(heap, 0)
196 return returnitem
197
198 def _heapify_max(x):
199 """Transform list into a maxheap, in-place, in O(len(x)) time."""
200 n = len(x)
201 for i in reversed(range(n//2)):
202 _siftup_max(x, i)
203
204 # 'heap' is a heap at all indices >= startpos, except possibly for pos. pos
205 # is the index of a leaf with a possibly out-of-order value. Restore the
206 # heap invariant.
207 def _siftdown(heap, startpos, pos):
208 newitem = heap[pos]
209 # Follow the path to the root, moving parents down until finding a place
210 # newitem fits.
211 while pos > startpos:
212 parentpos = (pos - 1) >> 1
213 parent = heap[parentpos]
214 if newitem < parent:
215 heap[pos] = parent
216 pos = parentpos
217 continue
218 break
219 heap[pos] = newitem
220
221 # The child indices of heap index pos are already heaps, and we want to make
222 # a heap at index pos too. We do this by bubbling the smaller child of
223 # pos up (and so on with that child's children, etc) until hitting a leaf,
224 # then using _siftdown to move the oddball originally at index pos into place.
225 #
226 # We *could* break out of the loop as soon as we find a pos where newitem <=
227 # both its children, but turns out that's not a good idea, and despite that
228 # many books write the algorithm that way. During a heap pop, the last array
229 # element is sifted in, and that tends to be large, so that comparing it
230 # against values starting from the root usually doesn't pay (= usually doesn't
231 # get us out of the loop early). See Knuth, Volume 3, where this is
232 # explained and quantified in an exercise.
233 #
234 # Cutting the # of comparisons is important, since these routines have no
235 # way to extract "the priority" from an array element, so that intelligence
236 # is likely to be hiding in custom comparison methods, or in array elements
237 # storing (priority, record) tuples. Comparisons are thus potentially
238 # expensive.
239 #
240 # On random arrays of length 1000, making this change cut the number of
241 # comparisons made by heapify() a little, and those made by exhaustive
242 # heappop() a lot, in accord with theory. Here are typical results from 3
243 # runs (3 just to demonstrate how small the variance is):
244 #
245 # Compares needed by heapify Compares needed by 1000 heappops
246 # -------------------------- --------------------------------
247 # 1837 cut to 1663 14996 cut to 8680
248 # 1855 cut to 1659 14966 cut to 8678
249 # 1847 cut to 1660 15024 cut to 8703
250 #
251 # Building the heap by using heappush() 1000 times instead required
252 # 2198, 2148, and 2219 compares: heapify() is more efficient, when
253 # you can use it.
254 #
255 # The total compares needed by list.sort() on the same lists were 8627,
256 # 8627, and 8632 (this should be compared to the sum of heapify() and
257 # heappop() compares): list.sort() is (unsurprisingly!) more efficient
258 # for sorting.
259
260 def _siftup(heap, pos):
261 endpos = len(heap)
262 startpos = pos
263 newitem = heap[pos]
264 # Bubble up the smaller child until hitting a leaf.
265 childpos = 2*pos + 1 # leftmost child position
266 while childpos < endpos:
267 # Set childpos to index of smaller child.
268 rightpos = childpos + 1
269 if rightpos < endpos and not heap[childpos] < heap[rightpos]:
270 childpos = rightpos
271 # Move the smaller child up.
272 heap[pos] = heap[childpos]
273 pos = childpos
274 childpos = 2*pos + 1
275 # The leaf at pos is empty now. Put newitem there, and bubble it up
276 # to its final resting place (by sifting its parents down).
277 heap[pos] = newitem
278 _siftdown(heap, startpos, pos)
279
280 def _siftdown_max(heap, startpos, pos):
281 'Maxheap variant of _siftdown'
282 newitem = heap[pos]
283 # Follow the path to the root, moving parents down until finding a place
284 # newitem fits.
285 while pos > startpos:
286 parentpos = (pos - 1) >> 1
287 parent = heap[parentpos]
288 if parent < newitem:
289 heap[pos] = parent
290 pos = parentpos
291 continue
292 break
293 heap[pos] = newitem
294
295 def _siftup_max(heap, pos):
296 'Maxheap variant of _siftup'
297 endpos = len(heap)
298 startpos = pos
299 newitem = heap[pos]
300 # Bubble up the larger child until hitting a leaf.
301 childpos = 2*pos + 1 # leftmost child position
302 while childpos < endpos:
303 # Set childpos to index of larger child.
304 rightpos = childpos + 1
305 if rightpos < endpos and not heap[rightpos] < heap[childpos]:
306 childpos = rightpos
307 # Move the larger child up.
308 heap[pos] = heap[childpos]
309 pos = childpos
310 childpos = 2*pos + 1
311 # The leaf at pos is empty now. Put newitem there, and bubble it up
312 # to its final resting place (by sifting its parents down).
313 heap[pos] = newitem
314 _siftdown_max(heap, startpos, pos)
315
316 def merge(*iterables, key=None, reverse=False):
317 '''Merge multiple sorted inputs into a single sorted output.
318
319 Similar to sorted(itertools.chain(*iterables)) but returns a generator,
320 does not pull the data into memory all at once, and assumes that each of
321 the input streams is already sorted (smallest to largest).
322
323 >>> list(merge([1,3,5,7], [0,2,4,8], [5,10,15,20], [], [25]))
324 [0, 1, 2, 3, 4, 5, 5, 7, 8, 10, 15, 20, 25]
325
326 If *key* is not None, applies a key function to each element to determine
327 its sort order.
328
329 >>> list(merge(['dog', 'horse'], ['cat', 'fish', 'kangaroo'], key=len))
330 ['dog', 'cat', 'fish', 'horse', 'kangaroo']
331
332 '''
333
334 h = []
335 h_append = h.append
336
337 if reverse:
338 _heapify = _heapify_max
339 _heappop = _heappop_max
340 _heapreplace = _heapreplace_max
341 direction = -1
342 else:
343 _heapify = heapify
344 _heappop = heappop
345 _heapreplace = heapreplace
346 direction = 1
347
348 if key is None:
349 for order, it in enumerate(map(iter, iterables)):
350 try:
351 next = it.__next__
352 h_append([next(), order * direction, next])
353 except StopIteration:
354 pass
355 _heapify(h)
356 while len(h) > 1:
357 try:
358 while True:
359 value, order, next = s = h[0]
360 yield value
361 s[0] = next() # raises StopIteration when exhausted
362 _heapreplace(h, s) # restore heap condition
363 except StopIteration:
364 _heappop(h) # remove empty iterator
365 if h:
366 # fast case when only a single iterator remains
367 value, order, next = h[0]
368 yield value
369 yield from next.__self__
370 return
371
372 for order, it in enumerate(map(iter, iterables)):
373 try:
374 next = it.__next__
375 value = next()
376 h_append([key(value), order * direction, value, next])
377 except StopIteration:
378 pass
379 _heapify(h)
380 while len(h) > 1:
381 try:
382 while True:
383 key_value, order, value, next = s = h[0]
384 yield value
385 value = next()
386 s[0] = key(value)
387 s[2] = value
388 _heapreplace(h, s)
389 except StopIteration:
390 _heappop(h)
391 if h:
392 key_value, order, value, next = h[0]
393 yield value
394 yield from next.__self__
395
396
397 # Algorithm notes for nlargest() and nsmallest()
398 # ==============================================
399 #
400 # Make a single pass over the data while keeping the k most extreme values
401 # in a heap. Memory consumption is limited to keeping k values in a list.
402 #
403 # Measured performance for random inputs:
404 #
405 # number of comparisons
406 # n inputs k-extreme values (average of 5 trials) % more than min()
407 # ------------- ---------------- --------------------- -----------------
408 # 1,000 100 3,317 231.7%
409 # 10,000 100 14,046 40.5%
410 # 100,000 100 105,749 5.7%
411 # 1,000,000 100 1,007,751 0.8%
412 # 10,000,000 100 10,009,401 0.1%
413 #
414 # Theoretical number of comparisons for k smallest of n random inputs:
415 #
416 # Step Comparisons Action
417 # ---- -------------------------- ---------------------------
418 # 1 1.66 * k heapify the first k-inputs
419 # 2 n - k compare remaining elements to top of heap
420 # 3 k * (1 + lg2(k)) * ln(n/k) replace the topmost value on the heap
421 # 4 k * lg2(k) - (k/2) final sort of the k most extreme values
422 #
423 # Combining and simplifying for a rough estimate gives:
424 #
425 # comparisons = n + k * (log(k, 2) * log(n/k) + log(k, 2) + log(n/k))
426 #
427 # Computing the number of comparisons for step 3:
428 # -----------------------------------------------
429 # * For the i-th new value from the iterable, the probability of being in the
430 # k most extreme values is k/i. For example, the probability of the 101st
431 # value seen being in the 100 most extreme values is 100/101.
432 # * If the value is a new extreme value, the cost of inserting it into the
433 # heap is 1 + log(k, 2).
434 # * The probability times the cost gives:
435 # (k/i) * (1 + log(k, 2))
436 # * Summing across the remaining n-k elements gives:
437 # sum((k/i) * (1 + log(k, 2)) for i in range(k+1, n+1))
438 # * This reduces to:
439 # (H(n) - H(k)) * k * (1 + log(k, 2))
440 # * Where H(n) is the n-th harmonic number estimated by:
441 # gamma = 0.5772156649
442 # H(n) = log(n, e) + gamma + 1 / (2 * n)
443 # http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Rate_of_divergence
444 # * Substituting the H(n) formula:
445 # comparisons = k * (1 + log(k, 2)) * (log(n/k, e) + (1/n - 1/k) / 2)
446 #
447 # Worst-case for step 3:
448 # ----------------------
449 # In the worst case, the input data is reversed sorted so that every new element
450 # must be inserted in the heap:
451 #
452 # comparisons = 1.66 * k + log(k, 2) * (n - k)
453 #
454 # Alternative Algorithms
455 # ----------------------
456 # Other algorithms were not used because they:
457 # 1) Took much more auxiliary memory,
458 # 2) Made multiple passes over the data.
459 # 3) Made more comparisons in common cases (small k, large n, semi-random input).
460 # See the more detailed comparison of approach at:
461 # http://code.activestate.com/recipes/577573-compare-algorithms-for-heapqsmallest
462
463 def nsmallest(n, iterable, key=None):
464 """Find the n smallest elements in a dataset.
465
466 Equivalent to: sorted(iterable, key=key)[:n]
467 """
468
469 # Short-cut for n==1 is to use min()
470 if n == 1:
471 it = iter(iterable)
472 sentinel = object()
473 result = min(it, default=sentinel, key=key)
474 return [] if result is sentinel else [result]
475
476 # When n>=size, it's faster to use sorted()
477 try:
478 size = len(iterable)
479 except (TypeError, AttributeError):
480 pass
481 else:
482 if n >= size:
483 return sorted(iterable, key=key)[:n]
484
485 # When key is none, use simpler decoration
486 if key is None:
487 it = iter(iterable)
488 # put the range(n) first so that zip() doesn't
489 # consume one too many elements from the iterator
490 result = [(elem, i) for i, elem in zip(range(n), it)]
491 if not result:
492 return result
493 _heapify_max(result)
494 top = result[0][0]
495 order = n
496 _heapreplace = _heapreplace_max
497 for elem in it:
498 if elem < top:
499 _heapreplace(result, (elem, order))
500 top, _order = result[0]
501 order += 1
502 result.sort()
503 return [elem for (elem, order) in result]
504
505 # General case, slowest method
506 it = iter(iterable)
507 result = [(key(elem), i, elem) for i, elem in zip(range(n), it)]
508 if not result:
509 return result
510 _heapify_max(result)
511 top = result[0][0]
512 order = n
513 _heapreplace = _heapreplace_max
514 for elem in it:
515 k = key(elem)
516 if k < top:
517 _heapreplace(result, (k, order, elem))
518 top, _order, _elem = result[0]
519 order += 1
520 result.sort()
521 return [elem for (k, order, elem) in result]
522
523 def nlargest(n, iterable, key=None):
524 """Find the n largest elements in a dataset.
525
526 Equivalent to: sorted(iterable, key=key, reverse=True)[:n]
527 """
528
529 # Short-cut for n==1 is to use max()
530 if n == 1:
531 it = iter(iterable)
532 sentinel = object()
533 result = max(it, default=sentinel, key=key)
534 return [] if result is sentinel else [result]
535
536 # When n>=size, it's faster to use sorted()
537 try:
538 size = len(iterable)
539 except (TypeError, AttributeError):
540 pass
541 else:
542 if n >= size:
543 return sorted(iterable, key=key, reverse=True)[:n]
544
545 # When key is none, use simpler decoration
546 if key is None:
547 it = iter(iterable)
548 result = [(elem, i) for i, elem in zip(range(0, -n, -1), it)]
549 if not result:
550 return result
551 heapify(result)
552 top = result[0][0]
553 order = -n
554 _heapreplace = heapreplace
555 for elem in it:
556 if top < elem:
557 _heapreplace(result, (elem, order))
558 top, _order = result[0]
559 order -= 1
560 result.sort(reverse=True)
561 return [elem for (elem, order) in result]
562
563 # General case, slowest method
564 it = iter(iterable)
565 result = [(key(elem), i, elem) for i, elem in zip(range(0, -n, -1), it)]
566 if not result:
567 return result
568 heapify(result)
569 top = result[0][0]
570 order = -n
571 _heapreplace = heapreplace
572 for elem in it:
573 k = key(elem)
574 if top < k:
575 _heapreplace(result, (k, order, elem))
576 top, _order, _elem = result[0]
577 order -= 1
578 result.sort(reverse=True)
579 return [elem for (k, order, elem) in result]
580
581 # If available, use C implementation
582 try:
583 from _heapq import *
584 except ImportError:
585 pass
586 try:
587 from _heapq import _heapreplace_max
588 except ImportError:
589 pass
590 try:
591 from _heapq import _heapify_max
592 except ImportError:
593 pass
594 try:
595 from _heapq import _heappop_max
596 except ImportError:
597 pass
598
599
600 if __name__ == "__main__":
601
602 import doctest # pragma: no cover
603 print(doctest.testmod()) # pragma: no cover