/Lib/heapq.py

http://unladen-swallow.googlecode.com/ · Python · 393 lines · 236 code · 23 blank · 134 comment · 32 complexity · 33f8f0eba1662815249e87b96194fa55 MD5 · raw file

  1. # -*- coding: Latin-1 -*-
  2. """Heap queue algorithm (a.k.a. priority queue).
  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. Usage:
  8. heap = [] # creates an empty heap
  9. heappush(heap, item) # pushes a new item on the heap
  10. item = heappop(heap) # pops the smallest item from the heap
  11. item = heap[0] # smallest item on the heap without popping it
  12. heapify(x) # transforms list into a heap, in-place, in linear time
  13. item = heapreplace(heap, item) # pops and returns smallest item, and adds
  14. # new item; the heap size is unchanged
  15. Our API differs from textbook heap algorithms as follows:
  16. - We use 0-based indexing. This makes the relationship between the
  17. index for a node and the indexes for its children slightly less
  18. obvious, but is more suitable since Python uses 0-based indexing.
  19. - Our heappop() method returns the smallest item, not the largest.
  20. These two make it possible to view the heap as a regular Python list
  21. without surprises: heap[0] is the smallest item, and heap.sort()
  22. maintains the heap invariant!
  23. """
  24. # Original code by Kevin O'Connor, augmented by Tim Peters and Raymond Hettinger
  25. __about__ = """Heap queues
  26. [explanation by François Pinard]
  27. Heaps are arrays for which a[k] <= a[2*k+1] and a[k] <= a[2*k+2] for
  28. all k, counting elements from 0. For the sake of comparison,
  29. non-existing elements are considered to be infinite. The interesting
  30. property of a heap is that a[0] is always its smallest element.
  31. The strange invariant above is meant to be an efficient memory
  32. representation for a tournament. The numbers below are `k', not a[k]:
  33. 0
  34. 1 2
  35. 3 4 5 6
  36. 7 8 9 10 11 12 13 14
  37. 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
  38. In the tree above, each cell `k' is topping `2*k+1' and `2*k+2'. In
  39. an usual binary tournament we see in sports, each cell is the winner
  40. over the two cells it tops, and we can trace the winner down the tree
  41. to see all opponents s/he had. However, in many computer applications
  42. of such tournaments, we do not need to trace the history of a winner.
  43. To be more memory efficient, when a winner is promoted, we try to
  44. replace it by something else at a lower level, and the rule becomes
  45. that a cell and the two cells it tops contain three different items,
  46. but the top cell "wins" over the two topped cells.
  47. If this heap invariant is protected at all time, index 0 is clearly
  48. the overall winner. The simplest algorithmic way to remove it and
  49. find the "next" winner is to move some loser (let's say cell 30 in the
  50. diagram above) into the 0 position, and then percolate this new 0 down
  51. the tree, exchanging values, until the invariant is re-established.
  52. This is clearly logarithmic on the total number of items in the tree.
  53. By iterating over all items, you get an O(n ln n) sort.
  54. A nice feature of this sort is that you can efficiently insert new
  55. items while the sort is going on, provided that the inserted items are
  56. not "better" than the last 0'th element you extracted. This is
  57. especially useful in simulation contexts, where the tree holds all
  58. incoming events, and the "win" condition means the smallest scheduled
  59. time. When an event schedule other events for execution, they are
  60. scheduled into the future, so they can easily go into the heap. So, a
  61. heap is a good structure for implementing schedulers (this is what I
  62. used for my MIDI sequencer :-).
  63. Various structures for implementing schedulers have been extensively
  64. studied, and heaps are good for this, as they are reasonably speedy,
  65. the speed is almost constant, and the worst case is not much different
  66. than the average case. However, there are other representations which
  67. are more efficient overall, yet the worst cases might be terrible.
  68. Heaps are also very useful in big disk sorts. You most probably all
  69. know that a big sort implies producing "runs" (which are pre-sorted
  70. sequences, which size is usually related to the amount of CPU memory),
  71. followed by a merging passes for these runs, which merging is often
  72. very cleverly organised[1]. It is very important that the initial
  73. sort produces the longest runs possible. Tournaments are a good way
  74. to that. If, using all the memory available to hold a tournament, you
  75. replace and percolate items that happen to fit the current run, you'll
  76. produce runs which are twice the size of the memory for random input,
  77. and much better for input fuzzily ordered.
  78. Moreover, if you output the 0'th item on disk and get an input which
  79. may not fit in the current tournament (because the value "wins" over
  80. the last output value), it cannot fit in the heap, so the size of the
  81. heap decreases. The freed memory could be cleverly reused immediately
  82. for progressively building a second heap, which grows at exactly the
  83. same rate the first heap is melting. When the first heap completely
  84. vanishes, you switch heaps and start a new run. Clever and quite
  85. effective!
  86. In a word, heaps are useful memory structures to know. I use them in
  87. a few applications, and I think it is good to keep a `heap' module
  88. around. :-)
  89. --------------------
  90. [1] The disk balancing algorithms which are current, nowadays, are
  91. more annoying than clever, and this is a consequence of the seeking
  92. capabilities of the disks. On devices which cannot seek, like big
  93. tape drives, the story was quite different, and one had to be very
  94. clever to ensure (far in advance) that each tape movement will be the
  95. most effective possible (that is, will best participate at
  96. "progressing" the merge). Some tapes were even able to read
  97. backwards, and this was also used to avoid the rewinding time.
  98. Believe me, real good tape sorts were quite spectacular to watch!
  99. From all times, sorting has always been a Great Art! :-)
  100. """
  101. __all__ = ['heappush', 'heappop', 'heapify', 'heapreplace', 'merge',
  102. 'nlargest', 'nsmallest', 'heappushpop']
  103. from itertools import islice, repeat, count, imap, izip, tee
  104. from operator import itemgetter, neg
  105. import bisect
  106. def heappush(heap, item):
  107. """Push item onto heap, maintaining the heap invariant."""
  108. heap.append(item)
  109. _siftdown(heap, 0, len(heap)-1)
  110. def heappop(heap):
  111. """Pop the smallest item off the heap, maintaining the heap invariant."""
  112. lastelt = heap.pop() # raises appropriate IndexError if heap is empty
  113. if heap:
  114. returnitem = heap[0]
  115. heap[0] = lastelt
  116. _siftup(heap, 0)
  117. else:
  118. returnitem = lastelt
  119. return returnitem
  120. def heapreplace(heap, item):
  121. """Pop and return the current smallest value, and add the new item.
  122. This is more efficient than heappop() followed by heappush(), and can be
  123. more appropriate when using a fixed-size heap. Note that the value
  124. returned may be larger than item! That constrains reasonable uses of
  125. this routine unless written as part of a conditional replacement:
  126. if item > heap[0]:
  127. item = heapreplace(heap, item)
  128. """
  129. returnitem = heap[0] # raises appropriate IndexError if heap is empty
  130. heap[0] = item
  131. _siftup(heap, 0)
  132. return returnitem
  133. def heappushpop(heap, item):
  134. """Fast version of a heappush followed by a heappop."""
  135. if heap and heap[0] < item:
  136. item, heap[0] = heap[0], item
  137. _siftup(heap, 0)
  138. return item
  139. def heapify(x):
  140. """Transform list into a heap, in-place, in O(len(heap)) time."""
  141. n = len(x)
  142. # Transform bottom-up. The largest index there's any point to looking at
  143. # is the largest with a child index in-range, so must have 2*i + 1 < n,
  144. # or i < (n-1)/2. If n is even = 2*j, this is (2*j-1)/2 = j-1/2 so
  145. # j-1 is the largest, which is n//2 - 1. If n is odd = 2*j+1, this is
  146. # (2*j+1-1)/2 = j so j-1 is the largest, and that's again n//2-1.
  147. for i in reversed(xrange(n//2)):
  148. _siftup(x, i)
  149. def nlargest(n, iterable):
  150. """Find the n largest elements in a dataset.
  151. Equivalent to: sorted(iterable, reverse=True)[:n]
  152. """
  153. it = iter(iterable)
  154. result = list(islice(it, n))
  155. if not result:
  156. return result
  157. heapify(result)
  158. _heappushpop = heappushpop
  159. for elem in it:
  160. _heappushpop(result, elem)
  161. result.sort(reverse=True)
  162. return result
  163. def nsmallest(n, iterable):
  164. """Find the n smallest elements in a dataset.
  165. Equivalent to: sorted(iterable)[:n]
  166. """
  167. if hasattr(iterable, '__len__') and n * 10 <= len(iterable):
  168. # For smaller values of n, the bisect method is faster than a minheap.
  169. # It is also memory efficient, consuming only n elements of space.
  170. it = iter(iterable)
  171. result = sorted(islice(it, 0, n))
  172. if not result:
  173. return result
  174. insort = bisect.insort
  175. pop = result.pop
  176. los = result[-1] # los --> Largest of the nsmallest
  177. for elem in it:
  178. if los <= elem:
  179. continue
  180. insort(result, elem)
  181. pop()
  182. los = result[-1]
  183. return result
  184. # An alternative approach manifests the whole iterable in memory but
  185. # saves comparisons by heapifying all at once. Also, saves time
  186. # over bisect.insort() which has O(n) data movement time for every
  187. # insertion. Finding the n smallest of an m length iterable requires
  188. # O(m) + O(n log m) comparisons.
  189. h = list(iterable)
  190. heapify(h)
  191. return map(heappop, repeat(h, min(n, len(h))))
  192. # 'heap' is a heap at all indices >= startpos, except possibly for pos. pos
  193. # is the index of a leaf with a possibly out-of-order value. Restore the
  194. # heap invariant.
  195. def _siftdown(heap, startpos, pos):
  196. newitem = heap[pos]
  197. # Follow the path to the root, moving parents down until finding a place
  198. # newitem fits.
  199. while pos > startpos:
  200. parentpos = (pos - 1) >> 1
  201. parent = heap[parentpos]
  202. if newitem < parent:
  203. heap[pos] = parent
  204. pos = parentpos
  205. continue
  206. break
  207. heap[pos] = newitem
  208. # The child indices of heap index pos are already heaps, and we want to make
  209. # a heap at index pos too. We do this by bubbling the smaller child of
  210. # pos up (and so on with that child's children, etc) until hitting a leaf,
  211. # then using _siftdown to move the oddball originally at index pos into place.
  212. #
  213. # We *could* break out of the loop as soon as we find a pos where newitem <=
  214. # both its children, but turns out that's not a good idea, and despite that
  215. # many books write the algorithm that way. During a heap pop, the last array
  216. # element is sifted in, and that tends to be large, so that comparing it
  217. # against values starting from the root usually doesn't pay (= usually doesn't
  218. # get us out of the loop early). See Knuth, Volume 3, where this is
  219. # explained and quantified in an exercise.
  220. #
  221. # Cutting the # of comparisons is important, since these routines have no
  222. # way to extract "the priority" from an array element, so that intelligence
  223. # is likely to be hiding in custom __cmp__ methods, or in array elements
  224. # storing (priority, record) tuples. Comparisons are thus potentially
  225. # expensive.
  226. #
  227. # On random arrays of length 1000, making this change cut the number of
  228. # comparisons made by heapify() a little, and those made by exhaustive
  229. # heappop() a lot, in accord with theory. Here are typical results from 3
  230. # runs (3 just to demonstrate how small the variance is):
  231. #
  232. # Compares needed by heapify Compares needed by 1000 heappops
  233. # -------------------------- --------------------------------
  234. # 1837 cut to 1663 14996 cut to 8680
  235. # 1855 cut to 1659 14966 cut to 8678
  236. # 1847 cut to 1660 15024 cut to 8703
  237. #
  238. # Building the heap by using heappush() 1000 times instead required
  239. # 2198, 2148, and 2219 compares: heapify() is more efficient, when
  240. # you can use it.
  241. #
  242. # The total compares needed by list.sort() on the same lists were 8627,
  243. # 8627, and 8632 (this should be compared to the sum of heapify() and
  244. # heappop() compares): list.sort() is (unsurprisingly!) more efficient
  245. # for sorting.
  246. def _siftup(heap, pos):
  247. endpos = len(heap)
  248. startpos = pos
  249. newitem = heap[pos]
  250. # Bubble up the smaller child until hitting a leaf.
  251. childpos = 2*pos + 1 # leftmost child position
  252. while childpos < endpos:
  253. # Set childpos to index of smaller child.
  254. rightpos = childpos + 1
  255. if rightpos < endpos and not heap[childpos] < heap[rightpos]:
  256. childpos = rightpos
  257. # Move the smaller child up.
  258. heap[pos] = heap[childpos]
  259. pos = childpos
  260. childpos = 2*pos + 1
  261. # The leaf at pos is empty now. Put newitem there, and bubble it up
  262. # to its final resting place (by sifting its parents down).
  263. heap[pos] = newitem
  264. _siftdown(heap, startpos, pos)
  265. # If available, use C implementation
  266. try:
  267. from _heapq import heappush, heappop, heapify, heapreplace, nlargest, nsmallest, heappushpop
  268. except ImportError:
  269. pass
  270. def merge(*iterables):
  271. '''Merge multiple sorted inputs into a single sorted output.
  272. Similar to sorted(itertools.chain(*iterables)) but returns a generator,
  273. does not pull the data into memory all at once, and assumes that each of
  274. the input streams is already sorted (smallest to largest).
  275. >>> list(merge([1,3,5,7], [0,2,4,8], [5,10,15,20], [], [25]))
  276. [0, 1, 2, 3, 4, 5, 5, 7, 8, 10, 15, 20, 25]
  277. '''
  278. _heappop, _heapreplace, _StopIteration = heappop, heapreplace, StopIteration
  279. h = []
  280. h_append = h.append
  281. for itnum, it in enumerate(map(iter, iterables)):
  282. try:
  283. next = it.next
  284. h_append([next(), itnum, next])
  285. except _StopIteration:
  286. pass
  287. heapify(h)
  288. while 1:
  289. try:
  290. while 1:
  291. v, itnum, next = s = h[0] # raises IndexError when h is empty
  292. yield v
  293. s[0] = next() # raises StopIteration when exhausted
  294. _heapreplace(h, s) # restore heap condition
  295. except _StopIteration:
  296. _heappop(h) # remove empty iterator
  297. except IndexError:
  298. return
  299. # Extend the implementations of nsmallest and nlargest to use a key= argument
  300. _nsmallest = nsmallest
  301. def nsmallest(n, iterable, key=None):
  302. """Find the n smallest elements in a dataset.
  303. Equivalent to: sorted(iterable, key=key)[:n]
  304. """
  305. if key is None:
  306. it = izip(iterable, count()) # decorate
  307. result = _nsmallest(n, it)
  308. return map(itemgetter(0), result) # undecorate
  309. in1, in2 = tee(iterable)
  310. it = izip(imap(key, in1), count(), in2) # decorate
  311. result = _nsmallest(n, it)
  312. return map(itemgetter(2), result) # undecorate
  313. _nlargest = nlargest
  314. def nlargest(n, iterable, key=None):
  315. """Find the n largest elements in a dataset.
  316. Equivalent to: sorted(iterable, key=key, reverse=True)[:n]
  317. """
  318. if key is None:
  319. it = izip(iterable, imap(neg, count())) # decorate
  320. result = _nlargest(n, it)
  321. return map(itemgetter(0), result) # undecorate
  322. in1, in2 = tee(iterable)
  323. it = izip(imap(key, in1), imap(neg, count()), in2) # decorate
  324. result = _nlargest(n, it)
  325. return map(itemgetter(2), result) # undecorate
  326. if __name__ == "__main__":
  327. # Simple sanity test
  328. heap = []
  329. data = [1, 3, 5, 7, 9, 2, 4, 6, 8, 0]
  330. for item in data:
  331. heappush(heap, item)
  332. sort = []
  333. while heap:
  334. sort.append(heappop(heap))
  335. print sort
  336. import doctest
  337. doctest.testmod()