Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 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 = heapreplace(heap, item) # pops and returns smallest item, and adds |
| 16 | # new item; the heap size is unchanged |
| 17 | |
| 18 | Our API differs from textbook heap algorithms as follows: |
| 19 | |
| 20 | - We use 0-based indexing. This makes the relationship between the |
| 21 | index for a node and the indexes for its children slightly less |
| 22 | obvious, but is more suitable since Python uses 0-based indexing. |
| 23 | |
| 24 | - Our heappop() method returns the smallest item, not the largest. |
| 25 | |
| 26 | These two make it possible to view the heap as a regular Python list |
| 27 | without surprises: heap[0] is the smallest item, and heap.sort() |
| 28 | maintains the heap invariant! |
| 29 | """ |
| 30 | |
Raymond Hettinger | 33ecffb | 2004-06-10 05:03:17 +0000 | [diff] [blame] | 31 | # Original code by Kevin O'Connor, augmented by Tim Peters and Raymond Hettinger |
Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 32 | |
| 33 | __about__ = """Heap queues |
| 34 | |
Mark Dickinson | b4a17a8 | 2010-07-04 19:23:49 +0000 | [diff] [blame] | 35 | [explanation by François Pinard] |
Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 36 | |
| 37 | Heaps are arrays for which a[k] <= a[2*k+1] and a[k] <= a[2*k+2] for |
| 38 | all k, counting elements from 0. For the sake of comparison, |
| 39 | non-existing elements are considered to be infinite. The interesting |
| 40 | property of a heap is that a[0] is always its smallest element. |
| 41 | |
| 42 | The strange invariant above is meant to be an efficient memory |
| 43 | representation for a tournament. The numbers below are `k', not a[k]: |
| 44 | |
| 45 | 0 |
| 46 | |
| 47 | 1 2 |
| 48 | |
| 49 | 3 4 5 6 |
| 50 | |
| 51 | 7 8 9 10 11 12 13 14 |
| 52 | |
| 53 | 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 |
| 54 | |
| 55 | |
| 56 | In the tree above, each cell `k' is topping `2*k+1' and `2*k+2'. In |
| 57 | an usual binary tournament we see in sports, each cell is the winner |
| 58 | over the two cells it tops, and we can trace the winner down the tree |
| 59 | to see all opponents s/he had. However, in many computer applications |
| 60 | of such tournaments, we do not need to trace the history of a winner. |
| 61 | To be more memory efficient, when a winner is promoted, we try to |
| 62 | replace it by something else at a lower level, and the rule becomes |
| 63 | that a cell and the two cells it tops contain three different items, |
| 64 | but the top cell "wins" over the two topped cells. |
| 65 | |
| 66 | If this heap invariant is protected at all time, index 0 is clearly |
| 67 | the overall winner. The simplest algorithmic way to remove it and |
| 68 | find the "next" winner is to move some loser (let's say cell 30 in the |
| 69 | diagram above) into the 0 position, and then percolate this new 0 down |
| 70 | the tree, exchanging values, until the invariant is re-established. |
| 71 | This is clearly logarithmic on the total number of items in the tree. |
| 72 | By iterating over all items, you get an O(n ln n) sort. |
| 73 | |
| 74 | A nice feature of this sort is that you can efficiently insert new |
| 75 | items while the sort is going on, provided that the inserted items are |
| 76 | not "better" than the last 0'th element you extracted. This is |
| 77 | especially useful in simulation contexts, where the tree holds all |
| 78 | incoming events, and the "win" condition means the smallest scheduled |
| 79 | time. When an event schedule other events for execution, they are |
| 80 | scheduled into the future, so they can easily go into the heap. So, a |
| 81 | heap is a good structure for implementing schedulers (this is what I |
| 82 | used for my MIDI sequencer :-). |
| 83 | |
| 84 | Various structures for implementing schedulers have been extensively |
| 85 | studied, and heaps are good for this, as they are reasonably speedy, |
| 86 | the speed is almost constant, and the worst case is not much different |
| 87 | than the average case. However, there are other representations which |
| 88 | are more efficient overall, yet the worst cases might be terrible. |
| 89 | |
| 90 | Heaps are also very useful in big disk sorts. You most probably all |
| 91 | know that a big sort implies producing "runs" (which are pre-sorted |
| 92 | sequences, which size is usually related to the amount of CPU memory), |
| 93 | followed by a merging passes for these runs, which merging is often |
| 94 | very cleverly organised[1]. It is very important that the initial |
| 95 | sort produces the longest runs possible. Tournaments are a good way |
| 96 | to that. If, using all the memory available to hold a tournament, you |
| 97 | replace and percolate items that happen to fit the current run, you'll |
| 98 | produce runs which are twice the size of the memory for random input, |
| 99 | and much better for input fuzzily ordered. |
| 100 | |
| 101 | Moreover, if you output the 0'th item on disk and get an input which |
| 102 | may not fit in the current tournament (because the value "wins" over |
| 103 | the last output value), it cannot fit in the heap, so the size of the |
| 104 | heap decreases. The freed memory could be cleverly reused immediately |
| 105 | for progressively building a second heap, which grows at exactly the |
| 106 | same rate the first heap is melting. When the first heap completely |
| 107 | vanishes, you switch heaps and start a new run. Clever and quite |
| 108 | effective! |
| 109 | |
| 110 | In a word, heaps are useful memory structures to know. I use them in |
| 111 | a few applications, and I think it is good to keep a `heap' module |
| 112 | around. :-) |
| 113 | |
| 114 | -------------------- |
| 115 | [1] The disk balancing algorithms which are current, nowadays, are |
| 116 | more annoying than clever, and this is a consequence of the seeking |
| 117 | capabilities of the disks. On devices which cannot seek, like big |
| 118 | tape drives, the story was quite different, and one had to be very |
| 119 | clever to ensure (far in advance) that each tape movement will be the |
| 120 | most effective possible (that is, will best participate at |
| 121 | "progressing" the merge). Some tapes were even able to read |
| 122 | backwards, and this was also used to avoid the rewinding time. |
| 123 | Believe me, real good tape sorts were quite spectacular to watch! |
| 124 | From all times, sorting has always been a Great Art! :-) |
| 125 | """ |
| 126 | |
Thomas Wouters | cf297e4 | 2007-02-23 15:07:44 +0000 | [diff] [blame] | 127 | __all__ = ['heappush', 'heappop', 'heapify', 'heapreplace', 'merge', |
Christian Heimes | dd15f6c | 2008-03-16 00:07:10 +0000 | [diff] [blame] | 128 | 'nlargest', 'nsmallest', 'heappushpop'] |
Raymond Hettinger | 33ecffb | 2004-06-10 05:03:17 +0000 | [diff] [blame] | 129 | |
Raymond Hettinger | 234fb2d | 2014-05-11 14:21:23 -0700 | [diff] [blame] | 130 | from itertools import islice, count |
Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 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 | else: |
| 145 | returnitem = lastelt |
| 146 | return returnitem |
| 147 | |
| 148 | def heapreplace(heap, item): |
| 149 | """Pop and return the current smallest value, and add the new item. |
| 150 | |
| 151 | This is more efficient than heappop() followed by heappush(), and can be |
| 152 | more appropriate when using a fixed-size heap. Note that the value |
| 153 | returned may be larger than item! That constrains reasonable uses of |
Raymond Hettinger | 8158e84 | 2004-09-06 07:04:09 +0000 | [diff] [blame] | 154 | this routine unless written as part of a conditional replacement: |
Raymond Hettinger | 28224f8 | 2004-06-20 09:07:53 +0000 | [diff] [blame] | 155 | |
Raymond Hettinger | 8158e84 | 2004-09-06 07:04:09 +0000 | [diff] [blame] | 156 | if item > heap[0]: |
| 157 | item = heapreplace(heap, item) |
Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 158 | """ |
| 159 | returnitem = heap[0] # raises appropriate IndexError if heap is empty |
| 160 | heap[0] = item |
| 161 | _siftup(heap, 0) |
| 162 | return returnitem |
| 163 | |
Christian Heimes | dd15f6c | 2008-03-16 00:07:10 +0000 | [diff] [blame] | 164 | def heappushpop(heap, item): |
| 165 | """Fast version of a heappush followed by a heappop.""" |
Georg Brandl | f78e02b | 2008-06-10 17:40:04 +0000 | [diff] [blame] | 166 | if heap and heap[0] < item: |
Christian Heimes | dd15f6c | 2008-03-16 00:07:10 +0000 | [diff] [blame] | 167 | item, heap[0] = heap[0], item |
| 168 | _siftup(heap, 0) |
| 169 | return item |
| 170 | |
Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 171 | def heapify(x): |
Éric Araujo | 395ba35 | 2011-04-15 23:34:31 +0200 | [diff] [blame] | 172 | """Transform list into a heap, in-place, in O(len(x)) time.""" |
Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 173 | n = len(x) |
| 174 | # Transform bottom-up. The largest index there's any point to looking at |
| 175 | # is the largest with a child index in-range, so must have 2*i + 1 < n, |
| 176 | # or i < (n-1)/2. If n is even = 2*j, this is (2*j-1)/2 = j-1/2 so |
| 177 | # j-1 is the largest, which is n//2 - 1. If n is odd = 2*j+1, this is |
| 178 | # (2*j+1-1)/2 = j so j-1 is the largest, and that's again n//2-1. |
Guido van Rossum | 805365e | 2007-05-07 22:24:25 +0000 | [diff] [blame] | 179 | for i in reversed(range(n//2)): |
Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 180 | _siftup(x, i) |
| 181 | |
Raymond Hettinger | 234fb2d | 2014-05-11 14:21:23 -0700 | [diff] [blame] | 182 | def _heapreplace_max(heap, item): |
| 183 | """Maxheap version of a heappop followed by a heappush.""" |
| 184 | returnitem = heap[0] # raises appropriate IndexError if heap is empty |
| 185 | heap[0] = item |
| 186 | _siftup_max(heap, 0) |
| 187 | return returnitem |
Raymond Hettinger | f6b2667 | 2013-03-05 01:36:30 -0500 | [diff] [blame] | 188 | |
| 189 | def _heapify_max(x): |
| 190 | """Transform list into a maxheap, in-place, in O(len(x)) time.""" |
| 191 | n = len(x) |
| 192 | for i in reversed(range(n//2)): |
| 193 | _siftup_max(x, i) |
| 194 | |
Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 195 | # 'heap' is a heap at all indices >= startpos, except possibly for pos. pos |
| 196 | # is the index of a leaf with a possibly out-of-order value. Restore the |
| 197 | # heap invariant. |
| 198 | def _siftdown(heap, startpos, pos): |
| 199 | newitem = heap[pos] |
| 200 | # Follow the path to the root, moving parents down until finding a place |
| 201 | # newitem fits. |
| 202 | while pos > startpos: |
| 203 | parentpos = (pos - 1) >> 1 |
| 204 | parent = heap[parentpos] |
Georg Brandl | f78e02b | 2008-06-10 17:40:04 +0000 | [diff] [blame] | 205 | if newitem < parent: |
| 206 | heap[pos] = parent |
| 207 | pos = parentpos |
| 208 | continue |
| 209 | break |
Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 210 | heap[pos] = newitem |
| 211 | |
| 212 | # The child indices of heap index pos are already heaps, and we want to make |
| 213 | # a heap at index pos too. We do this by bubbling the smaller child of |
| 214 | # pos up (and so on with that child's children, etc) until hitting a leaf, |
| 215 | # then using _siftdown to move the oddball originally at index pos into place. |
| 216 | # |
| 217 | # We *could* break out of the loop as soon as we find a pos where newitem <= |
| 218 | # both its children, but turns out that's not a good idea, and despite that |
| 219 | # many books write the algorithm that way. During a heap pop, the last array |
| 220 | # element is sifted in, and that tends to be large, so that comparing it |
| 221 | # against values starting from the root usually doesn't pay (= usually doesn't |
| 222 | # get us out of the loop early). See Knuth, Volume 3, where this is |
| 223 | # explained and quantified in an exercise. |
| 224 | # |
| 225 | # Cutting the # of comparisons is important, since these routines have no |
| 226 | # way to extract "the priority" from an array element, so that intelligence |
Mark Dickinson | a56c467 | 2009-01-27 18:17:45 +0000 | [diff] [blame] | 227 | # is likely to be hiding in custom comparison methods, or in array elements |
Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 228 | # storing (priority, record) tuples. Comparisons are thus potentially |
| 229 | # expensive. |
| 230 | # |
| 231 | # On random arrays of length 1000, making this change cut the number of |
| 232 | # comparisons made by heapify() a little, and those made by exhaustive |
| 233 | # heappop() a lot, in accord with theory. Here are typical results from 3 |
| 234 | # runs (3 just to demonstrate how small the variance is): |
| 235 | # |
| 236 | # Compares needed by heapify Compares needed by 1000 heappops |
| 237 | # -------------------------- -------------------------------- |
| 238 | # 1837 cut to 1663 14996 cut to 8680 |
| 239 | # 1855 cut to 1659 14966 cut to 8678 |
| 240 | # 1847 cut to 1660 15024 cut to 8703 |
| 241 | # |
| 242 | # Building the heap by using heappush() 1000 times instead required |
| 243 | # 2198, 2148, and 2219 compares: heapify() is more efficient, when |
| 244 | # you can use it. |
| 245 | # |
| 246 | # The total compares needed by list.sort() on the same lists were 8627, |
| 247 | # 8627, and 8632 (this should be compared to the sum of heapify() and |
| 248 | # heappop() compares): list.sort() is (unsurprisingly!) more efficient |
| 249 | # for sorting. |
| 250 | |
| 251 | def _siftup(heap, pos): |
| 252 | endpos = len(heap) |
| 253 | startpos = pos |
| 254 | newitem = heap[pos] |
| 255 | # Bubble up the smaller child until hitting a leaf. |
| 256 | childpos = 2*pos + 1 # leftmost child position |
| 257 | while childpos < endpos: |
| 258 | # Set childpos to index of smaller child. |
| 259 | rightpos = childpos + 1 |
Georg Brandl | f78e02b | 2008-06-10 17:40:04 +0000 | [diff] [blame] | 260 | if rightpos < endpos and not heap[childpos] < heap[rightpos]: |
Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 261 | childpos = rightpos |
| 262 | # Move the smaller child up. |
| 263 | heap[pos] = heap[childpos] |
| 264 | pos = childpos |
| 265 | childpos = 2*pos + 1 |
| 266 | # The leaf at pos is empty now. Put newitem there, and bubble it up |
| 267 | # to its final resting place (by sifting its parents down). |
| 268 | heap[pos] = newitem |
| 269 | _siftdown(heap, startpos, pos) |
| 270 | |
Raymond Hettinger | f6b2667 | 2013-03-05 01:36:30 -0500 | [diff] [blame] | 271 | def _siftdown_max(heap, startpos, pos): |
| 272 | 'Maxheap variant of _siftdown' |
| 273 | newitem = heap[pos] |
| 274 | # Follow the path to the root, moving parents down until finding a place |
| 275 | # newitem fits. |
| 276 | while pos > startpos: |
| 277 | parentpos = (pos - 1) >> 1 |
| 278 | parent = heap[parentpos] |
| 279 | if parent < newitem: |
| 280 | heap[pos] = parent |
| 281 | pos = parentpos |
| 282 | continue |
| 283 | break |
| 284 | heap[pos] = newitem |
| 285 | |
| 286 | def _siftup_max(heap, pos): |
Raymond Hettinger | 2e8d9a7 | 2013-03-05 02:11:10 -0500 | [diff] [blame] | 287 | 'Maxheap variant of _siftup' |
Raymond Hettinger | f6b2667 | 2013-03-05 01:36:30 -0500 | [diff] [blame] | 288 | endpos = len(heap) |
| 289 | startpos = pos |
| 290 | newitem = heap[pos] |
| 291 | # Bubble up the larger child until hitting a leaf. |
| 292 | childpos = 2*pos + 1 # leftmost child position |
| 293 | while childpos < endpos: |
| 294 | # Set childpos to index of larger child. |
| 295 | rightpos = childpos + 1 |
| 296 | if rightpos < endpos and not heap[rightpos] < heap[childpos]: |
| 297 | childpos = rightpos |
| 298 | # Move the larger child up. |
| 299 | heap[pos] = heap[childpos] |
| 300 | pos = childpos |
| 301 | childpos = 2*pos + 1 |
| 302 | # The leaf at pos is empty now. Put newitem there, and bubble it up |
| 303 | # to its final resting place (by sifting its parents down). |
| 304 | heap[pos] = newitem |
| 305 | _siftdown_max(heap, startpos, pos) |
| 306 | |
Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 307 | # If available, use C implementation |
| 308 | try: |
Raymond Hettinger | 0dd737b | 2009-03-29 19:30:50 +0000 | [diff] [blame] | 309 | from _heapq import * |
Brett Cannon | cd171c8 | 2013-07-04 17:43:24 -0400 | [diff] [blame] | 310 | except ImportError: |
Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 311 | pass |
Raymond Hettinger | 234fb2d | 2014-05-11 14:21:23 -0700 | [diff] [blame] | 312 | try: |
| 313 | from _heapq import _heapreplace_max |
| 314 | except ImportError: |
| 315 | pass |
Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 316 | |
Thomas Wouters | cf297e4 | 2007-02-23 15:07:44 +0000 | [diff] [blame] | 317 | def merge(*iterables): |
| 318 | '''Merge multiple sorted inputs into a single sorted output. |
| 319 | |
Guido van Rossum | d8faa36 | 2007-04-27 19:54:29 +0000 | [diff] [blame] | 320 | Similar to sorted(itertools.chain(*iterables)) but returns a generator, |
Thomas Wouters | cf297e4 | 2007-02-23 15:07:44 +0000 | [diff] [blame] | 321 | does not pull the data into memory all at once, and assumes that each of |
| 322 | the input streams is already sorted (smallest to largest). |
| 323 | |
| 324 | >>> list(merge([1,3,5,7], [0,2,4,8], [5,10,15,20], [], [25])) |
| 325 | [0, 1, 2, 3, 4, 5, 5, 7, 8, 10, 15, 20, 25] |
| 326 | |
| 327 | ''' |
| 328 | _heappop, _heapreplace, _StopIteration = heappop, heapreplace, StopIteration |
Raymond Hettinger | f276232 | 2013-09-11 01:15:40 -0500 | [diff] [blame] | 329 | _len = len |
Thomas Wouters | cf297e4 | 2007-02-23 15:07:44 +0000 | [diff] [blame] | 330 | |
| 331 | h = [] |
| 332 | h_append = h.append |
| 333 | for itnum, it in enumerate(map(iter, iterables)): |
| 334 | try: |
Georg Brandl | a18af4e | 2007-04-21 15:47:16 +0000 | [diff] [blame] | 335 | next = it.__next__ |
Thomas Wouters | cf297e4 | 2007-02-23 15:07:44 +0000 | [diff] [blame] | 336 | h_append([next(), itnum, next]) |
| 337 | except _StopIteration: |
| 338 | pass |
| 339 | heapify(h) |
| 340 | |
Raymond Hettinger | f276232 | 2013-09-11 01:15:40 -0500 | [diff] [blame] | 341 | while _len(h) > 1: |
Thomas Wouters | cf297e4 | 2007-02-23 15:07:44 +0000 | [diff] [blame] | 342 | try: |
Raymond Hettinger | f276232 | 2013-09-11 01:15:40 -0500 | [diff] [blame] | 343 | while True: |
| 344 | v, itnum, next = s = h[0] |
Thomas Wouters | cf297e4 | 2007-02-23 15:07:44 +0000 | [diff] [blame] | 345 | yield v |
| 346 | s[0] = next() # raises StopIteration when exhausted |
| 347 | _heapreplace(h, s) # restore heap condition |
| 348 | except _StopIteration: |
| 349 | _heappop(h) # remove empty iterator |
Raymond Hettinger | f276232 | 2013-09-11 01:15:40 -0500 | [diff] [blame] | 350 | if h: |
| 351 | # fast case when only a single iterator remains |
| 352 | v, itnum, next = h[0] |
| 353 | yield v |
| 354 | yield from next.__self__ |
Thomas Wouters | cf297e4 | 2007-02-23 15:07:44 +0000 | [diff] [blame] | 355 | |
Raymond Hettinger | 234fb2d | 2014-05-11 14:21:23 -0700 | [diff] [blame] | 356 | |
| 357 | # Algorithm notes for nlargest() and nsmallest() |
| 358 | # ============================================== |
| 359 | # |
| 360 | # Makes just a single pass over the data while keeping the k most extreme values |
| 361 | # in a heap. Memory consumption is limited to keeping k values in a list. |
| 362 | # |
| 363 | # Measured performance for random inputs: |
| 364 | # |
| 365 | # number of comparisons |
| 366 | # n inputs k-extreme values (average of 5 trials) % more than min() |
| 367 | # ------------- ---------------- - ------------------- ----------------- |
| 368 | # 1,000 100 3,317 133.2% |
| 369 | # 10,000 100 14,046 40.5% |
| 370 | # 100,000 100 105,749 5.7% |
| 371 | # 1,000,000 100 1,007,751 0.8% |
| 372 | # 10,000,000 100 10,009,401 0.1% |
| 373 | # |
| 374 | # Theoretical number of comparisons for k smallest of n random inputs: |
| 375 | # |
| 376 | # Step Comparisons Action |
| 377 | # ---- -------------------------- --------------------------- |
| 378 | # 1 1.66 * k heapify the first k-inputs |
| 379 | # 2 n - k compare remaining elements to top of heap |
| 380 | # 3 k * (1 + lg2(k)) * ln(n/k) replace the topmost value on the heap |
| 381 | # 4 k * lg2(k) - (k/2) final sort of the k most extreme values |
| 382 | # Combining and simplifying for a rough estimate gives: |
| 383 | # comparisons = n + k * (1 + log(n/k)) * (1 + log(k, 2)) |
| 384 | # |
| 385 | # Computing the number of comparisons for step 3: |
| 386 | # ----------------------------------------------- |
| 387 | # * For the i-th new value from the iterable, the probability of being in the |
| 388 | # k most extreme values is k/i. For example, the probability of the 101st |
| 389 | # value seen being in the 100 most extreme values is 100/101. |
| 390 | # * If the value is a new extreme value, the cost of inserting it into the |
| 391 | # heap is 1 + log(k, 2). |
| 392 | # * The probabilty times the cost gives: |
| 393 | # (k/i) * (1 + log(k, 2)) |
| 394 | # * Summing across the remaining n-k elements gives: |
| 395 | # sum((k/i) * (1 + log(k, 2)) for xrange(k+1, n+1)) |
| 396 | # * This reduces to: |
| 397 | # (H(n) - H(k)) * k * (1 + log(k, 2)) |
| 398 | # * Where H(n) is the n-th harmonic number estimated by: |
| 399 | # gamma = 0.5772156649 |
| 400 | # H(n) = log(n, e) + gamma + 1.0 / (2.0 * n) |
| 401 | # http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Rate_of_divergence |
| 402 | # * Substituting the H(n) formula: |
| 403 | # comparisons = k * (1 + log(k, 2)) * (log(n/k, e) + (1/n - 1/k) / 2) |
| 404 | # |
| 405 | # Worst-case for step 3: |
| 406 | # ---------------------- |
| 407 | # In the worst case, the input data is reversed sorted so that every new element |
| 408 | # must be inserted in the heap: |
| 409 | # |
| 410 | # comparisons = 1.66 * k + log(k, 2) * (n - k) |
| 411 | # |
| 412 | # Alternative Algorithms |
| 413 | # ---------------------- |
| 414 | # Other algorithms were not used because they: |
| 415 | # 1) Took much more auxiliary memory, |
| 416 | # 2) Made multiple passes over the data. |
| 417 | # 3) Made more comparisons in common cases (small k, large n, semi-random input). |
| 418 | # See the more detailed comparison of approach at: |
| 419 | # http://code.activestate.com/recipes/577573-compare-algorithms-for-heapqsmallest |
| 420 | |
Raymond Hettinger | 4901a1f | 2004-12-02 08:59:14 +0000 | [diff] [blame] | 421 | def nsmallest(n, iterable, key=None): |
| 422 | """Find the n smallest elements in a dataset. |
| 423 | |
| 424 | Equivalent to: sorted(iterable, key=key)[:n] |
| 425 | """ |
Raymond Hettinger | 234fb2d | 2014-05-11 14:21:23 -0700 | [diff] [blame] | 426 | |
Benjamin Peterson | 18e9512 | 2009-01-18 22:46:33 +0000 | [diff] [blame] | 427 | # Short-cut for n==1 is to use min() when len(iterable)>0 |
| 428 | if n == 1: |
| 429 | it = iter(iterable) |
Raymond Hettinger | 234fb2d | 2014-05-11 14:21:23 -0700 | [diff] [blame] | 430 | sentinel = object() |
Benjamin Peterson | 18e9512 | 2009-01-18 22:46:33 +0000 | [diff] [blame] | 431 | if key is None: |
Raymond Hettinger | 234fb2d | 2014-05-11 14:21:23 -0700 | [diff] [blame] | 432 | result = min(it, default=sentinel) |
| 433 | else: |
| 434 | result = min(it, default=sentinel, key=key) |
| 435 | return [] if result is sentinel else [result] |
Benjamin Peterson | 18e9512 | 2009-01-18 22:46:33 +0000 | [diff] [blame] | 436 | |
Éric Araujo | 395ba35 | 2011-04-15 23:34:31 +0200 | [diff] [blame] | 437 | # When n>=size, it's faster to use sorted() |
Benjamin Peterson | 18e9512 | 2009-01-18 22:46:33 +0000 | [diff] [blame] | 438 | try: |
| 439 | size = len(iterable) |
| 440 | except (TypeError, AttributeError): |
| 441 | pass |
| 442 | else: |
| 443 | if n >= size: |
| 444 | return sorted(iterable, key=key)[:n] |
| 445 | |
| 446 | # When key is none, use simpler decoration |
Georg Brandl | 3a9b062 | 2009-01-03 22:07:57 +0000 | [diff] [blame] | 447 | if key is None: |
Raymond Hettinger | 234fb2d | 2014-05-11 14:21:23 -0700 | [diff] [blame] | 448 | it = iter(iterable) |
| 449 | result = list(islice(zip(it, count()), n)) |
| 450 | if not result: |
| 451 | return result |
| 452 | _heapify_max(result) |
| 453 | order = n |
| 454 | top = result[0][0] |
| 455 | _heapreplace = _heapreplace_max |
| 456 | for elem in it: |
| 457 | if elem < top: |
| 458 | _heapreplace(result, (elem, order)) |
| 459 | top = result[0][0] |
| 460 | order += 1 |
| 461 | result.sort() |
| 462 | return [r[0] for r in result] |
Benjamin Peterson | 18e9512 | 2009-01-18 22:46:33 +0000 | [diff] [blame] | 463 | |
| 464 | # General case, slowest method |
Raymond Hettinger | 234fb2d | 2014-05-11 14:21:23 -0700 | [diff] [blame] | 465 | it = iter(iterable) |
| 466 | result = [(key(elem), i, elem) for i, elem in zip(range(n), it)] |
| 467 | if not result: |
| 468 | return result |
| 469 | _heapify_max(result) |
| 470 | order = n |
| 471 | top = result[0][0] |
| 472 | _heapreplace = _heapreplace_max |
| 473 | for elem in it: |
| 474 | k = key(elem) |
| 475 | if k < top: |
| 476 | _heapreplace(result, (k, order, elem)) |
| 477 | top = result[0][0] |
| 478 | order += 1 |
| 479 | result.sort() |
| 480 | return [r[2] for r in result] |
Raymond Hettinger | 4901a1f | 2004-12-02 08:59:14 +0000 | [diff] [blame] | 481 | |
Raymond Hettinger | 4901a1f | 2004-12-02 08:59:14 +0000 | [diff] [blame] | 482 | def nlargest(n, iterable, key=None): |
| 483 | """Find the n largest elements in a dataset. |
| 484 | |
| 485 | Equivalent to: sorted(iterable, key=key, reverse=True)[:n] |
| 486 | """ |
Benjamin Peterson | 18e9512 | 2009-01-18 22:46:33 +0000 | [diff] [blame] | 487 | |
| 488 | # Short-cut for n==1 is to use max() when len(iterable)>0 |
| 489 | if n == 1: |
| 490 | it = iter(iterable) |
Raymond Hettinger | 277842e | 2014-05-11 01:55:46 -0700 | [diff] [blame] | 491 | sentinel = object() |
Benjamin Peterson | 18e9512 | 2009-01-18 22:46:33 +0000 | [diff] [blame] | 492 | if key is None: |
Raymond Hettinger | 277842e | 2014-05-11 01:55:46 -0700 | [diff] [blame] | 493 | result = max(it, default=sentinel) |
| 494 | else: |
| 495 | result = max(it, default=sentinel, key=key) |
| 496 | return [] if result is sentinel else [result] |
Benjamin Peterson | 18e9512 | 2009-01-18 22:46:33 +0000 | [diff] [blame] | 497 | |
Éric Araujo | 395ba35 | 2011-04-15 23:34:31 +0200 | [diff] [blame] | 498 | # When n>=size, it's faster to use sorted() |
Benjamin Peterson | 18e9512 | 2009-01-18 22:46:33 +0000 | [diff] [blame] | 499 | try: |
| 500 | size = len(iterable) |
| 501 | except (TypeError, AttributeError): |
| 502 | pass |
| 503 | else: |
| 504 | if n >= size: |
| 505 | return sorted(iterable, key=key, reverse=True)[:n] |
| 506 | |
| 507 | # When key is none, use simpler decoration |
Georg Brandl | 3a9b062 | 2009-01-03 22:07:57 +0000 | [diff] [blame] | 508 | if key is None: |
Raymond Hettinger | 277842e | 2014-05-11 01:55:46 -0700 | [diff] [blame] | 509 | it = iter(iterable) |
| 510 | result = list(islice(zip(it, count(0, -1)), n)) |
| 511 | if not result: |
| 512 | return result |
| 513 | heapify(result) |
| 514 | order = -n |
| 515 | top = result[0][0] |
| 516 | _heapreplace = heapreplace |
| 517 | for elem in it: |
| 518 | if top < elem: |
Raymond Hettinger | 277842e | 2014-05-11 01:55:46 -0700 | [diff] [blame] | 519 | _heapreplace(result, (elem, order)) |
| 520 | top = result[0][0] |
Raymond Hettinger | 234fb2d | 2014-05-11 14:21:23 -0700 | [diff] [blame] | 521 | order -= 1 |
Raymond Hettinger | 277842e | 2014-05-11 01:55:46 -0700 | [diff] [blame] | 522 | result.sort(reverse=True) |
| 523 | return [r[0] for r in result] |
Benjamin Peterson | 18e9512 | 2009-01-18 22:46:33 +0000 | [diff] [blame] | 524 | |
| 525 | # General case, slowest method |
Raymond Hettinger | 277842e | 2014-05-11 01:55:46 -0700 | [diff] [blame] | 526 | it = iter(iterable) |
| 527 | result = [(key(elem), i, elem) for i, elem in zip(range(0, -n, -1), it)] |
| 528 | if not result: |
| 529 | return result |
| 530 | heapify(result) |
| 531 | order = -n |
| 532 | top = result[0][0] |
| 533 | _heapreplace = heapreplace |
| 534 | for elem in it: |
| 535 | k = key(elem) |
| 536 | if top < k: |
Raymond Hettinger | 277842e | 2014-05-11 01:55:46 -0700 | [diff] [blame] | 537 | _heapreplace(result, (k, order, elem)) |
| 538 | top = result[0][0] |
Raymond Hettinger | 234fb2d | 2014-05-11 14:21:23 -0700 | [diff] [blame] | 539 | order -= 1 |
Raymond Hettinger | 277842e | 2014-05-11 01:55:46 -0700 | [diff] [blame] | 540 | result.sort(reverse=True) |
| 541 | return [r[2] for r in result] |
| 542 | |
Raymond Hettinger | 4901a1f | 2004-12-02 08:59:14 +0000 | [diff] [blame] | 543 | |
Raymond Hettinger | c46cb2a | 2004-04-19 19:06:21 +0000 | [diff] [blame] | 544 | if __name__ == "__main__": |
| 545 | # Simple sanity test |
| 546 | heap = [] |
| 547 | data = [1, 3, 5, 7, 9, 2, 4, 6, 8, 0] |
| 548 | for item in data: |
| 549 | heappush(heap, item) |
| 550 | sort = [] |
| 551 | while heap: |
| 552 | sort.append(heappop(heap)) |
Guido van Rossum | be19ed7 | 2007-02-09 05:37:30 +0000 | [diff] [blame] | 553 | print(sort) |
Thomas Wouters | cf297e4 | 2007-02-23 15:07:44 +0000 | [diff] [blame] | 554 | |
| 555 | import doctest |
| 556 | doctest.testmod() |