github.com/grumpyhome/grumpy@v0.3.1-0.20201208125205-7b775405bdf1/grumpy-runtime-src/third_party/stdlib/heapq.py (about) 1 # -*- coding: latin-1 -*- 2 3 """Heap queue algorithm (a.k.a. priority queue). 4 5 Heaps are arrays for which a[k] <= a[2*k+1] and a[k] <= a[2*k+2] for 6 all k, counting elements from 0. For the sake of comparison, 7 non-existing elements are considered to be infinite. The interesting 8 property of a heap is that a[0] is always its smallest element. 9 10 Usage: 11 12 heap = [] # creates an empty heap 13 heappush(heap, item) # pushes a new item on the heap 14 item = heappop(heap) # pops the smallest item from the heap 15 item = heap[0] # smallest item on the heap without popping it 16 heapify(x) # transforms list into a heap, in-place, in linear time 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 import itertools 133 islice = itertools.islice 134 count = itertools.count 135 imap = itertools.imap 136 izip = itertools.izip 137 tee = itertools.tee 138 chain = itertools.chain 139 import operator 140 itemgetter = operator.itemgetter 141 142 def cmp_lt(x, y): 143 # Use __lt__ if available; otherwise, try __le__. 144 # In Py3.x, only __lt__ will be called. 145 return (x < y) if hasattr(x, '__lt__') else (not y <= x) 146 147 def heappush(heap, item): 148 """Push item onto heap, maintaining the heap invariant.""" 149 heap.append(item) 150 _siftdown(heap, 0, len(heap)-1) 151 152 def heappop(heap): 153 """Pop the smallest item off the heap, maintaining the heap invariant.""" 154 lastelt = heap.pop() # raises appropriate IndexError if heap is empty 155 if heap: 156 returnitem = heap[0] 157 heap[0] = lastelt 158 _siftup(heap, 0) 159 else: 160 returnitem = lastelt 161 return returnitem 162 163 def heapreplace(heap, item): 164 """Pop and return the current smallest value, and add the new item. 165 166 This is more efficient than heappop() followed by heappush(), and can be 167 more appropriate when using a fixed-size heap. Note that the value 168 returned may be larger than item! That constrains reasonable uses of 169 this routine unless written as part of a conditional replacement: 170 171 if item > heap[0]: 172 item = heapreplace(heap, item) 173 """ 174 returnitem = heap[0] # raises appropriate IndexError if heap is empty 175 heap[0] = item 176 _siftup(heap, 0) 177 return returnitem 178 179 def heappushpop(heap, item): 180 """Fast version of a heappush followed by a heappop.""" 181 if heap and cmp_lt(heap[0], item): 182 item, heap[0] = heap[0], item 183 _siftup(heap, 0) 184 return item 185 186 def heapify(x): 187 """Transform list into a heap, in-place, in O(len(x)) time.""" 188 n = len(x) 189 # Transform bottom-up. The largest index there's any point to looking at 190 # is the largest with a child index in-range, so must have 2*i + 1 < n, 191 # or i < (n-1)/2. If n is even = 2*j, this is (2*j-1)/2 = j-1/2 so 192 # j-1 is the largest, which is n//2 - 1. If n is odd = 2*j+1, this is 193 # (2*j+1-1)/2 = j so j-1 is the largest, and that's again n//2-1. 194 for i in reversed(xrange(n//2)): 195 _siftup(x, i) 196 197 def _heappushpop_max(heap, item): 198 """Maxheap version of a heappush followed by a heappop.""" 199 if heap and cmp_lt(item, heap[0]): 200 item, heap[0] = heap[0], item 201 _siftup_max(heap, 0) 202 return item 203 204 def _heapify_max(x): 205 """Transform list into a maxheap, in-place, in O(len(x)) time.""" 206 n = len(x) 207 for i in reversed(range(n//2)): 208 _siftup_max(x, i) 209 210 def nlargest(n, iterable): 211 """Find the n largest elements in a dataset. 212 213 Equivalent to: sorted(iterable, reverse=True)[:n] 214 """ 215 if n < 0: 216 return [] 217 it = iter(iterable) 218 result = list(islice(it, n)) 219 if not result: 220 return result 221 heapify(result) 222 _heappushpop = heappushpop 223 for elem in it: 224 _heappushpop(result, elem) 225 result.sort(reverse=True) 226 return result 227 228 def nsmallest(n, iterable): 229 """Find the n smallest elements in a dataset. 230 231 Equivalent to: sorted(iterable)[:n] 232 """ 233 if n < 0: 234 return [] 235 it = iter(iterable) 236 result = list(islice(it, n)) 237 if not result: 238 return result 239 _heapify_max(result) 240 _heappushpop = _heappushpop_max 241 for elem in it: 242 _heappushpop(result, elem) 243 result.sort() 244 return result 245 246 # 'heap' is a heap at all indices >= startpos, except possibly for pos. pos 247 # is the index of a leaf with a possibly out-of-order value. Restore the 248 # heap invariant. 249 def _siftdown(heap, startpos, pos): 250 newitem = heap[pos] 251 # Follow the path to the root, moving parents down until finding a place 252 # newitem fits. 253 while pos > startpos: 254 parentpos = (pos - 1) >> 1 255 parent = heap[parentpos] 256 if cmp_lt(newitem, parent): 257 heap[pos] = parent 258 pos = parentpos 259 continue 260 break 261 heap[pos] = newitem 262 263 # The child indices of heap index pos are already heaps, and we want to make 264 # a heap at index pos too. We do this by bubbling the smaller child of 265 # pos up (and so on with that child's children, etc) until hitting a leaf, 266 # then using _siftdown to move the oddball originally at index pos into place. 267 # 268 # We *could* break out of the loop as soon as we find a pos where newitem <= 269 # both its children, but turns out that's not a good idea, and despite that 270 # many books write the algorithm that way. During a heap pop, the last array 271 # element is sifted in, and that tends to be large, so that comparing it 272 # against values starting from the root usually doesn't pay (= usually doesn't 273 # get us out of the loop early). See Knuth, Volume 3, where this is 274 # explained and quantified in an exercise. 275 # 276 # Cutting the # of comparisons is important, since these routines have no 277 # way to extract "the priority" from an array element, so that intelligence 278 # is likely to be hiding in custom __cmp__ methods, or in array elements 279 # storing (priority, record) tuples. Comparisons are thus potentially 280 # expensive. 281 # 282 # On random arrays of length 1000, making this change cut the number of 283 # comparisons made by heapify() a little, and those made by exhaustive 284 # heappop() a lot, in accord with theory. Here are typical results from 3 285 # runs (3 just to demonstrate how small the variance is): 286 # 287 # Compares needed by heapify Compares needed by 1000 heappops 288 # -------------------------- -------------------------------- 289 # 1837 cut to 1663 14996 cut to 8680 290 # 1855 cut to 1659 14966 cut to 8678 291 # 1847 cut to 1660 15024 cut to 8703 292 # 293 # Building the heap by using heappush() 1000 times instead required 294 # 2198, 2148, and 2219 compares: heapify() is more efficient, when 295 # you can use it. 296 # 297 # The total compares needed by list.sort() on the same lists were 8627, 298 # 8627, and 8632 (this should be compared to the sum of heapify() and 299 # heappop() compares): list.sort() is (unsurprisingly!) more efficient 300 # for sorting. 301 302 def _siftup(heap, pos): 303 endpos = len(heap) 304 startpos = pos 305 newitem = heap[pos] 306 # Bubble up the smaller child until hitting a leaf. 307 childpos = 2*pos + 1 # leftmost child position 308 while childpos < endpos: 309 # Set childpos to index of smaller child. 310 rightpos = childpos + 1 311 if rightpos < endpos and not cmp_lt(heap[childpos], heap[rightpos]): 312 childpos = rightpos 313 # Move the smaller child up. 314 heap[pos] = heap[childpos] 315 pos = childpos 316 childpos = 2*pos + 1 317 # The leaf at pos is empty now. Put newitem there, and bubble it up 318 # to its final resting place (by sifting its parents down). 319 heap[pos] = newitem 320 _siftdown(heap, startpos, pos) 321 322 def _siftdown_max(heap, startpos, pos): 323 'Maxheap variant of _siftdown' 324 newitem = heap[pos] 325 # Follow the path to the root, moving parents down until finding a place 326 # newitem fits. 327 while pos > startpos: 328 parentpos = (pos - 1) >> 1 329 parent = heap[parentpos] 330 if cmp_lt(parent, newitem): 331 heap[pos] = parent 332 pos = parentpos 333 continue 334 break 335 heap[pos] = newitem 336 337 def _siftup_max(heap, pos): 338 'Maxheap variant of _siftup' 339 endpos = len(heap) 340 startpos = pos 341 newitem = heap[pos] 342 # Bubble up the larger child until hitting a leaf. 343 childpos = 2*pos + 1 # leftmost child position 344 while childpos < endpos: 345 # Set childpos to index of larger child. 346 rightpos = childpos + 1 347 if rightpos < endpos and not cmp_lt(heap[rightpos], heap[childpos]): 348 childpos = rightpos 349 # Move the larger child up. 350 heap[pos] = heap[childpos] 351 pos = childpos 352 childpos = 2*pos + 1 353 # The leaf at pos is empty now. Put newitem there, and bubble it up 354 # to its final resting place (by sifting its parents down). 355 heap[pos] = newitem 356 _siftdown_max(heap, startpos, pos) 357 358 # If available, use C implementation 359 #try: 360 # import _heapq 361 #except ImportError: 362 # pass 363 364 def merge(*iterables): 365 '''Merge multiple sorted inputs into a single sorted output. 366 367 Similar to sorted(itertools.chain(*iterables)) but returns a generator, 368 does not pull the data into memory all at once, and assumes that each of 369 the input streams is already sorted (smallest to largest). 370 371 >>> list(merge([1,3,5,7], [0,2,4,8], [5,10,15,20], [], [25])) 372 [0, 1, 2, 3, 4, 5, 5, 7, 8, 10, 15, 20, 25] 373 374 ''' 375 _heappop, _heapreplace, _StopIteration = heappop, heapreplace, StopIteration 376 _len = len 377 378 h = [] 379 h_append = h.append 380 for itnum, it in enumerate(map(iter, iterables)): 381 try: 382 next = it.next 383 h_append([next(), itnum, next]) 384 except _StopIteration: 385 pass 386 heapify(h) 387 388 while _len(h) > 1: 389 try: 390 while 1: 391 v, itnum, next = s = h[0] 392 yield v 393 s[0] = next() # raises StopIteration when exhausted 394 _heapreplace(h, s) # restore heap condition 395 except _StopIteration: 396 _heappop(h) # remove empty iterator 397 if h: 398 # fast case when only a single iterator remains 399 v, itnum, next = h[0] 400 yield v 401 for v in next.__self__: 402 yield v 403 404 # Extend the implementations of nsmallest and nlargest to use a key= argument 405 _nsmallest = nsmallest 406 def nsmallest(n, iterable, key=None): 407 """Find the n smallest elements in a dataset. 408 409 Equivalent to: sorted(iterable, key=key)[:n] 410 """ 411 # Short-cut for n==1 is to use min() when len(iterable)>0 412 if n == 1: 413 it = iter(iterable) 414 head = list(islice(it, 1)) 415 if not head: 416 return [] 417 if key is None: 418 return [min(chain(head, it))] 419 return [min(chain(head, it), key=key)] 420 421 # When n>=size, it's faster to use sorted() 422 try: 423 size = len(iterable) 424 except (TypeError, AttributeError): 425 pass 426 else: 427 if n >= size: 428 return sorted(iterable, key=key)[:n] 429 430 # When key is none, use simpler decoration 431 if key is None: 432 it = izip(iterable, count()) # decorate 433 result = _nsmallest(n, it) 434 return map(itemgetter(0), result) # undecorate 435 436 # General case, slowest method 437 in1, in2 = tee(iterable) 438 it = izip(imap(key, in1), count(), in2) # decorate 439 result = _nsmallest(n, it) 440 return map(itemgetter(2), result) # undecorate 441 442 _nlargest = nlargest 443 def nlargest(n, iterable, key=None): 444 """Find the n largest elements in a dataset. 445 446 Equivalent to: sorted(iterable, key=key, reverse=True)[:n] 447 """ 448 449 # Short-cut for n==1 is to use max() when len(iterable)>0 450 if n == 1: 451 it = iter(iterable) 452 head = list(islice(it, 1)) 453 if not head: 454 return [] 455 if key is None: 456 return [max(chain(head, it))] 457 return [max(chain(head, it), key=key)] 458 459 # When n>=size, it's faster to use sorted() 460 try: 461 size = len(iterable) 462 except (TypeError, AttributeError): 463 pass 464 else: 465 if n >= size: 466 return sorted(iterable, key=key, reverse=True)[:n] 467 468 # When key is none, use simpler decoration 469 if key is None: 470 it = izip(iterable, count(0,-1)) # decorate 471 result = _nlargest(n, it) 472 return map(itemgetter(0), result) # undecorate 473 474 # General case, slowest method 475 in1, in2 = tee(iterable) 476 it = izip(imap(key, in1), count(0,-1), in2) # decorate 477 result = _nlargest(n, it) 478 return map(itemgetter(2), result) # undecorate 479 480 #if __name__ == "__main__": 481 # # Simple sanity test 482 # heap = [] 483 # data = [1, 3, 5, 7, 9, 2, 4, 6, 8, 0] 484 # for item in data: 485 # heappush(heap, item) 486 # sort = [] 487 # while heap: 488 # sort.append(heappop(heap)) 489 # print sort 490 # 491 # import doctest 492 # doctest.testmod()