Mercurial > cpython-lru_cache-weakref
annotate Lib/heapq.py @ 79675:c0a2294de787 2.7
Issue #16199: Backport 2to3 typo fix from 3.2.
| author | Chris Jerdonek <chris.jerdonek@gmail.com> |
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| date | Thu, 11 Oct 2012 16:15:46 -0700 |
| parents | 57f73b0f921c |
| children | aa9ed98a49cb |
| rev | line source |
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1 # -*- coding: latin-1 -*- |
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2 |
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3 """Heap queue algorithm (a.k.a. priority queue). |
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4 |
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5 Heaps are arrays for which a[k] <= a[2*k+1] and a[k] <= a[2*k+2] for |
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6 all k, counting elements from 0. For the sake of comparison, |
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7 non-existing elements are considered to be infinite. The interesting |
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8 property of a heap is that a[0] is always its smallest element. |
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9 |
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10 Usage: |
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11 |
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12 heap = [] # creates an empty heap |
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13 heappush(heap, item) # pushes a new item on the heap |
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14 item = heappop(heap) # pops the smallest item from the heap |
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15 item = heap[0] # smallest item on the heap without popping it |
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16 heapify(x) # transforms list into a heap, in-place, in linear time |
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17 item = heapreplace(heap, item) # pops and returns smallest item, and adds |
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18 # new item; the heap size is unchanged |
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19 |
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20 Our API differs from textbook heap algorithms as follows: |
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21 |
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22 - We use 0-based indexing. This makes the relationship between the |
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23 index for a node and the indexes for its children slightly less |
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24 obvious, but is more suitable since Python uses 0-based indexing. |
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25 |
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26 - Our heappop() method returns the smallest item, not the largest. |
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27 |
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28 These two make it possible to view the heap as a regular Python list |
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29 without surprises: heap[0] is the smallest item, and heap.sort() |
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30 maintains the heap invariant! |
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31 """ |
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32 |
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SF patch #969791: Add nlargest() and nsmallest() to heapq.
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33 # Original code by Kevin O'Connor, augmented by Tim Peters and Raymond Hettinger |
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34 |
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35 __about__ = """Heap queues |
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36 |
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37 [explanation by François Pinard] |
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38 |
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39 Heaps are arrays for which a[k] <= a[2*k+1] and a[k] <= a[2*k+2] for |
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40 all k, counting elements from 0. For the sake of comparison, |
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41 non-existing elements are considered to be infinite. The interesting |
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42 property of a heap is that a[0] is always its smallest element. |
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43 |
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44 The strange invariant above is meant to be an efficient memory |
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45 representation for a tournament. The numbers below are `k', not a[k]: |
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46 |
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47 0 |
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48 |
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49 1 2 |
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50 |
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51 3 4 5 6 |
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52 |
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53 7 8 9 10 11 12 13 14 |
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54 |
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55 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 |
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56 |
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57 |
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58 In the tree above, each cell `k' is topping `2*k+1' and `2*k+2'. In |
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59 an usual binary tournament we see in sports, each cell is the winner |
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60 over the two cells it tops, and we can trace the winner down the tree |
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61 to see all opponents s/he had. However, in many computer applications |
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62 of such tournaments, we do not need to trace the history of a winner. |
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63 To be more memory efficient, when a winner is promoted, we try to |
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64 replace it by something else at a lower level, and the rule becomes |
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65 that a cell and the two cells it tops contain three different items, |
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66 but the top cell "wins" over the two topped cells. |
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67 |
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68 If this heap invariant is protected at all time, index 0 is clearly |
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69 the overall winner. The simplest algorithmic way to remove it and |
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70 find the "next" winner is to move some loser (let's say cell 30 in the |
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71 diagram above) into the 0 position, and then percolate this new 0 down |
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72 the tree, exchanging values, until the invariant is re-established. |
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73 This is clearly logarithmic on the total number of items in the tree. |
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74 By iterating over all items, you get an O(n ln n) sort. |
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75 |
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76 A nice feature of this sort is that you can efficiently insert new |
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77 items while the sort is going on, provided that the inserted items are |
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78 not "better" than the last 0'th element you extracted. This is |
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79 especially useful in simulation contexts, where the tree holds all |
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80 incoming events, and the "win" condition means the smallest scheduled |
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81 time. When an event schedule other events for execution, they are |
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82 scheduled into the future, so they can easily go into the heap. So, a |
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83 heap is a good structure for implementing schedulers (this is what I |
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84 used for my MIDI sequencer :-). |
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85 |
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86 Various structures for implementing schedulers have been extensively |
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87 studied, and heaps are good for this, as they are reasonably speedy, |
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88 the speed is almost constant, and the worst case is not much different |
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89 than the average case. However, there are other representations which |
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90 are more efficient overall, yet the worst cases might be terrible. |
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91 |
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92 Heaps are also very useful in big disk sorts. You most probably all |
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93 know that a big sort implies producing "runs" (which are pre-sorted |
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94 sequences, which size is usually related to the amount of CPU memory), |
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95 followed by a merging passes for these runs, which merging is often |
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96 very cleverly organised[1]. It is very important that the initial |
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97 sort produces the longest runs possible. Tournaments are a good way |
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98 to that. If, using all the memory available to hold a tournament, you |
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99 replace and percolate items that happen to fit the current run, you'll |
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100 produce runs which are twice the size of the memory for random input, |
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101 and much better for input fuzzily ordered. |
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102 |
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103 Moreover, if you output the 0'th item on disk and get an input which |
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104 may not fit in the current tournament (because the value "wins" over |
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105 the last output value), it cannot fit in the heap, so the size of the |
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106 heap decreases. The freed memory could be cleverly reused immediately |
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107 for progressively building a second heap, which grows at exactly the |
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108 same rate the first heap is melting. When the first heap completely |
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109 vanishes, you switch heaps and start a new run. Clever and quite |
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110 effective! |
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111 |
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112 In a word, heaps are useful memory structures to know. I use them in |
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113 a few applications, and I think it is good to keep a `heap' module |
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114 around. :-) |
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115 |
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116 -------------------- |
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117 [1] The disk balancing algorithms which are current, nowadays, are |
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118 more annoying than clever, and this is a consequence of the seeking |
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119 capabilities of the disks. On devices which cannot seek, like big |
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120 tape drives, the story was quite different, and one had to be very |
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121 clever to ensure (far in advance) that each tape movement will be the |
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122 most effective possible (that is, will best participate at |
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123 "progressing" the merge). Some tapes were even able to read |
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124 backwards, and this was also used to avoid the rewinding time. |
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125 Believe me, real good tape sorts were quite spectacular to watch! |
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126 From all times, sorting has always been a Great Art! :-) |
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127 """ |
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128 |
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129 __all__ = ['heappush', 'heappop', 'heapify', 'heapreplace', 'merge', |
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130 'nlargest', 'nsmallest', 'heappushpop'] |
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131 |
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132 from itertools import islice, repeat, count, imap, izip, tee, chain |
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133 from operator import itemgetter |
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134 import bisect |
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135 |
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136 def cmp_lt(x, y): |
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137 # Use __lt__ if available; otherwise, try __le__. |
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138 # In Py3.x, only __lt__ will be called. |
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139 return (x < y) if hasattr(x, '__lt__') else (not y <= x) |
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140 |
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141 def heappush(heap, item): |
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142 """Push item onto heap, maintaining the heap invariant.""" |
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143 heap.append(item) |
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144 _siftdown(heap, 0, len(heap)-1) |
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145 |
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146 def heappop(heap): |
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147 """Pop the smallest item off the heap, maintaining the heap invariant.""" |
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148 lastelt = heap.pop() # raises appropriate IndexError if heap is empty |
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149 if heap: |
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150 returnitem = heap[0] |
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151 heap[0] = lastelt |
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152 _siftup(heap, 0) |
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153 else: |
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154 returnitem = lastelt |
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155 return returnitem |
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156 |
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157 def heapreplace(heap, item): |
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158 """Pop and return the current smallest value, and add the new item. |
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159 |
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160 This is more efficient than heappop() followed by heappush(), and can be |
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161 more appropriate when using a fixed-size heap. Note that the value |
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162 returned may be larger than item! That constrains reasonable uses of |
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163 this routine unless written as part of a conditional replacement: |
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164 |
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165 if item > heap[0]: |
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166 item = heapreplace(heap, item) |
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167 """ |
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168 returnitem = heap[0] # raises appropriate IndexError if heap is empty |
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169 heap[0] = item |
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170 _siftup(heap, 0) |
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171 return returnitem |
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172 |
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173 def heappushpop(heap, item): |
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174 """Fast version of a heappush followed by a heappop.""" |
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175 if heap and cmp_lt(heap[0], item): |
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176 item, heap[0] = heap[0], item |
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177 _siftup(heap, 0) |
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178 return item |
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179 |
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180 def heapify(x): |
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181 """Transform list into a heap, in-place, in O(len(x)) time.""" |
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182 n = len(x) |
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183 # Transform bottom-up. The largest index there's any point to looking at |
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184 # is the largest with a child index in-range, so must have 2*i + 1 < n, |
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185 # or i < (n-1)/2. If n is even = 2*j, this is (2*j-1)/2 = j-1/2 so |
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186 # j-1 is the largest, which is n//2 - 1. If n is odd = 2*j+1, this is |
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187 # (2*j+1-1)/2 = j so j-1 is the largest, and that's again n//2-1. |
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188 for i in reversed(xrange(n//2)): |
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189 _siftup(x, i) |
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190 |
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191 def nlargest(n, iterable): |
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192 """Find the n largest elements in a dataset. |
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193 |
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194 Equivalent to: sorted(iterable, reverse=True)[:n] |
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195 """ |
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196 if n < 0: |
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197 return [] |
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198 it = iter(iterable) |
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199 result = list(islice(it, n)) |
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200 if not result: |
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201 return result |
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202 heapify(result) |
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203 _heappushpop = heappushpop |
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204 for elem in it: |
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205 _heappushpop(result, elem) |
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206 result.sort(reverse=True) |
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207 return result |
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208 |
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209 def nsmallest(n, iterable): |
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210 """Find the n smallest elements in a dataset. |
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211 |
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212 Equivalent to: sorted(iterable)[:n] |
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213 """ |
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214 if n < 0: |
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215 return [] |
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216 if hasattr(iterable, '__len__') and n * 10 <= len(iterable): |
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217 # For smaller values of n, the bisect method is faster than a minheap. |
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218 # It is also memory efficient, consuming only n elements of space. |
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219 it = iter(iterable) |
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220 result = sorted(islice(it, 0, n)) |
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221 if not result: |
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222 return result |
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223 insort = bisect.insort |
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224 pop = result.pop |
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225 los = result[-1] # los --> Largest of the nsmallest |
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226 for elem in it: |
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227 if cmp_lt(elem, los): |
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228 insort(result, elem) |
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229 pop() |
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230 los = result[-1] |
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231 return result |
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232 # An alternative approach manifests the whole iterable in memory but |
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233 # saves comparisons by heapifying all at once. Also, saves time |
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234 # over bisect.insort() which has O(n) data movement time for every |
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235 # insertion. Finding the n smallest of an m length iterable requires |
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236 # O(m) + O(n log m) comparisons. |
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237 h = list(iterable) |
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238 heapify(h) |
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239 return map(heappop, repeat(h, min(n, len(h)))) |
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240 |
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241 # 'heap' is a heap at all indices >= startpos, except possibly for pos. pos |
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242 # is the index of a leaf with a possibly out-of-order value. Restore the |
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243 # heap invariant. |
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244 def _siftdown(heap, startpos, pos): |
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245 newitem = heap[pos] |
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246 # Follow the path to the root, moving parents down until finding a place |
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247 # newitem fits. |
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248 while pos > startpos: |
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249 parentpos = (pos - 1) >> 1 |
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250 parent = heap[parentpos] |
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251 if cmp_lt(newitem, parent): |
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252 heap[pos] = parent |
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253 pos = parentpos |
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254 continue |
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255 break |
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256 heap[pos] = newitem |
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257 |
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258 # The child indices of heap index pos are already heaps, and we want to make |
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259 # a heap at index pos too. We do this by bubbling the smaller child of |
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260 # pos up (and so on with that child's children, etc) until hitting a leaf, |
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261 # then using _siftdown to move the oddball originally at index pos into place. |
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262 # |
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263 # We *could* break out of the loop as soon as we find a pos where newitem <= |
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264 # both its children, but turns out that's not a good idea, and despite that |
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265 # many books write the algorithm that way. During a heap pop, the last array |
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266 # element is sifted in, and that tends to be large, so that comparing it |
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267 # against values starting from the root usually doesn't pay (= usually doesn't |
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268 # get us out of the loop early). See Knuth, Volume 3, where this is |
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269 # explained and quantified in an exercise. |
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270 # |
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271 # Cutting the # of comparisons is important, since these routines have no |
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272 # way to extract "the priority" from an array element, so that intelligence |
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273 # is likely to be hiding in custom __cmp__ methods, or in array elements |
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274 # storing (priority, record) tuples. Comparisons are thus potentially |
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275 # expensive. |
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276 # |
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277 # On random arrays of length 1000, making this change cut the number of |
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278 # comparisons made by heapify() a little, and those made by exhaustive |
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279 # heappop() a lot, in accord with theory. Here are typical results from 3 |
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280 # runs (3 just to demonstrate how small the variance is): |
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281 # |
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282 # Compares needed by heapify Compares needed by 1000 heappops |
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283 # -------------------------- -------------------------------- |
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284 # 1837 cut to 1663 14996 cut to 8680 |
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285 # 1855 cut to 1659 14966 cut to 8678 |
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286 # 1847 cut to 1660 15024 cut to 8703 |
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287 # |
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288 # Building the heap by using heappush() 1000 times instead required |
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289 # 2198, 2148, and 2219 compares: heapify() is more efficient, when |
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290 # you can use it. |
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291 # |
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292 # The total compares needed by list.sort() on the same lists were 8627, |
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293 # 8627, and 8632 (this should be compared to the sum of heapify() and |
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294 # heappop() compares): list.sort() is (unsurprisingly!) more efficient |
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295 # for sorting. |
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296 |
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297 def _siftup(heap, pos): |
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298 endpos = len(heap) |
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299 startpos = pos |
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300 newitem = heap[pos] |
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301 # Bubble up the smaller child until hitting a leaf. |
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302 childpos = 2*pos + 1 # leftmost child position |
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303 while childpos < endpos: |
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304 # Set childpos to index of smaller child. |
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305 rightpos = childpos + 1 |
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306 if rightpos < endpos and not cmp_lt(heap[childpos], heap[rightpos]): |
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307 childpos = rightpos |
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308 # Move the smaller child up. |
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309 heap[pos] = heap[childpos] |
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310 pos = childpos |
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311 childpos = 2*pos + 1 |
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312 # The leaf at pos is empty now. Put newitem there, and bubble it up |
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313 # to its final resting place (by sifting its parents down). |
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314 heap[pos] = newitem |
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315 _siftdown(heap, startpos, pos) |
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316 |
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317 # If available, use C implementation |
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318 try: |
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319 from _heapq import * |
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320 except ImportError: |
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321 pass |
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322 |
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323 def merge(*iterables): |
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324 '''Merge multiple sorted inputs into a single sorted output. |
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325 |
| 40988 | 326 Similar to sorted(itertools.chain(*iterables)) but returns a generator, |
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327 does not pull the data into memory all at once, and assumes that each of |
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328 the input streams is already sorted (smallest to largest). |
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329 |
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330 >>> list(merge([1,3,5,7], [0,2,4,8], [5,10,15,20], [], [25])) |
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331 [0, 1, 2, 3, 4, 5, 5, 7, 8, 10, 15, 20, 25] |
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332 |
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333 ''' |
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334 _heappop, _heapreplace, _StopIteration = heappop, heapreplace, StopIteration |
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335 |
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336 h = [] |
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337 h_append = h.append |
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338 for itnum, it in enumerate(map(iter, iterables)): |
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339 try: |
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340 next = it.next |
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341 h_append([next(), itnum, next]) |
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342 except _StopIteration: |
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343 pass |
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344 heapify(h) |
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345 |
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346 while 1: |
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347 try: |
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348 while 1: |
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349 v, itnum, next = s = h[0] # raises IndexError when h is empty |
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350 yield v |
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351 s[0] = next() # raises StopIteration when exhausted |
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352 _heapreplace(h, s) # restore heap condition |
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353 except _StopIteration: |
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354 _heappop(h) # remove empty iterator |
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355 except IndexError: |
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356 return |
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357 |
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358 # Extend the implementations of nsmallest and nlargest to use a key= argument |
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359 _nsmallest = nsmallest |
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360 def nsmallest(n, iterable, key=None): |
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361 """Find the n smallest elements in a dataset. |
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362 |
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363 Equivalent to: sorted(iterable, key=key)[:n] |
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364 """ |
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365 # Short-cut for n==1 is to use min() when len(iterable)>0 |
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366 if n == 1: |
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367 it = iter(iterable) |
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368 head = list(islice(it, 1)) |
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369 if not head: |
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370 return [] |
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371 if key is None: |
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372 return [min(chain(head, it))] |
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373 return [min(chain(head, it), key=key)] |
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374 |
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375 # When n>=size, it's faster to use sorted() |
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376 try: |
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377 size = len(iterable) |
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378 except (TypeError, AttributeError): |
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379 pass |
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380 else: |
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381 if n >= size: |
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382 return sorted(iterable, key=key)[:n] |
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383 |
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384 # When key is none, use simpler decoration |
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385 if key is None: |
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386 it = izip(iterable, count()) # decorate |
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387 result = _nsmallest(n, it) |
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388 return map(itemgetter(0), result) # undecorate |
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389 |
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390 # General case, slowest method |
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391 in1, in2 = tee(iterable) |
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392 it = izip(imap(key, in1), count(), in2) # decorate |
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393 result = _nsmallest(n, it) |
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394 return map(itemgetter(2), result) # undecorate |
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395 |
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396 _nlargest = nlargest |
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397 def nlargest(n, iterable, key=None): |
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398 """Find the n largest elements in a dataset. |
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399 |
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400 Equivalent to: sorted(iterable, key=key, reverse=True)[:n] |
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401 """ |
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402 |
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403 # Short-cut for n==1 is to use max() when len(iterable)>0 |
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404 if n == 1: |
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405 it = iter(iterable) |
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406 head = list(islice(it, 1)) |
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407 if not head: |
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408 return [] |
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409 if key is None: |
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410 return [max(chain(head, it))] |
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411 return [max(chain(head, it), key=key)] |
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412 |
|
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413 # When n>=size, it's faster to use sorted() |
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414 try: |
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415 size = len(iterable) |
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416 except (TypeError, AttributeError): |
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417 pass |
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418 else: |
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419 if n >= size: |
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420 return sorted(iterable, key=key, reverse=True)[:n] |
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421 |
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422 # When key is none, use simpler decoration |
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423 if key is None: |
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424 it = izip(iterable, count(0,-1)) # decorate |
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425 result = _nlargest(n, it) |
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426 return map(itemgetter(0), result) # undecorate |
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427 |
|
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428 # General case, slowest method |
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429 in1, in2 = tee(iterable) |
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430 it = izip(imap(key, in1), count(0,-1), in2) # decorate |
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431 result = _nlargest(n, it) |
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432 return map(itemgetter(2), result) # undecorate |
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433 |
|
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434 if __name__ == "__main__": |
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435 # Simple sanity test |
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436 heap = [] |
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437 data = [1, 3, 5, 7, 9, 2, 4, 6, 8, 0] |
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438 for item in data: |
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439 heappush(heap, item) |
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440 sort = [] |
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63b68cb2b442
* Restore the pure python version of heapq.py.
Raymond Hettinger <python@rcn.com>
parents:
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441 while heap: |
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63b68cb2b442
* Restore the pure python version of heapq.py.
Raymond Hettinger <python@rcn.com>
parents:
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442 sort.append(heappop(heap)) |
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63b68cb2b442
* Restore the pure python version of heapq.py.
Raymond Hettinger <python@rcn.com>
parents:
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443 print sort |
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40866
36d7b3b6ee59
Add merge() function to heapq.
Raymond Hettinger <python@rcn.com>
parents:
40541
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444 |
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36d7b3b6ee59
Add merge() function to heapq.
Raymond Hettinger <python@rcn.com>
parents:
40541
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445 import doctest |
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36d7b3b6ee59
Add merge() function to heapq.
Raymond Hettinger <python@rcn.com>
parents:
40541
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446 doctest.testmod() |
