1 | from __future__ import print_function # for OPy compiler
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2 | """Heap queue algorithm (a.k.a. priority queue).
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3 |
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4 | Heaps are arrays for which a[k] <= a[2*k+1] and a[k] <= a[2*k+2] for
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5 | all k, counting elements from 0. For the sake of comparison,
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6 | non-existing elements are considered to be infinite. The interesting
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7 | property of a heap is that a[0] is always its smallest element.
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8 |
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9 | Usage:
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10 |
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11 | heap = [] # creates an empty heap
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12 | heappush(heap, item) # pushes a new item on the heap
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13 | item = heappop(heap) # pops the smallest item from the heap
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14 | item = heap[0] # smallest item on the heap without popping it
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15 | heapify(x) # transforms list into a heap, in-place, in linear time
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16 | item = heapreplace(heap, item) # pops and returns smallest item, and adds
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17 | # new item; the heap size is unchanged
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18 |
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19 | Our API differs from textbook heap algorithms as follows:
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20 |
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21 | - We use 0-based indexing. This makes the relationship between the
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22 | index for a node and the indexes for its children slightly less
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23 | obvious, but is more suitable since Python uses 0-based indexing.
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24 |
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25 | - Our heappop() method returns the smallest item, not the largest.
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26 |
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27 | These two make it possible to view the heap as a regular Python list
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28 | without surprises: heap[0] is the smallest item, and heap.sort()
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29 | maintains the heap invariant!
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30 | """
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31 |
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32 | # Original code by Kevin O'Connor, augmented by Tim Peters and Raymond Hettinger
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33 |
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34 | __about__ = """Heap queues
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35 |
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36 | [explanation by Francois Pinard]
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37 |
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38 | Heaps are arrays for which a[k] <= a[2*k+1] and a[k] <= a[2*k+2] for
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39 | all k, counting elements from 0. For the sake of comparison,
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40 | non-existing elements are considered to be infinite. The interesting
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41 | property of a heap is that a[0] is always its smallest element.
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42 |
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43 | The strange invariant above is meant to be an efficient memory
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44 | representation for a tournament. The numbers below are `k', not a[k]:
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45 |
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46 | 0
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47 |
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48 | 1 2
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49 |
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50 | 3 4 5 6
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51 |
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52 | 7 8 9 10 11 12 13 14
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53 |
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54 | 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
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55 |
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56 |
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57 | In the tree above, each cell `k' is topping `2*k+1' and `2*k+2'. In
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58 | a usual binary tournament we see in sports, each cell is the winner
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59 | over the two cells it tops, and we can trace the winner down the tree
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60 | to see all opponents s/he had. However, in many computer applications
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61 | of such tournaments, we do not need to trace the history of a winner.
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62 | To be more memory efficient, when a winner is promoted, we try to
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63 | replace it by something else at a lower level, and the rule becomes
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64 | that a cell and the two cells it tops contain three different items,
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65 | but the top cell "wins" over the two topped cells.
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66 |
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67 | If this heap invariant is protected at all time, index 0 is clearly
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68 | the overall winner. The simplest algorithmic way to remove it and
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69 | find the "next" winner is to move some loser (let's say cell 30 in the
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70 | diagram above) into the 0 position, and then percolate this new 0 down
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71 | the tree, exchanging values, until the invariant is re-established.
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72 | This is clearly logarithmic on the total number of items in the tree.
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73 | By iterating over all items, you get an O(n ln n) sort.
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74 |
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75 | A nice feature of this sort is that you can efficiently insert new
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76 | items while the sort is going on, provided that the inserted items are
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77 | not "better" than the last 0'th element you extracted. This is
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78 | especially useful in simulation contexts, where the tree holds all
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79 | incoming events, and the "win" condition means the smallest scheduled
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80 | time. When an event schedule other events for execution, they are
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81 | scheduled into the future, so they can easily go into the heap. So, a
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82 | heap is a good structure for implementing schedulers (this is what I
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83 | used for my MIDI sequencer :-).
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84 |
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85 | Various structures for implementing schedulers have been extensively
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86 | studied, and heaps are good for this, as they are reasonably speedy,
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87 | the speed is almost constant, and the worst case is not much different
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88 | than the average case. However, there are other representations which
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89 | are more efficient overall, yet the worst cases might be terrible.
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90 |
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91 | Heaps are also very useful in big disk sorts. You most probably all
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92 | know that a big sort implies producing "runs" (which are pre-sorted
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93 | sequences, which size is usually related to the amount of CPU memory),
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94 | followed by a merging passes for these runs, which merging is often
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95 | very cleverly organised[1]. It is very important that the initial
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96 | sort produces the longest runs possible. Tournaments are a good way
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97 | to that. If, using all the memory available to hold a tournament, you
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98 | replace and percolate items that happen to fit the current run, you'll
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99 | produce runs which are twice the size of the memory for random input,
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100 | and much better for input fuzzily ordered.
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101 |
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102 | Moreover, if you output the 0'th item on disk and get an input which
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103 | may not fit in the current tournament (because the value "wins" over
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104 | the last output value), it cannot fit in the heap, so the size of the
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105 | heap decreases. The freed memory could be cleverly reused immediately
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106 | for progressively building a second heap, which grows at exactly the
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107 | same rate the first heap is melting. When the first heap completely
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108 | vanishes, you switch heaps and start a new run. Clever and quite
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109 | effective!
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110 |
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111 | In a word, heaps are useful memory structures to know. I use them in
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112 | a few applications, and I think it is good to keep a `heap' module
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113 | around. :-)
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114 |
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115 | --------------------
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116 | [1] The disk balancing algorithms which are current, nowadays, are
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117 | more annoying than clever, and this is a consequence of the seeking
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118 | capabilities of the disks. On devices which cannot seek, like big
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119 | tape drives, the story was quite different, and one had to be very
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120 | clever to ensure (far in advance) that each tape movement will be the
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121 | most effective possible (that is, will best participate at
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122 | "progressing" the merge). Some tapes were even able to read
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123 | backwards, and this was also used to avoid the rewinding time.
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124 | Believe me, real good tape sorts were quite spectacular to watch!
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125 | From all times, sorting has always been a Great Art! :-)
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126 | """
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127 |
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128 | __all__ = ['heappush', 'heappop', 'heapify', 'heapreplace', 'merge',
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129 | 'nlargest', 'nsmallest', 'heappushpop']
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130 |
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131 | from itertools import islice, count, imap, izip, tee, chain
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132 | from operator import itemgetter
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133 |
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134 | def cmp_lt(x, y):
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135 | # Use __lt__ if available; otherwise, try __le__.
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136 | # In Py3.x, only __lt__ will be called.
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137 | return (x < y) if hasattr(x, '__lt__') else (not y <= x)
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138 |
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139 | def heappush(heap, item):
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140 | """Push item onto heap, maintaining the heap invariant."""
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141 | heap.append(item)
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142 | _siftdown(heap, 0, len(heap)-1)
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143 |
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144 | def heappop(heap):
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145 | """Pop the smallest item off the heap, maintaining the heap invariant."""
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146 | lastelt = heap.pop() # raises appropriate IndexError if heap is empty
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147 | if heap:
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148 | returnitem = heap[0]
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149 | heap[0] = lastelt
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150 | _siftup(heap, 0)
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151 | else:
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152 | returnitem = lastelt
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153 | return returnitem
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154 |
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155 | def heapreplace(heap, item):
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156 | """Pop and return the current smallest value, and add the new item.
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157 |
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158 | This is more efficient than heappop() followed by heappush(), and can be
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159 | more appropriate when using a fixed-size heap. Note that the value
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160 | returned may be larger than item! That constrains reasonable uses of
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161 | this routine unless written as part of a conditional replacement:
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162 |
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163 | if item > heap[0]:
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164 | item = heapreplace(heap, item)
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165 | """
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166 | returnitem = heap[0] # raises appropriate IndexError if heap is empty
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167 | heap[0] = item
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168 | _siftup(heap, 0)
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169 | return returnitem
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170 |
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171 | def heappushpop(heap, item):
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172 | """Fast version of a heappush followed by a heappop."""
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173 | if heap and cmp_lt(heap[0], item):
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174 | item, heap[0] = heap[0], item
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175 | _siftup(heap, 0)
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176 | return item
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177 |
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178 | def heapify(x):
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179 | """Transform list into a heap, in-place, in O(len(x)) time."""
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180 | n = len(x)
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181 | # Transform bottom-up. The largest index there's any point to looking at
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182 | # is the largest with a child index in-range, so must have 2*i + 1 < n,
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183 | # 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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184 | # j-1 is the largest, which is n//2 - 1. If n is odd = 2*j+1, this is
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185 | # (2*j+1-1)/2 = j so j-1 is the largest, and that's again n//2-1.
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186 | for i in reversed(xrange(n//2)):
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187 | _siftup(x, i)
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188 |
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189 | def _heappushpop_max(heap, item):
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190 | """Maxheap version of a heappush followed by a heappop."""
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191 | if heap and cmp_lt(item, heap[0]):
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192 | item, heap[0] = heap[0], item
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193 | _siftup_max(heap, 0)
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194 | return item
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195 |
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196 | def _heapify_max(x):
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197 | """Transform list into a maxheap, in-place, in O(len(x)) time."""
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198 | n = len(x)
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199 | for i in reversed(range(n//2)):
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200 | _siftup_max(x, i)
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201 |
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202 | def nlargest(n, iterable):
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203 | """Find the n largest elements in a dataset.
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204 |
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205 | Equivalent to: sorted(iterable, reverse=True)[:n]
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206 | """
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207 | if n < 0:
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208 | return []
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209 | it = iter(iterable)
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210 | result = list(islice(it, n))
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211 | if not result:
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212 | return result
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213 | heapify(result)
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214 | _heappushpop = heappushpop
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215 | for elem in it:
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216 | _heappushpop(result, elem)
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217 | result.sort(reverse=True)
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218 | return result
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219 |
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220 | def nsmallest(n, iterable):
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221 | """Find the n smallest elements in a dataset.
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222 |
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223 | Equivalent to: sorted(iterable)[:n]
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224 | """
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225 | if n < 0:
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226 | return []
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227 | it = iter(iterable)
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228 | result = list(islice(it, n))
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229 | if not result:
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230 | return result
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231 | _heapify_max(result)
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232 | _heappushpop = _heappushpop_max
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233 | for elem in it:
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234 | _heappushpop(result, elem)
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235 | result.sort()
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236 | return result
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237 |
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238 | # 'heap' is a heap at all indices >= startpos, except possibly for pos. pos
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239 | # is the index of a leaf with a possibly out-of-order value. Restore the
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240 | # heap invariant.
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241 | def _siftdown(heap, startpos, pos):
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242 | newitem = heap[pos]
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243 | # Follow the path to the root, moving parents down until finding a place
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244 | # newitem fits.
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245 | while pos > startpos:
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246 | parentpos = (pos - 1) >> 1
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247 | parent = heap[parentpos]
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248 | if cmp_lt(newitem, parent):
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249 | heap[pos] = parent
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250 | pos = parentpos
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251 | continue
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252 | break
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253 | heap[pos] = newitem
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254 |
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255 | # The child indices of heap index pos are already heaps, and we want to make
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256 | # a heap at index pos too. We do this by bubbling the smaller child of
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257 | # pos up (and so on with that child's children, etc) until hitting a leaf,
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258 | # then using _siftdown to move the oddball originally at index pos into place.
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259 | #
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260 | # We *could* break out of the loop as soon as we find a pos where newitem <=
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261 | # both its children, but turns out that's not a good idea, and despite that
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262 | # many books write the algorithm that way. During a heap pop, the last array
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263 | # element is sifted in, and that tends to be large, so that comparing it
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264 | # against values starting from the root usually doesn't pay (= usually doesn't
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265 | # get us out of the loop early). See Knuth, Volume 3, where this is
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266 | # explained and quantified in an exercise.
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267 | #
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268 | # Cutting the # of comparisons is important, since these routines have no
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269 | # way to extract "the priority" from an array element, so that intelligence
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270 | # is likely to be hiding in custom __cmp__ methods, or in array elements
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271 | # storing (priority, record) tuples. Comparisons are thus potentially
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272 | # expensive.
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273 | #
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274 | # On random arrays of length 1000, making this change cut the number of
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275 | # comparisons made by heapify() a little, and those made by exhaustive
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276 | # heappop() a lot, in accord with theory. Here are typical results from 3
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277 | # runs (3 just to demonstrate how small the variance is):
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278 | #
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279 | # Compares needed by heapify Compares needed by 1000 heappops
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280 | # -------------------------- --------------------------------
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281 | # 1837 cut to 1663 14996 cut to 8680
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282 | # 1855 cut to 1659 14966 cut to 8678
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283 | # 1847 cut to 1660 15024 cut to 8703
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284 | #
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285 | # Building the heap by using heappush() 1000 times instead required
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286 | # 2198, 2148, and 2219 compares: heapify() is more efficient, when
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287 | # you can use it.
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288 | #
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289 | # The total compares needed by list.sort() on the same lists were 8627,
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290 | # 8627, and 8632 (this should be compared to the sum of heapify() and
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291 | # heappop() compares): list.sort() is (unsurprisingly!) more efficient
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292 | # for sorting.
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293 |
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294 | def _siftup(heap, pos):
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295 | endpos = len(heap)
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296 | startpos = pos
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297 | newitem = heap[pos]
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298 | # Bubble up the smaller child until hitting a leaf.
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299 | childpos = 2*pos + 1 # leftmost child position
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300 | while childpos < endpos:
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301 | # Set childpos to index of smaller child.
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302 | rightpos = childpos + 1
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303 | if rightpos < endpos and not cmp_lt(heap[childpos], heap[rightpos]):
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304 | childpos = rightpos
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305 | # Move the smaller child up.
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306 | heap[pos] = heap[childpos]
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307 | pos = childpos
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308 | childpos = 2*pos + 1
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309 | # The leaf at pos is empty now. Put newitem there, and bubble it up
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310 | # to its final resting place (by sifting its parents down).
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311 | heap[pos] = newitem
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312 | _siftdown(heap, startpos, pos)
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313 |
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314 | def _siftdown_max(heap, startpos, pos):
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315 | 'Maxheap variant of _siftdown'
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316 | newitem = heap[pos]
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317 | # Follow the path to the root, moving parents down until finding a place
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318 | # newitem fits.
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319 | while pos > startpos:
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320 | parentpos = (pos - 1) >> 1
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321 | parent = heap[parentpos]
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322 | if cmp_lt(parent, newitem):
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323 | heap[pos] = parent
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324 | pos = parentpos
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325 | continue
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326 | break
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327 | heap[pos] = newitem
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328 |
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329 | def _siftup_max(heap, pos):
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330 | 'Maxheap variant of _siftup'
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331 | endpos = len(heap)
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332 | startpos = pos
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333 | newitem = heap[pos]
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334 | # Bubble up the larger child until hitting a leaf.
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335 | childpos = 2*pos + 1 # leftmost child position
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336 | while childpos < endpos:
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337 | # Set childpos to index of larger child.
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338 | rightpos = childpos + 1
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339 | if rightpos < endpos and not cmp_lt(heap[rightpos], heap[childpos]):
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340 | childpos = rightpos
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341 | # Move the larger child up.
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342 | heap[pos] = heap[childpos]
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343 | pos = childpos
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344 | childpos = 2*pos + 1
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345 | # The leaf at pos is empty now. Put newitem there, and bubble it up
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346 | # to its final resting place (by sifting its parents down).
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347 | heap[pos] = newitem
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348 | _siftdown_max(heap, startpos, pos)
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349 |
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350 | # If available, use C implementation
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351 | try:
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352 | from _heapq import *
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353 | except ImportError:
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354 | pass
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355 |
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356 | def merge(*iterables):
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357 | '''Merge multiple sorted inputs into a single sorted output.
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358 |
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359 | Similar to sorted(itertools.chain(*iterables)) but returns a generator,
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360 | does not pull the data into memory all at once, and assumes that each of
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361 | the input streams is already sorted (smallest to largest).
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362 |
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363 | >>> list(merge([1,3,5,7], [0,2,4,8], [5,10,15,20], [], [25]))
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364 | [0, 1, 2, 3, 4, 5, 5, 7, 8, 10, 15, 20, 25]
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365 |
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366 | '''
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367 | _heappop, _heapreplace, _StopIteration = heappop, heapreplace, StopIteration
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368 | _len = len
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369 |
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370 | h = []
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371 | h_append = h.append
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372 | for itnum, it in enumerate(map(iter, iterables)):
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373 | try:
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374 | next = it.next
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375 | h_append([next(), itnum, next])
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376 | except _StopIteration:
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377 | pass
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378 | heapify(h)
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379 |
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380 | while _len(h) > 1:
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381 | try:
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382 | while 1:
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383 | v, itnum, next = s = h[0]
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384 | yield v
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385 | s[0] = next() # raises StopIteration when exhausted
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386 | _heapreplace(h, s) # restore heap condition
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387 | except _StopIteration:
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388 | _heappop(h) # remove empty iterator
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389 | if h:
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390 | # fast case when only a single iterator remains
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391 | v, itnum, next = h[0]
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392 | yield v
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393 | for v in next.__self__:
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394 | yield v
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395 |
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396 | # Extend the implementations of nsmallest and nlargest to use a key= argument
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397 | _nsmallest = nsmallest
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398 | def nsmallest(n, iterable, key=None):
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399 | """Find the n smallest elements in a dataset.
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400 |
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401 | Equivalent to: sorted(iterable, key=key)[:n]
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402 | """
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403 | # Short-cut for n==1 is to use min() 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 [min(chain(head, it))]
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411 | return [min(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
|
418 | else:
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419 | if n >= size:
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420 | return sorted(iterable, key=key)[: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()) # decorate
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425 | result = _nsmallest(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(), in2) # decorate
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431 | result = _nsmallest(n, it)
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432 | return map(itemgetter(2), result) # undecorate
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433 |
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434 | _nlargest = nlargest
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435 | def nlargest(n, iterable, key=None):
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436 | """Find the n largest elements in a dataset.
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437 |
|
438 | Equivalent to: sorted(iterable, key=key, reverse=True)[:n]
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439 | """
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440 |
|
441 | # Short-cut for n==1 is to use max() when len(iterable)>0
|
442 | if n == 1:
|
443 | it = iter(iterable)
|
444 | head = list(islice(it, 1))
|
445 | if not head:
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446 | return []
|
447 | if key is None:
|
448 | return [max(chain(head, it))]
|
449 | return [max(chain(head, it), key=key)]
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450 |
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451 | # When n>=size, it's faster to use sorted()
|
452 | try:
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453 | size = len(iterable)
|
454 | except (TypeError, AttributeError):
|
455 | pass
|
456 | else:
|
457 | if n >= size:
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458 | return sorted(iterable, key=key, reverse=True)[:n]
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459 |
|
460 | # When key is none, use simpler decoration
|
461 | if key is None:
|
462 | it = izip(iterable, count(0,-1)) # decorate
|
463 | result = _nlargest(n, it)
|
464 | return map(itemgetter(0), result) # undecorate
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465 |
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466 | # General case, slowest method
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467 | in1, in2 = tee(iterable)
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468 | it = izip(imap(key, in1), count(0,-1), in2) # decorate
|
469 | result = _nlargest(n, it)
|
470 | return map(itemgetter(2), result) # undecorate
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471 |
|
472 | if __name__ == "__main__":
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473 | # Simple sanity test
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474 | heap = []
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475 | data = [1, 3, 5, 7, 9, 2, 4, 6, 8, 0]
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476 | for item in data:
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477 | heappush(heap, item)
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478 | sort = []
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479 | while heap:
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480 | sort.append(heappop(heap))
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481 | print(sort)
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482 |
|
483 | import doctest
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484 | doctest.testmod()
|