Issue #21424: Optimize heaqp.nlargest() to make fewer tuple comparisons.

Consolidates the logic for nlargest() into a single function so that
decoration tuples (elem,order) or (key, order, elem) only need to
be formed when a new element is added to the heap.  Formerly, a tuple
was created for every element regardless of whether it was added to
the heap.

The change reduces the number of tuples created, the number of ordering
integers created, and total number of tuple comparisons.

1 files changed, 37 insertions, 88 deletions

diff --git a/Lib/heapq.py b/Lib/heapq.py
index 88c7019..fc73df9 100644
--- a/Lib/heapq.py
+++ b/Lib/heapq.py
@@ -192,81 +192,6 @@ def _heapify_max(x):
     for i in reversed(range(n//2)):
         _siftup_max(x, i)
 
-
-# Algorithm notes for nlargest() and nsmallest()
-# ==============================================
-#
-# Makes just one pass over the data while keeping the n most extreme values
-# in a heap.  Memory consumption is limited to keeping n values in a list.
-#
-# Number of comparisons for n random inputs, keeping the k smallest values:
-# -----------------------------------------------------------
-# Step   Comparisons                 Action
-#  1        1.66*k                   heapify the first k-inputs
-#  2        n - k                    compare new input elements to top of heap
-#  3        k*lg2(k)*(ln(n)-ln(k))   add new extreme values to the heap
-#  4        k*lg2(k)                 final sort of the k most extreme values
-#
-#                                      number of comparisons
-# n-random inputs   k-extreme values    average of 5 trials     % more than min()
-# ---------------   ----------------    -------------------     -----------------
-#       10,000            100                   14,046                 40.5%
-#      100,000            100                  105,749                  5.7%
-#    1,000,000            100                1,007,751                  0.8%
-#
-# Computing the number of comparisons for step 3:
-# -----------------------------------------------
-# * For the i-th new value from the iterable, the probability of being in the
-#   k most extreme values is k/i.  For example, the probability of the 101st
-#   value seen being in the 100 most extreme values is 100/101.
-# * If the value is a new extreme value, the cost of inserting it into the
-#   heap is log(k, 2).
-# * The probabilty times the cost gives:
-#            (k/i) * log(k, 2)
-# * Summing across the remaining n-k elements gives:
-#            sum((k/i) * log(k, 2) for xrange(k+1, n+1))
-# * This reduces to:
-#            (H(n) - H(k)) * k * log(k, 2)
-# * Where H(n) is the n-th harmonic number estimated by:
-#            H(n) = log(n, e) + gamma + 1.0 / (2.0 * n)
-#            gamma = 0.5772156649
-#   http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Rate_of_divergence
-# * Substituting the H(n) formula and ignoring the (1/2*n) fraction gives:
-#            comparisons = k * log(k, 2) * (log(n,e) - log(k, e))
-#
-# Worst-case for step 3:
-# ----------------------
-# In the worst case, the input data is reversed sorted so that every new element
-# must be inserted in the heap:
-#             comparisons = log(k, 2) * (n - k)
-#
-# Alternative Algorithms
-# ----------------------
-# Other algorithms were not used because they:
-# 1) Took much more auxiliary memory,
-# 2) Made multiple passes over the data.
-# 3) Made more comparisons in common cases (small k, large n, semi-random input).
-# See detailed comparisons at:
-# http://code.activestate.com/recipes/577573-compare-algorithms-for-heapqsmallest
-
-def nlargest(n, iterable):
-    """Find the n largest elements in a dataset.
-
-    Equivalent to:  sorted(iterable, reverse=True)[:n]
-    """
-    if n <= 0:
-        return []
-    it = iter(iterable)
-    result = list(islice(it, n))
-    if not result:
-        return result
-    heapify(result)
-    _heappushpop = heappushpop
-    for elem in it:
-        _heappushpop(result, elem)
-    result.sort(reverse=True)
-    return result
-
 def nsmallest(n, iterable):
     """Find the n smallest elements in a dataset.
 
@@ -480,7 +405,6 @@ def nsmallest(n, iterable, key=None):
     result = _nsmallest(n, it)
     return [r[2] for r in result]                           # undecorate
 
-_nlargest = nlargest
 def nlargest(n, iterable, key=None):
     """Find the n largest elements in a dataset.
 
@@ -490,12 +414,12 @@ def nlargest(n, iterable, key=None):
     # Short-cut for n==1 is to use max() when len(iterable)>0
     if n == 1:
         it = iter(iterable)
-        head = list(islice(it, 1))
-        if not head:
-            return []
+        sentinel = object()
         if key is None:
-            return [max(chain(head, it))]
-        return [max(chain(head, it), key=key)]
+            result = max(it, default=sentinel)
+        else:
+            result = max(it, default=sentinel, key=key)
+        return [] if result is sentinel else [result]
 
     # When n>=size, it's faster to use sorted()
     try:
@@ -508,15 +432,40 @@ def nlargest(n, iterable, key=None):
 
     # When key is none, use simpler decoration
     if key is None:
-        it = zip(iterable, count(0,-1))                     # decorate
-        result = _nlargest(n, it)
-        return [r[0] for r in result]                       # undecorate
+        it = iter(iterable)
+        result = list(islice(zip(it, count(0, -1)), n))
+        if not result:
+            return result
+        heapify(result)
+        order = -n
+        top = result[0][0]
+        _heapreplace = heapreplace
+        for elem in it:
+            if top < elem:
+                order -= 1
+                _heapreplace(result, (elem, order))
+                top = result[0][0]
+        result.sort(reverse=True)
+        return [r[0] for r in result]
 
     # General case, slowest method
-    in1, in2 = tee(iterable)
-    it = zip(map(key, in1), count(0,-1), in2)               # decorate
-    result = _nlargest(n, it)
-    return [r[2] for r in result]                           # undecorate
+    it = iter(iterable)
+    result = [(key(elem), i, elem) for i, elem in zip(range(0, -n, -1), it)]
+    if not result:
+        return result
+    heapify(result)
+    order = -n
+    top = result[0][0]
+    _heapreplace = heapreplace
+    for elem in it:
+        k = key(elem)
+        if top < k:
+            order -= 1
+            _heapreplace(result, (k, order, elem))
+            top = result[0][0]
+    result.sort(reverse=True)
+    return [r[2] for r in result]
+
 
 if __name__ == "__main__":
     # Simple sanity test