Hmm! I thought I checked this in before! Oh well.

Added new heapify() function, which transforms an arbitrary list into a
heap in linear time; that's a fundamental tool for using heaps in real
life <wink>.

Added heapyify() test.  Added a "less naive" N-best algorithm to the test
suite, and noted that this could actually go much faster (building on
heapify()) if we had max-heaps instead of min-heaps (the iterative method
is appropriate when all the data isn't known in advance, but when it is
known in advance the tradeoffs get murkier).

Lib/heapq.py

@@ -13,6 +13,7 @@ heap = []            # creates an empty heap
 heappush(heap, item) # pushes a new item on the heap
 item = heappop(heap) # pops the smallest item from the heap
 item = heap[0]       # smallest item on the heap without popping it
+heapify(heap)        # transform list into a heap, in-place, in linear time
 
 Our API differs from textbook heap algorithms as follows:
 
@@ -136,15 +137,13 @@ def heappush(heap, item):
         pos = parentpos
     heap[pos] = item
 
-def heappop(heap):
-    """Pop the smallest item off the heap, maintaining the heap invariant."""
-    endpos = len(heap) - 1
-    if endpos <= 0:
-        return heap.pop()
-    returnitem = heap[0]
-    item = heap.pop()
-    pos = 0
-    # Sift item into position, down from the root, moving the smaller
+# The child indices of heap index pos are already heaps, and we want to make
+# a heap at index pos too.
+def _siftdown(heap, pos):
+    endpos = len(heap)
+    assert pos < endpos
+    item = heap[pos]
+    # Sift item into position, down from pos, moving the smaller
     # child up, until finding pos such that item <= pos's children.
     childpos = 2*pos + 1    # leftmost child position
     while childpos < endpos:
@@ -164,8 +163,29 @@ def heappop(heap):
         pos = childpos
         childpos = 2*pos + 1
     heap[pos] = item
+
+def heappop(heap):
+    """Pop the smallest item off the heap, maintaining the heap invariant."""
+    lastelt = heap.pop()    # raises appropriate IndexError if heap is empty
+    if heap:
+        returnitem = heap[0]
+        heap[0] = lastelt
+        _siftdown(heap, 0)
+    else:
+        returnitem = lastelt
     return returnitem
 
+def heapify(heap):
+    """Transform list heap into a heap, in-place, in O(len(heap)) time."""
+    n = len(heap)
+    # Transform bottom-up.  The largest index there's any point to looking at
+    # is the largest with a child index in-range, so must have 2*i + 1 < n,
+    # or i < (n-1)/2.  If n is even = 2*j, this is (2*j-1)/2 = j-1/2 so
+    # j-1 is the largest, which is n//2 - 1.  If n is odd = 2*j+1, this is
+    # (2*j+1-1)/2 = j so j-1 is the largest, and that's again n//2-1.
+    for i in xrange(n//2 - 1, -1, -1):
+        _siftdown(heap, i)
+
 if __name__ == "__main__":
     # Simple sanity test
     heap = []

Lib/test/test_heapq.py

@@ -2,7 +2,7 @@
 
 from test.test_support import verify, vereq, verbose, TestFailed
 
-from heapq import heappush, heappop
+from heapq import heappush, heappop, heapify
 import random
 
 def check_invariant(heap):
@@ -40,6 +40,24 @@ def test_main():
             heappop(heap)
     heap.sort()
     vereq(heap, data_sorted[-10:])
+    # 4) Test heapify.
+    for size in range(30):
+        heap = [random.random() for dummy in range(size)]
+        heapify(heap)
+        check_invariant(heap)
+    # 5) Less-naive "N-best" algorithm, much faster (if len(data) is big
+    #    enough <wink>) than sorting all of data.  However, if we had a max
+    #    heap instead of a min heap, it would go much faster still via
+    #    heapify'ing all of data (linear time), then doing 10 heappops
+    #    (10 log-time steps).
+    heap = data[:10]
+    heapify(heap)
+    for item in data[10:]:
+        if item > heap[0]:  # this gets rarer and rarer the longer we run
+            heappush(heap, item)
+            heappop(heap)
+    heap.sort()
+    vereq(heap, data_sorted[-10:])
     # Make user happy
     if verbose:
         print "All OK"