1. Combined the base and length arrays into a single array of structs.

   This is friendlier for caches.

2. Cut MIN_GALLOP to 7, but added a per-sort min_gallop vrbl that adapts
   the "get into galloping mode" threshold higher when galloping isn't
   paying, and lower when it is.  There's no known case where this hurts.
   It's (of course) neutral for /sort, \sort and =sort.  It also happens
   to be neutral for !sort.  It cuts a tiny # of compares in 3sort and +sort.
   For *sort, it reduces the # of compares to better than what this used to
   do when MIN_GALLOP was hardcoded to 10 (it did about 0.1% more *sort
   compares before, but given how close we are to the limit, this is "a
   lot"!).  %sort used to do about 1.5% more compares, and ~sort about
   3.6% more.  Here are exact counts:

 i    *sort    3sort    +sort    %sort    ~sort    !sort
15   449235    33019    33016    51328   188720    65534  before
     448885    33016    33007    50426   182083    65534  after
      0.08%    0.01%    0.03%    1.79%    3.65%    0.00%  %ch from after

16   963714    65824    65809   103409   377634   131070
     962991    65821    65808   101667   364341   131070
      0.08%    0.00%    0.00%    1.71%    3.65%    0.00%

17  2059092   131413   131362   209130   755476   262142
    2057533   131410   131361   206193   728871   262142
      0.08%    0.00%    0.00%    1.42%    3.65%    0.00%

18  4380687   262440   262460   421998  1511174   524286
    4377402   262437   262459   416347  1457945   524286
      0.08%    0.00%    0.00%    1.36%    3.65%    0.00%

19  9285709   524581   524634   848590  3022584  1048574
    9278734   524580   524633   837947  2916107  1048574
      0.08%    0.00%    0.00%    1.27%    3.65%    0.00%

20 19621118  1048960  1048942  1715806  6045418  2097150
   19606028  1048958  1048941  1694896  5832445  2097150
      0.08%    0.00%    0.00%    1.23%    3.65%    0.00%

3. Added some key asserts I overlooked before.

4. Updated the doc file.

Objects/listobject.c

@@ -1122,11 +1122,10 @@ fail:
  */
 #define MAX_MERGE_PENDING 85
 
-/* If a run wins MIN_GALLOP times in a row, we switch to galloping mode,
- * and stay there until both runs win less often than MIN_GALLOP
- * consecutive times.  See listsort.txt for more info.
+/* When we get into galloping mode, we stay there until both runs win less
+ * often than MIN_GALLOP consecutive times.  See listsort.txt for more info.
  */
-#define MIN_GALLOP 8
+#define MIN_GALLOP 7
 
 /* Avoid malloc for small temp arrays. */
 #define MERGESTATE_TEMP_SIZE 256
@@ -1134,10 +1133,21 @@ fail:
 /* One MergeState exists on the stack per invocation of mergesort.  It's just
  * a convenient way to pass state around among the helper functions.
  */
+struct s_slice {
+	PyObject **base;
+	int len;
+};
+
 typedef struct s_MergeState {
 	/* The user-supplied comparison function. or NULL if none given. */
 	PyObject *compare;
 
+	/* This controls when we get *into* galloping mode.  It's initialized
+	 * to MIN_GALLOP.  merge_lo and merge_hi tend to nudge it higher for
+	 * random data, and lower for highly structured data.
+	 */
+	int min_gallop;
+
 	/* 'a' is temp storage to help with merges.  It contains room for
 	 * alloced entries.
 	 */
@@ -1148,14 +1158,13 @@ typedef struct s_MergeState {
 	 * address base[i] and extends for len[i] elements.  It's always
 	 * true (so long as the indices are in bounds) that
 	 *
-	 *     base[i] + len[i] == base[i+1]
+	 *     pending[i].base + pending[i].len == pending[i+1].base
 	 *
 	 * so we could cut the storage for this, but it's a minor amount,
 	 * and keeping all the info explicit simplifies the code.
 	 */
 	int n;
-	PyObject **base[MAX_MERGE_PENDING];
-	int len[MAX_MERGE_PENDING];
+	struct s_slice pending[MAX_MERGE_PENDING];
 
 	/* 'a' points to this when possible, rather than muck with malloc. */
 	PyObject *temparray[MERGESTATE_TEMP_SIZE];
@@ -1170,6 +1179,7 @@ merge_init(MergeState *ms, PyObject *compare)
 	ms->a = ms->temparray;
 	ms->alloced = MERGESTATE_TEMP_SIZE;
 	ms->n = 0;
+	ms->min_gallop = MIN_GALLOP;
 }
 
 /* Free all the temp memory owned by the MergeState.  This must be called
@@ -1224,6 +1234,7 @@ merge_lo(MergeState *ms, PyObject **pa, int na, PyObject **pb, int nb)
 	PyObject *compare;
 	PyObject **dest;
 	int result = -1;	/* guilty until proved innocent */
+	int min_gallop = ms->min_gallop;
 
 	assert(ms && pa && pb && na > 0 && nb > 0 && pa + na == pb);
 	if (MERGE_GETMEM(ms, na) < 0)
@@ -1248,6 +1259,7 @@ merge_lo(MergeState *ms, PyObject **pa, int na, PyObject **pb, int nb)
 		 * appears to win consistently.
 		 */
  		for (;;) {
+ 			assert(na > 1 && nb > 0);
 	 		k = ISLT(*pb, *pa, compare);
 			if (k) {
 				if (k < 0)
@@ -1258,7 +1270,7 @@ merge_lo(MergeState *ms, PyObject **pa, int na, PyObject **pb, int nb)
 				--nb;
 				if (nb == 0)
 					goto Succeed;
-				if (bcount >= MIN_GALLOP)
+				if (bcount >= min_gallop)
 					break;
 			}
 			else {
@@ -1268,7 +1280,7 @@ merge_lo(MergeState *ms, PyObject **pa, int na, PyObject **pb, int nb)
 				--na;
 				if (na == 1)
 					goto CopyB;
-				if (acount >= MIN_GALLOP)
+				if (acount >= min_gallop)
 					break;
 			}
  		}
@@ -1278,7 +1290,11 @@ merge_lo(MergeState *ms, PyObject **pa, int na, PyObject **pb, int nb)
 		 * (if ever) neither run appears to be winning consistently
 		 * anymore.
 		 */
+		++min_gallop;
 		do {
+ 			assert(na > 1 && nb > 0);
+			min_gallop -= min_gallop > 1;
+	 		ms->min_gallop = min_gallop;
 			k = gallop_right(*pb, pa, na, 0, compare);
 			acount = k;
 			if (k) {
@@ -1319,6 +1335,8 @@ merge_lo(MergeState *ms, PyObject **pa, int na, PyObject **pb, int nb)
 			if (na == 1)
 				goto CopyB;
  		} while (acount >= MIN_GALLOP || bcount >= MIN_GALLOP);
+ 		++min_gallop;	/* penalize it for leaving galloping mode */
+ 		ms->min_gallop = min_gallop;
  	}
 Succeed:
 	result = 0;
@@ -1349,6 +1367,7 @@ merge_hi(MergeState *ms, PyObject **pa, int na, PyObject **pb, int nb)
 	int result = -1;	/* guilty until proved innocent */
 	PyObject **basea;
 	PyObject **baseb;
+	int min_gallop = ms->min_gallop;
 
 	assert(ms && pa && pb && na > 0 && nb > 0 && pa + na == pb);
 	if (MERGE_GETMEM(ms, nb) < 0)
@@ -1376,6 +1395,7 @@ merge_hi(MergeState *ms, PyObject **pa, int na, PyObject **pb, int nb)
 		 * appears to win consistently.
 		 */
  		for (;;) {
+ 			assert(na > 0 && nb > 1);
 	 		k = ISLT(*pb, *pa, compare);
 			if (k) {
 				if (k < 0)
@@ -1386,7 +1406,7 @@ merge_hi(MergeState *ms, PyObject **pa, int na, PyObject **pb, int nb)
 				--na;
 				if (na == 0)
 					goto Succeed;
-				if (acount >= MIN_GALLOP)
+				if (acount >= min_gallop)
 					break;
 			}
 			else {
@@ -1396,7 +1416,7 @@ merge_hi(MergeState *ms, PyObject **pa, int na, PyObject **pb, int nb)
 				--nb;
 				if (nb == 1)
 					goto CopyA;
-				if (bcount >= MIN_GALLOP)
+				if (bcount >= min_gallop)
 					break;
 			}
  		}
@@ -1406,7 +1426,11 @@ merge_hi(MergeState *ms, PyObject **pa, int na, PyObject **pb, int nb)
 		 * (if ever) neither run appears to be winning consistently
 		 * anymore.
 		 */
+		++min_gallop;
 		do {
+ 			assert(na > 0 && nb > 1);
+			min_gallop -= min_gallop > 1;
+	 		ms->min_gallop = min_gallop;
 			k = gallop_right(*pb, basea, na, na-1, compare);
 			if (k < 0)
 				goto Fail;
@@ -1449,6 +1473,8 @@ merge_hi(MergeState *ms, PyObject **pa, int na, PyObject **pb, int nb)
 			if (na == 0)
 				goto Succeed;
  		} while (acount >= MIN_GALLOP || bcount >= MIN_GALLOP);
+ 		++min_gallop;	/* penalize it for leaving galloping mode */
+ 		ms->min_gallop = min_gallop;
  	}
 Succeed:
 	result = 0;
@@ -1482,10 +1508,10 @@ merge_at(MergeState *ms, int i)
 	assert(i >= 0);
 	assert(i == ms->n - 2 || i == ms->n - 3);
 
-	pa = ms->base[i];
-	pb = ms->base[i+1];
-	na = ms->len[i];
-	nb = ms->len[i+1];
+	pa = ms->pending[i].base;
+	na = ms->pending[i].len;
+	pb = ms->pending[i+1].base;
+	nb = ms->pending[i+1].len;
 	assert(na > 0 && nb > 0);
 	assert(pa + na == pb);
 
@@ -1493,11 +1519,9 @@ merge_at(MergeState *ms, int i)
 	 * run now, also slide over the last run (which isn't involved
 	 * in this merge).  The current run i+1 goes away in any case.
 	 */
-	if (i == ms->n - 3) {
-		ms->len[i+1] = ms->len[i+2];
-		ms->base[i+1] = ms->base[i+2];
-	}
-	ms->len[i] = na + nb;
+	ms->pending[i].len = na + nb;
+	if (i == ms->n - 3)
+		ms->pending[i+1] = ms->pending[i+2];
 	--ms->n;
 
 	/* Where does b start in a?  Elements in a before that can be
@@ -1541,18 +1565,18 @@ merge_at(MergeState *ms, int i)
 static int
 merge_collapse(MergeState *ms)
 {
-	int *len = ms->len;
+	struct s_slice *p = ms->pending;
 
 	assert(ms);
 	while (ms->n > 1) {
 		int n = ms->n - 2;
-		if (n > 0 && len[n-1] <= len[n] + len[n+1]) {
-		    	if (len[n-1] < len[n+1])
+		if (n > 0 && p[n-1].len <= p[n].len + p[n+1].len) {
+		    	if (p[n-1].len < p[n+1].len)
 		    		--n;
 			if (merge_at(ms, n) < 0)
 				return -1;
 		}
-		else if (len[n] <= len[n+1]) {
+		else if (p[n].len <= p[n+1].len) {
 			 if (merge_at(ms, n) < 0)
 			 	return -1;
 		}
@@ -1570,12 +1594,12 @@ merge_collapse(MergeState *ms)
 static int
 merge_force_collapse(MergeState *ms)
 {
-	int *len = ms->len;
+	struct s_slice *p = ms->pending;
 
 	assert(ms);
 	while (ms->n > 1) {
 		int n = ms->n - 2;
-		if (n > 0 && len[n-1] < len[n+1])
+		if (n > 0 && p[n-1].len < p[n+1].len)
 			--n;
 		if (merge_at(ms, n) < 0)
 			return -1;
@@ -1664,8 +1688,8 @@ listsort(PyListObject *self, PyObject *args)
 		}
 		/* Push run onto pending-runs stack, and maybe merge. */
 		assert(ms.n < MAX_MERGE_PENDING);
-		ms.base[ms.n] = lo;
-		ms.len[ms.n] = n;
+		ms.pending[ms.n].base = lo;
+		ms.pending[ms.n].len = n;
 		++ms.n;
 		if (merge_collapse(&ms) < 0)
 			goto fail;
@@ -1678,8 +1702,8 @@ listsort(PyListObject *self, PyObject *args)
 	if (merge_force_collapse(&ms) < 0)
 		goto fail;
 	assert(ms.n == 1);
-	assert(ms.base[0] == self->ob_item);
-	assert(ms.len[0] == self->ob_size);
+	assert(ms.pending[0].base == self->ob_item);
+	assert(ms.pending[0].len == self->ob_size);
 
 succeed:
 	result = Py_None;

Objects/listsort.txt

@@ -95,31 +95,31 @@ Comparison with Python's Samplesort Hybrid
   below that, it's either astronomically lucky, or is finding exploitable
   structure in the data.
 
-      n   lg(n!)    *sort     3sort    +sort   %sort    ~sort     !sort
--------  -------   ------  --------  -------  ------  -------  --------
-  32768   444255   453096   453614    32908   452871   130491    469141  old
-                   449235    33019    33016    51328   188720     65534  new
-                    0.86% 1273.80%   -0.33%  782.31%  -30.85%   615.87%  %ch from new
+      n   lg(n!)    *sort    3sort     +sort   %sort    ~sort     !sort
+-------  -------   ------   -------  -------  ------  -------  --------
+  32768   444255   453096   453614    32908   452871   130491    469141 old
+                   448885    33016    33007    50426   182083     65534 new
+                    0.94% 1273.92%   -0.30%  798.09%  -28.33%   615.87% %ch from new
 
   65536   954037   972699   981940    65686   973104   260029   1004607
-                   963714    65824    65809   103409   377634    131070
-                    0.93% 1391.77%   -0.19%  841.02%  -31.14%   666.47%
+                   962991    65821    65808   101667   364341    131070
+                    1.01% 1391.83%   -0.19%  857.15%  -28.63%   666.47%
 
  131072  2039137  2101881  2091491   131232  2092894   554790   2161379
-                  2059092   131413   131362   209130   755476    262142
-                    2.08% 1491.54%   -0.10%  900.76%  -26.56%   724.51%
+                  2057533   131410   131361   206193   728871    262142
+                    2.16% 1491.58%   -0.10%  915.02%  -23.88%   724.51%
 
  262144  4340409  4464460  4403233   262314  4445884  1107842   4584560
-                  4380687   262440   262460   421998  1511174    524286
-                    1.91% 1577.81%   -0.06%  953.53%  -26.69%   774.44%
+                  4377402   262437   262459   416347  1457945    524286
+                    1.99% 1577.82%   -0.06%  967.83%  -24.01%   774.44%
 
  524288  9205096 9453356   9408463   524468  9441930  2218577   9692015
-                 9285709    524581   524634   848590  3022584   1048574
-                   1.81%  1693.52%   -0.03% 1012.66%  -26.60%   824.30%
+                 9278734    524580   524633   837947  2916107   1048574
+                   1.88%  1693.52%   -0.03% 1026.79%  -23.92%   824.30%
 
 1048576 19458756 19950272 19838588  1048766 19912134  4430649  20434212
-                 19621118  1048960  1048942  1715806  6045418   2097150
-                    1.68% 1791.26%   -0.02% 1060.51%  -26.71%   874.38%
+                 19606028  1048958  1048941  1694896  5832445   2097150
+                    1.76% 1791.27%   -0.02% 1074.83%  -24.03%   874.38%
 
   Discussion of cases:
 
@@ -171,7 +171,6 @@ Comparison with Python's Samplesort Hybrid
   bytes each on this box) needed by each test, again with arguments
   "15 20 1":
 
-
    2**i  *sort \sort /sort  3sort  +sort  %sort  ~sort  =sort  !sort
   32768  16384     0     0   6256      0  10821  12288      0  16383
   65536  32766     0     0  21652      0  31276  24576      0  32767
@@ -430,6 +429,11 @@ etc.  We stay in galloping mode until both searches find slices to copy
 less than MIN_GALLOP elements long, at which point we go back to one-pair-
 at-a-time mode.
 
+A refinement:  The MergeState struct contains the value of min_gallop that
+controls when we enter galloping mode, initialized to MIN_GALLOP.
+merge_lo() and merge_hi() adjust this higher when gallooping isn't paying
+off, and lower when it is.
+
 
 Galloping
 ---------
@@ -536,13 +540,21 @@ at the other values.  At and after i=6, galloping always wins.
 
 We can't guess in advance when it's going to win, though, so we do one pair
 at a time until the evidence seems strong that galloping may pay.  MIN_GALLOP
-is 8 as I type this, and that's pretty strong evidence.  However, if the data
-is random, it simply will trigger galloping mode purely by luck every now
-and again, and it's quite likely to hit one of the losing cases next.  8
-favors protecting against a slowdown on random data at the expense of giving
-up small wins on lightly clustered data, and tiny marginal wins on highly
-clustered data (they win huge anyway, and if you're getting a factor of
-10 speedup, another percent just isn't worth fighting for).
+is 7, and that's pretty strong evidence.  However, if the data is random, it
+simply will trigger galloping mode purely by luck every now and again, and
+it's quite likely to hit one of the losing cases next.  On the other hand,
+in cases like ~sort, galloping always pays, and MIN_GALLOP is larger than it
+"should be" then.  So the MergeState struct keeps a min_gallop variable
+that merge_lo and merge_hi adjust:  the longer we stay in galloping mode,
+the smaller min_gallop gets, making it easier to transition back to
+galloping mode (if we ever leave it in the current merge, and at the
+start of the next merge).  But whenever the gallop loop doesn't pay,
+min_gallop is increased by one, making it harder to transition to back
+to galloping mode (and again both within a merge and across merges).  For
+random data, this all but eliminates the gallop penalty:  min_gallop grows
+large enough that we almost never get into galloping mode.  And for cases
+like ~sort, min_gallop can fall to as low as 1.  This seems to work well,
+but in all it's a minor improvement over using a fixed MIN_GALLOP value.
 
 
 Galloping Complication
@@ -567,6 +579,9 @@ wildly unbalanced runs already enjoys excellent performance.
 
 Comparing Average # of Compares on Random Arrays
 ------------------------------------------------
+[NOTE:  This was done when the new algorithm used about 0.1% more compares
+ on random data than does its current incarnation.]
+
 Here list.sort() is samplesort, and list.msort() this sort:
 
 """