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667 lines
31 KiB
Text
Intro
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-----
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This describes an adaptive, stable, natural mergesort, modestly called
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timsort (hey, I earned it <wink>). It has supernatural performance on many
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kinds of partially ordered arrays (less than lg(N!) comparisons needed, and
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as few as N-1), yet as fast as Python's previous highly tuned samplesort
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hybrid on random arrays.
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In a nutshell, the main routine marches over the array once, left to right,
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alternately identifying the next run, then merging it into the previous
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runs "intelligently". Everything else is complication for speed, and some
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hard-won measure of memory efficiency.
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Comparison with Python's Samplesort Hybrid
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------------------------------------------
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+ timsort can require a temp array containing as many as N//2 pointers,
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which means as many as 2*N extra bytes on 32-bit boxes. It can be
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expected to require a temp array this large when sorting random data; on
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data with significant structure, it may get away without using any extra
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heap memory. This appears to be the strongest argument against it, but
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compared to the size of an object, 2 temp bytes worst-case (also expected-
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case for random data) doesn't scare me much.
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It turns out that Perl is moving to a stable mergesort, and the code for
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that appears always to require a temp array with room for at least N
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pointers. (Note that I wouldn't want to do that even if space weren't an
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issue; I believe its efforts at memory frugality also save timsort
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significant pointer-copying costs, and allow it to have a smaller working
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set.)
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+ Across about four hours of generating random arrays, and sorting them
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under both methods, samplesort required about 1.5% more comparisons
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(the program is at the end of this file).
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+ In real life, this may be faster or slower on random arrays than
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samplesort was, depending on platform quirks. Since it does fewer
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comparisons on average, it can be expected to do better the more
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expensive a comparison function is. OTOH, it does more data movement
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(pointer copying) than samplesort, and that may negate its small
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comparison advantage (depending on platform quirks) unless comparison
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is very expensive.
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+ On arrays with many kinds of pre-existing order, this blows samplesort out
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of the water. It's significantly faster than samplesort even on some
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cases samplesort was special-casing the snot out of. I believe that lists
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very often do have exploitable partial order in real life, and this is the
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strongest argument in favor of timsort (indeed, samplesort's special cases
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for extreme partial order are appreciated by real users, and timsort goes
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much deeper than those, in particular naturally covering every case where
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someone has suggested "and it would be cool if list.sort() had a special
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case for this too ... and for that ...").
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+ Here are exact comparison counts across all the tests in sortperf.py,
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when run with arguments "15 20 1".
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First the trivial cases, trivial for samplesort because it special-cased
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them, and trivial for timsort because it naturally works on runs. Within
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an "n" block, the first line gives the # of compares done by samplesort,
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the second line by timsort, and the third line is the percentage by
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which the samplesort count exceeds the timsort count:
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n \sort /sort =sort
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------- ------ ------ ------
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32768 32768 32767 32767 samplesort
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32767 32767 32767 timsort
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0.00% 0.00% 0.00% (samplesort - timsort) / timsort
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65536 65536 65535 65535
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65535 65535 65535
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0.00% 0.00% 0.00%
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131072 131072 131071 131071
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131071 131071 131071
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0.00% 0.00% 0.00%
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262144 262144 262143 262143
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262143 262143 262143
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0.00% 0.00% 0.00%
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524288 524288 524287 524287
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524287 524287 524287
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0.00% 0.00% 0.00%
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1048576 1048576 1048575 1048575
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1048575 1048575 1048575
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0.00% 0.00% 0.00%
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The algorithms are effectively identical in these cases, except that
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timsort does one less compare in \sort.
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Now for the more interesting cases. lg(n!) is the information-theoretic
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limit for the best any comparison-based sorting algorithm can do on
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average (across all permutations). When a method gets significantly
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below that, it's either astronomically lucky, or is finding exploitable
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structure in the data.
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n lg(n!) *sort 3sort +sort %sort ~sort !sort
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------- ------- ------ ------- ------- ------ ------- --------
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32768 444255 453096 453614 32908 452871 130491 469141 old
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448885 33016 33007 50426 182083 65534 new
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0.94% 1273.92% -0.30% 798.09% -28.33% 615.87% %ch from new
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65536 954037 972699 981940 65686 973104 260029 1004607
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962991 65821 65808 101667 364341 131070
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1.01% 1391.83% -0.19% 857.15% -28.63% 666.47%
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131072 2039137 2101881 2091491 131232 2092894 554790 2161379
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2057533 131410 131361 206193 728871 262142
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2.16% 1491.58% -0.10% 915.02% -23.88% 724.51%
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262144 4340409 4464460 4403233 262314 4445884 1107842 4584560
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4377402 262437 262459 416347 1457945 524286
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1.99% 1577.82% -0.06% 967.83% -24.01% 774.44%
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524288 9205096 9453356 9408463 524468 9441930 2218577 9692015
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9278734 524580 524633 837947 2916107 1048574
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1.88% 1693.52% -0.03% 1026.79% -23.92% 824.30%
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1048576 19458756 19950272 19838588 1048766 19912134 4430649 20434212
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19606028 1048958 1048941 1694896 5832445 2097150
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1.76% 1791.27% -0.02% 1074.83% -24.03% 874.38%
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Discussion of cases:
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*sort: There's no structure in random data to exploit, so the theoretical
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limit is lg(n!). Both methods get close to that, and timsort is hugging
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it (indeed, in a *marginal* sense, it's a spectacular improvement --
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there's only about 1% left before hitting the wall, and timsort knows
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darned well it's doing compares that won't pay on random data -- but so
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does the samplesort hybrid). For contrast, Hoare's original random-pivot
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quicksort does about 39% more compares than the limit, and the median-of-3
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variant about 19% more.
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3sort, %sort, and !sort: No contest; there's structure in this data, but
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not of the specific kinds samplesort special-cases. Note that structure
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in !sort wasn't put there on purpose -- it was crafted as a worst case for
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a previous quicksort implementation. That timsort nails it came as a
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surprise to me (although it's obvious in retrospect).
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+sort: samplesort special-cases this data, and does a few less compares
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than timsort. However, timsort runs this case significantly faster on all
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boxes we have timings for, because timsort is in the business of merging
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runs efficiently, while samplesort does much more data movement in this
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(for it) special case.
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~sort: samplesort's special cases for large masses of equal elements are
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extremely effective on ~sort's specific data pattern, and timsort just
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isn't going to get close to that, despite that it's clearly getting a
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great deal of benefit out of the duplicates (the # of compares is much less
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than lg(n!)). ~sort has a perfectly uniform distribution of just 4
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distinct values, and as the distribution gets more skewed, samplesort's
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equal-element gimmicks become less effective, while timsort's adaptive
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strategies find more to exploit; in a database supplied by Kevin Altis, a
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sort on its highly skewed "on which stock exchange does this company's
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stock trade?" field ran over twice as fast under timsort.
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However, despite that timsort does many more comparisons on ~sort, and
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that on several platforms ~sort runs highly significantly slower under
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timsort, on other platforms ~sort runs highly significantly faster under
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timsort. No other kind of data has shown this wild x-platform behavior,
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and we don't have an explanation for it. The only thing I can think of
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that could transform what "should be" highly significant slowdowns into
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highly significant speedups on some boxes are catastrophic cache effects
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in samplesort.
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But timsort "should be" slower than samplesort on ~sort, so it's hard
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to count that it isn't on some boxes as a strike against it <wink>.
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+ Here's the highwater mark for the number of heap-based temp slots (4
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bytes each on this box) needed by each test, again with arguments
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"15 20 1":
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2**i *sort \sort /sort 3sort +sort %sort ~sort =sort !sort
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32768 16384 0 0 6256 0 10821 12288 0 16383
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65536 32766 0 0 21652 0 31276 24576 0 32767
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131072 65534 0 0 17258 0 58112 49152 0 65535
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262144 131072 0 0 35660 0 123561 98304 0 131071
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524288 262142 0 0 31302 0 212057 196608 0 262143
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1048576 524286 0 0 312438 0 484942 393216 0 524287
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Discussion: The tests that end up doing (close to) perfectly balanced
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merges (*sort, !sort) need all N//2 temp slots (or almost all). ~sort
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also ends up doing balanced merges, but systematically benefits a lot from
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the preliminary pre-merge searches described under "Merge Memory" later.
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%sort approaches having a balanced merge at the end because the random
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selection of elements to replace is expected to produce an out-of-order
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element near the midpoint. \sort, /sort, =sort are the trivial one-run
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cases, needing no merging at all. +sort ends up having one very long run
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and one very short, and so gets all the temp space it needs from the small
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temparray member of the MergeState struct (note that the same would be
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true if the new random elements were prefixed to the sorted list instead,
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but not if they appeared "in the middle"). 3sort approaches N//3 temp
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slots twice, but the run lengths that remain after 3 random exchanges
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clearly has very high variance.
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A detailed description of timsort follows.
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Runs
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----
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count_run() returns the # of elements in the next run. A run is either
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"ascending", which means non-decreasing:
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a0 <= a1 <= a2 <= ...
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or "descending", which means strictly decreasing:
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a0 > a1 > a2 > ...
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Note that a run is always at least 2 long, unless we start at the array's
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last element.
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The definition of descending is strict, because the main routine reverses
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a descending run in-place, transforming a descending run into an ascending
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run. Reversal is done via the obvious fast "swap elements starting at each
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end, and converge at the middle" method, and that can violate stability if
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the slice contains any equal elements. Using a strict definition of
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descending ensures that a descending run contains distinct elements.
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If an array is random, it's very unlikely we'll see long runs. If a natural
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run contains less than minrun elements (see next section), the main loop
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artificially boosts it to minrun elements, via a stable binary insertion sort
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applied to the right number of array elements following the short natural
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run. In a random array, *all* runs are likely to be minrun long as a
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result. This has two primary good effects:
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1. Random data strongly tends then toward perfectly balanced (both runs have
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the same length) merges, which is the most efficient way to proceed when
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data is random.
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2. Because runs are never very short, the rest of the code doesn't make
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heroic efforts to shave a few cycles off per-merge overheads. For
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example, reasonable use of function calls is made, rather than trying to
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inline everything. Since there are no more than N/minrun runs to begin
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with, a few "extra" function calls per merge is barely measurable.
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Computing minrun
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----------------
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If N < 64, minrun is N. IOW, binary insertion sort is used for the whole
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array then; it's hard to beat that given the overheads of trying something
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fancier.
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When N is a power of 2, testing on random data showed that minrun values of
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16, 32, 64 and 128 worked about equally well. At 256 the data-movement cost
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in binary insertion sort clearly hurt, and at 8 the increase in the number
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of function calls clearly hurt. Picking *some* power of 2 is important
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here, so that the merges end up perfectly balanced (see next section). We
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pick 32 as a good value in the sweet range; picking a value at the low end
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allows the adaptive gimmicks more opportunity to exploit shorter natural
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runs.
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Because sortperf.py only tries powers of 2, it took a long time to notice
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that 32 isn't a good choice for the general case! Consider N=2112:
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>>> divmod(2112, 32)
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(66, 0)
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>>>
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If the data is randomly ordered, we're very likely to end up with 66 runs
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each of length 32. The first 64 of these trigger a sequence of perfectly
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balanced merges (see next section), leaving runs of lengths 2048 and 64 to
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merge at the end. The adaptive gimmicks can do that with fewer than 2048+64
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compares, but it's still more compares than necessary, and-- mergesort's
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bugaboo relative to samplesort --a lot more data movement (O(N) copies just
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to get 64 elements into place).
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If we take minrun=33 in this case, then we're very likely to end up with 64
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runs each of length 33, and then all merges are perfectly balanced. Better!
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What we want to avoid is picking minrun such that in
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q, r = divmod(N, minrun)
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q is a power of 2 and r>0 (then the last merge only gets r elements into
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place, and r < minrun is small compared to N), or q a little larger than a
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power of 2 regardless of r (then we've got a case similar to "2112", again
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leaving too little work for the last merge to do).
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Instead we pick a minrun in range(32, 65) such that N/minrun is exactly a
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power of 2, or if that isn't possible, is close to, but strictly less than,
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a power of 2. This is easier to do than it may sound: take the first 6
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bits of N, and add 1 if any of the remaining bits are set. In fact, that
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rule covers every case in this section, including small N and exact powers
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of 2; merge_compute_minrun() is a deceptively simple function.
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The Merge Pattern
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-----------------
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In order to exploit regularities in the data, we're merging on natural
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run lengths, and they can become wildly unbalanced. That's a Good Thing
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for this sort! It means we have to find a way to manage an assortment of
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potentially very different run lengths, though.
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Stability constrains permissible merging patterns. For example, if we have
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3 consecutive runs of lengths
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A:10000 B:20000 C:10000
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we dare not merge A with C first, because if A, B and C happen to contain
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a common element, it would get out of order wrt its occurence(s) in B. The
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merging must be done as (A+B)+C or A+(B+C) instead.
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So merging is always done on two consecutive runs at a time, and in-place,
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although this may require some temp memory (more on that later).
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When a run is identified, its base address and length are pushed on a stack
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in the MergeState struct. merge_collapse() is then called to see whether it
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should merge it with preceding run(s). We would like to delay merging as
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long as possible in order to exploit patterns that may come up later, but we
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like even more to do merging as soon as possible to exploit that the run just
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found is still high in the memory hierarchy. We also can't delay merging
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"too long" because it consumes memory to remember the runs that are still
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unmerged, and the stack has a fixed size.
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What turned out to be a good compromise maintains two invariants on the
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stack entries, where A, B and C are the lengths of the three righmost not-yet
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merged slices:
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1. A > B+C
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2. B > C
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Note that, by induction, #2 implies the lengths of pending runs form a
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decreasing sequence. #1 implies that, reading the lengths right to left,
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the pending-run lengths grow at least as fast as the Fibonacci numbers.
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Therefore the stack can never grow larger than about log_base_phi(N) entries,
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where phi = (1+sqrt(5))/2 ~= 1.618. Thus a small # of stack slots suffice
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for very large arrays.
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If A <= B+C, the smaller of A and C is merged with B (ties favor C, for the
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freshness-in-cache reason), and the new run replaces the A,B or B,C entries;
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e.g., if the last 3 entries are
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A:30 B:20 C:10
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then B is merged with C, leaving
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A:30 BC:30
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on the stack. Or if they were
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A:500 B:400: C:1000
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then A is merged with B, leaving
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AB:900 C:1000
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on the stack.
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In both examples, the stack configuration after the merge still violates
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invariant #2, and merge_collapse() goes on to continue merging runs until
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both invariants are satisfied. As an extreme case, suppose we didn't do the
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minrun gimmick, and natural runs were of lengths 128, 64, 32, 16, 8, 4, 2,
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and 2. Nothing would get merged until the final 2 was seen, and that would
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trigger 7 perfectly balanced merges.
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The thrust of these rules when they trigger merging is to balance the run
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lengths as closely as possible, while keeping a low bound on the number of
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runs we have to remember. This is maximally effective for random data,
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where all runs are likely to be of (artificially forced) length minrun, and
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then we get a sequence of perfectly balanced merges (with, perhaps, some
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oddballs at the end).
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OTOH, one reason this sort is so good for partly ordered data has to do
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with wildly unbalanced run lengths.
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Merge Memory
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------------
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Merging adjacent runs of lengths A and B in-place is very difficult.
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Theoretical constructions are known that can do it, but they're too difficult
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and slow for practical use. But if we have temp memory equal to min(A, B),
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it's easy.
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If A is smaller (function merge_lo), copy A to a temp array, leave B alone,
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and then we can do the obvious merge algorithm left to right, from the temp
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area and B, starting the stores into where A used to live. There's always a
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free area in the original area comprising a number of elements equal to the
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number not yet merged from the temp array (trivially true at the start;
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proceed by induction). The only tricky bit is that if a comparison raises an
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exception, we have to remember to copy the remaining elements back in from
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the temp area, lest the array end up with duplicate entries from B. But
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that's exactly the same thing we need to do if we reach the end of B first,
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so the exit code is pleasantly common to both the normal and error cases.
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If B is smaller (function merge_hi, which is merge_lo's "mirror image"),
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much the same, except that we need to merge right to left, copying B into a
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temp array and starting the stores at the right end of where B used to live.
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A refinement: When we're about to merge adjacent runs A and B, we first do
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a form of binary search (more on that later) to see where B[0] should end up
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in A. Elements in A preceding that point are already in their final
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positions, effectively shrinking the size of A. Likewise we also search to
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see where A[-1] should end up in B, and elements of B after that point can
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also be ignored. This cuts the amount of temp memory needed by the same
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amount.
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These preliminary searches may not pay off, and can be expected *not* to
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repay their cost if the data is random. But they can win huge in all of
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time, copying, and memory savings when they do pay, so this is one of the
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"per-merge overheads" mentioned above that we're happy to endure because
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there is at most one very short run. It's generally true in this algorithm
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that we're willing to gamble a little to win a lot, even though the net
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expectation is negative for random data.
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Merge Algorithms
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----------------
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merge_lo() and merge_hi() are where the bulk of the time is spent. merge_lo
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deals with runs where A <= B, and merge_hi where A > B. They don't know
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whether the data is clustered or uniform, but a lovely thing about merging
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is that many kinds of clustering "reveal themselves" by how many times in a
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row the winning merge element comes from the same run. We'll only discuss
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merge_lo here; merge_hi is exactly analogous.
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Merging begins in the usual, obvious way, comparing the first element of A
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to the first of B, and moving B[0] to the merge area if it's less than A[0],
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else moving A[0] to the merge area. Call that the "one pair at a time"
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mode. The only twist here is keeping track of how many times in a row "the
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winner" comes from the same run.
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If that count reaches MIN_GALLOP, we switch to "galloping mode". Here
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we *search* B for where A[0] belongs, and move over all the B's before
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that point in one chunk to the merge area, then move A[0] to the merge
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area. Then we search A for where B[0] belongs, and similarly move a
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slice of A in one chunk. Then back to searching B for where A[0] belongs,
|
||
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 galloping isn't paying
|
||
off, and lower when it is.
|
||
|
||
|
||
Galloping
|
||
---------
|
||
Still without loss of generality, assume A is the shorter run. In galloping
|
||
mode, we first look for A[0] in B. We do this via "galloping", comparing
|
||
A[0] in turn to B[0], B[1], B[3], B[7], ..., B[2**j - 1], ..., until finding
|
||
the k such that B[2**(k-1) - 1] < A[0] <= B[2**k - 1]. This takes at most
|
||
roughly lg(B) comparisons, and, unlike a straight binary search, favors
|
||
finding the right spot early in B (more on that later).
|
||
|
||
After finding such a k, the region of uncertainty is reduced to 2**(k-1) - 1
|
||
consecutive elements, and a straight binary search requires exactly k-1
|
||
additional comparisons to nail it. Then we copy all the B's up to that
|
||
point in one chunk, and then copy A[0]. Note that no matter where A[0]
|
||
belongs in B, the combination of galloping + binary search finds it in no
|
||
more than about 2*lg(B) comparisons.
|
||
|
||
If we did a straight binary search, we could find it in no more than
|
||
ceiling(lg(B+1)) comparisons -- but straight binary search takes that many
|
||
comparisons no matter where A[0] belongs. Straight binary search thus loses
|
||
to galloping unless the run is quite long, and we simply can't guess
|
||
whether it is in advance.
|
||
|
||
If data is random and runs have the same length, A[0] belongs at B[0] half
|
||
the time, at B[1] a quarter of the time, and so on: a consecutive winning
|
||
sub-run in B of length k occurs with probability 1/2**(k+1). So long
|
||
winning sub-runs are extremely unlikely in random data, and guessing that a
|
||
winning sub-run is going to be long is a dangerous game.
|
||
|
||
OTOH, if data is lopsided or lumpy or contains many duplicates, long
|
||
stretches of winning sub-runs are very likely, and cutting the number of
|
||
comparisons needed to find one from O(B) to O(log B) is a huge win.
|
||
|
||
Galloping compromises by getting out fast if there isn't a long winning
|
||
sub-run, yet finding such very efficiently when they exist.
|
||
|
||
I first learned about the galloping strategy in a related context; see:
|
||
|
||
"Adaptive Set Intersections, Unions, and Differences" (2000)
|
||
Erik D. Demaine, Alejandro L<>pez-Ortiz, J. Ian Munro
|
||
|
||
and its followup(s). An earlier paper called the same strategy
|
||
"exponential search":
|
||
|
||
"Optimistic Sorting and Information Theoretic Complexity"
|
||
Peter McIlroy
|
||
SODA (Fourth Annual ACM-SIAM Symposium on Discrete Algorithms), pp
|
||
467-474, Austin, Texas, 25-27 January 1993.
|
||
|
||
and it probably dates back to an earlier paper by Bentley and Yao. The
|
||
McIlory paper in particular has good analysis of a mergesort that's
|
||
probably strongly related to this one in its galloping strategy.
|
||
|
||
|
||
Galloping with a Broken Leg
|
||
---------------------------
|
||
So why don't we always gallop? Because it can lose, on two counts:
|
||
|
||
1. While we're willing to endure small per-merge overheads, per-comparison
|
||
overheads are a different story. Calling Yet Another Function per
|
||
comparison is expensive, and gallop_left() and gallop_right() are
|
||
too long-winded for sane inlining.
|
||
|
||
2. Galloping can-- alas --require more comparisons than linear one-at-time
|
||
search, depending on the data.
|
||
|
||
#2 requires details. If A[0] belongs before B[0], galloping requires 1
|
||
compare to determine that, same as linear search, except it costs more
|
||
to call the gallop function. If A[0] belongs right before B[1], galloping
|
||
requires 2 compares, again same as linear search. On the third compare,
|
||
galloping checks A[0] against B[3], and if it's <=, requires one more
|
||
compare to determine whether A[0] belongs at B[2] or B[3]. That's a total
|
||
of 4 compares, but if A[0] does belong at B[2], linear search would have
|
||
discovered that in only 3 compares, and that's a huge loss! Really. It's
|
||
an increase of 33% in the number of compares needed, and comparisons are
|
||
expensive in Python.
|
||
|
||
index in B where # compares linear # gallop # binary gallop
|
||
A[0] belongs search needs compares compares total
|
||
---------------- ----------------- -------- -------- ------
|
||
0 1 1 0 1
|
||
|
||
1 2 2 0 2
|
||
|
||
2 3 3 1 4
|
||
3 4 3 1 4
|
||
|
||
4 5 4 2 6
|
||
5 6 4 2 6
|
||
6 7 4 2 6
|
||
7 8 4 2 6
|
||
|
||
8 9 5 3 8
|
||
9 10 5 3 8
|
||
10 11 5 3 8
|
||
11 12 5 3 8
|
||
...
|
||
|
||
In general, if A[0] belongs at B[i], linear search requires i+1 comparisons
|
||
to determine that, and galloping a total of 2*floor(lg(i))+2 comparisons.
|
||
The advantage of galloping is unbounded as i grows, but it doesn't win at
|
||
all until i=6. Before then, it loses twice (at i=2 and i=4), and ties
|
||
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 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 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
|
||
----------------------
|
||
The description above was for merge_lo. merge_hi has to merge "from the
|
||
other end", and really needs to gallop starting at the last element in a run
|
||
instead of the first. Galloping from the first still works, but does more
|
||
comparisons than it should (this is significant -- I timed it both ways).
|
||
For this reason, the gallop_left() and gallop_right() functions have a
|
||
"hint" argument, which is the index at which galloping should begin. So
|
||
galloping can actually start at any index, and proceed at offsets of 1, 3,
|
||
7, 15, ... or -1, -3, -7, -15, ... from the starting index.
|
||
|
||
In the code as I type it's always called with either 0 or n-1 (where n is
|
||
the # of elements in a run). It's tempting to try to do something fancier,
|
||
melding galloping with some form of interpolation search; for example, if
|
||
we're merging a run of length 1 with a run of length 10000, index 5000 is
|
||
probably a better guess at the final result than either 0 or 9999. But
|
||
it's unclear how to generalize that intuition usefully, and merging of
|
||
wildly unbalanced runs already enjoys excellent performance.
|
||
|
||
~sort is a good example of when balanced runs could benefit from a better
|
||
hint value: to the extent possible, this would like to use a starting
|
||
offset equal to the previous value of acount/bcount. Doing so saves about
|
||
10% of the compares in ~sort. However, doing so is also a mixed bag,
|
||
hurting other cases.
|
||
|
||
|
||
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:
|
||
|
||
"""
|
||
import random
|
||
from time import clock as now
|
||
|
||
def fill(n):
|
||
from random import random
|
||
return [random() for i in xrange(n)]
|
||
|
||
def mycmp(x, y):
|
||
global ncmp
|
||
ncmp += 1
|
||
return cmp(x, y)
|
||
|
||
def timeit(values, method):
|
||
global ncmp
|
||
X = values[:]
|
||
bound = getattr(X, method)
|
||
ncmp = 0
|
||
t1 = now()
|
||
bound(mycmp)
|
||
t2 = now()
|
||
return t2-t1, ncmp
|
||
|
||
format = "%5s %9.2f %11d"
|
||
f2 = "%5s %9.2f %11.2f"
|
||
|
||
def drive():
|
||
count = sst = sscmp = mst = mscmp = nelts = 0
|
||
while True:
|
||
n = random.randrange(100000)
|
||
nelts += n
|
||
x = fill(n)
|
||
|
||
t, c = timeit(x, 'sort')
|
||
sst += t
|
||
sscmp += c
|
||
|
||
t, c = timeit(x, 'msort')
|
||
mst += t
|
||
mscmp += c
|
||
|
||
count += 1
|
||
if count % 10:
|
||
continue
|
||
|
||
print "count", count, "nelts", nelts
|
||
print format % ("sort", sst, sscmp)
|
||
print format % ("msort", mst, mscmp)
|
||
print f2 % ("", (sst-mst)*1e2/mst, (sscmp-mscmp)*1e2/mscmp)
|
||
|
||
drive()
|
||
"""
|
||
|
||
I ran this on Windows and kept using the computer lightly while it was
|
||
running. time.clock() is wall-clock time on Windows, with better than
|
||
microsecond resolution. samplesort started with a 1.52% #-of-comparisons
|
||
disadvantage, fell quickly to 1.48%, and then fluctuated within that small
|
||
range. Here's the last chunk of output before I killed the job:
|
||
|
||
count 2630 nelts 130906543
|
||
sort 6110.80 1937887573
|
||
msort 6002.78 1909389381
|
||
1.80 1.49
|
||
|
||
We've done nearly 2 billion comparisons apiece at Python speed there, and
|
||
that's enough <wink>.
|
||
|
||
For random arrays of size 2 (yes, there are only 2 interesting ones),
|
||
samplesort has a 50%(!) comparison disadvantage. This is a consequence of
|
||
samplesort special-casing at most one ascending run at the start, then
|
||
falling back to the general case if it doesn't find an ascending run
|
||
immediately. The consequence is that it ends up using two compares to sort
|
||
[2, 1]. Gratifyingly, timsort doesn't do any special-casing, so had to be
|
||
taught how to deal with mixtures of ascending and descending runs
|
||
efficiently in all cases.
|