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802 lines
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802 lines
38 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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Column Key:
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*sort: random data
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\sort: descending data
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/sort: ascending data
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3sort: ascending, then 3 random exchanges
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+sort: ascending, then 10 random at the end
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%sort: ascending, then randomly replace 1% of elements w/ random values
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~sort: many duplicates
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=sort: all equal
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!sort: worst case scenario
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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. Where lg(x) is the logarithm of x to
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the base 2 (e.g., lg(8)=3), lg(n!) is the information-theoretic limit for
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the best any comparison-based sorting algorithm can do on average (across
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all permutations). When a method gets significantly below that, it's
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either astronomically lucky, or is finding exploitable structure in the
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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 (see note BINSORT).
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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 occurrence(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 length is passed to found_new_run() to
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potentially merge runs on a stack of pending runs. We would like to delay
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merging as long as possible in order to exploit patterns that may come up
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later, but we like even more to do merging as soon as possible to exploit
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that the run just found is still high in the memory hierarchy. We also can't
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delay merging "too long" because it consumes memory to remember the runs that
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are still unmerged, and the stack has a fixed size.
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The original version of this code used the first thing I made up that didn't
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obviously suck ;-) It was loosely based on invariants involving the Fibonacci
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sequence.
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It worked OK, but it was hard to reason about, and was subtle enough that the
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intended invariants weren't actually preserved. Researchers discovered that
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when trying to complete a computer-generated correctness proof. That was
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easily-enough repaired, but the discovery spurred quite a bit of academic
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interest in truly good ways to manage incremental merging on the fly.
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At least a dozen different approaches were developed, some provably having
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near-optimal worst case behavior with respect to the entropy of the
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distribution of run lengths. Some details can be found in bpo-34561.
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The code now uses the "powersort" merge strategy from:
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"Nearly-Optimal Mergesorts: Fast, Practical Sorting Methods
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That Optimally Adapt to Existing Runs"
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J. Ian Munro and Sebastian Wild
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The code is pretty simple, but the justification is quite involved, as it's
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based on fast approximations to optimal binary search trees, which are
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substantial topics on their own.
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Here we'll just cover some pragmatic details:
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The `powerloop()` function computes a run's "power". Say two adjacent runs
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begin at index s1. The first run has length n1, and the second run (starting
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at index s1+n1, called "s2" below) has length n2. The list has total length n.
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The "power" of the first run is a small integer, the depth of the node
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connecting the two runs in an ideal binary merge tree, where power 1 is the
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root node, and the power increases by 1 for each level deeper in the tree.
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The power is the least integer L such that the "midpoint interval" contains
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a rational number of the form J/2**L. The midpoint interval is the semi-
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closed interval:
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((s1 + n1/2)/n, (s2 + n2/2)/n]
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Yes, that's brain-busting at first ;-) Concretely, if (s1 + n1/2)/n and
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(s2 + n2/2)/n are computed to infinite precision in binary, the power L is
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the first position at which the 2**-L bit differs between the expansions.
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Since the left end of the interval is less than the right end, the first
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differing bit must be a 0 bit in the left quotient and a 1 bit in the right
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quotient.
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`powerloop()` emulates these divisions, 1 bit at a time, using comparisons,
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subtractions, and shifts in a loop.
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You'll notice the paper uses an O(1) method instead, but that relies on two
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things we don't have:
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- An O(1) "count leading zeroes" primitive. We can find such a thing as a C
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extension on most platforms, but not all, and there's no uniform spelling
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on the platforms that support it.
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- Integer division on an integer type twice as wide as needed to hold the
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list length. But the latter is Py_ssize_t for us, and is typically the
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widest native signed integer type the platform supports.
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But since runs in our algorithm are almost never very short, the once-per-run
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overhead of `powerloop()` seems lost in the noise.
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Detail: why is Py_ssize_t "wide enough" in `powerloop()`? We do, after all,
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shift integers of that width left by 1. How do we know that won't spill into
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the sign bit? The trick is that we have some slop. `n` (the total list
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length) is the number of list elements, which is at most 4 times (on a 32-box,
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with 4-byte pointers) smaller than than the largest size_t. So at least the
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leading two bits of the integers we're using are clear.
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Since we can't compute a run's power before seeing the run that follows it,
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the most-recently identified run is never merged by `found_new_run()`.
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Instead a new run is only used to compute the 2nd-most-recent run's power.
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Then adjacent runs are merged so long as their saved power (tree depth) is
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greater than that newly computed power. When found_new_run() returns, only
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then is a new run pushed on to the stack of pending runs.
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A key invariant is that powers on the run stack are strictly decreasing
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(starting from the run at the top of the stack).
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Note that even powersort's strategy isn't always truly optimal. It can't be.
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Computing an optimal merge sequence can be done in time quadratic in the
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number of runs, which is very much slower, and also requires finding &
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remembering _all_ the runs' lengths (of which there may be billions) in
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advance. It's remarkable, though, how close to optimal this strategy gets.
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Curious factoid: of all the alternatives I've seen in the literature,
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powersort's is the only one that's always truly optimal for a collection of 3
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run lengths (for three lengths A B C, it's always optimal to first merge the
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shorter of A and C with B).
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Merge Memory
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------------
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Merging adjacent runs of lengths A and B in-place, and in linear time, is
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difficult. Theoretical constructions are known that can do it, but they're
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too difficult and slow for practical use. But if we have temp memory equal
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to min(A, B), it's easy.
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If A is smaller (function merge_lo), copy A to a temp array, leave B alone,
|
|
and then we can do the obvious merge algorithm left to right, from the temp
|
|
area and B, starting the stores into where A used to live. There's always a
|
|
free area in the original area comprising a number of elements equal to the
|
|
number not yet merged from the temp array (trivially true at the start;
|
|
proceed by induction). The only tricky bit is that if a comparison raises an
|
|
exception, we have to remember to copy the remaining elements back in from
|
|
the temp area, lest the array end up with duplicate entries from B. But
|
|
that's exactly the same thing we need to do if we reach the end of B first,
|
|
so the exit code is pleasantly common to both the normal and error cases.
|
|
|
|
If B is smaller (function merge_hi, which is merge_lo's "mirror image"),
|
|
much the same, except that we need to merge right to left, copying B into a
|
|
temp array and starting the stores at the right end of where B used to live.
|
|
|
|
A refinement: When we're about to merge adjacent runs A and B, we first do
|
|
a form of binary search (more on that later) to see where B[0] should end up
|
|
in A. Elements in A preceding that point are already in their final
|
|
positions, effectively shrinking the size of A. Likewise we also search to
|
|
see where A[-1] should end up in B, and elements of B after that point can
|
|
also be ignored. This cuts the amount of temp memory needed by the same
|
|
amount.
|
|
|
|
These preliminary searches may not pay off, and can be expected *not* to
|
|
repay their cost if the data is random. But they can win huge in all of
|
|
time, copying, and memory savings when they do pay, so this is one of the
|
|
"per-merge overheads" mentioned above that we're happy to endure because
|
|
there is at most one very short run. It's generally true in this algorithm
|
|
that we're willing to gamble a little to win a lot, even though the net
|
|
expectation is negative for random data.
|
|
|
|
|
|
Merge Algorithms
|
|
----------------
|
|
merge_lo() and merge_hi() are where the bulk of the time is spent. merge_lo
|
|
deals with runs where A <= B, and merge_hi where A > B. They don't know
|
|
whether the data is clustered or uniform, but a lovely thing about merging
|
|
is that many kinds of clustering "reveal themselves" by how many times in a
|
|
row the winning merge element comes from the same run. We'll only discuss
|
|
merge_lo here; merge_hi is exactly analogous.
|
|
|
|
Merging begins in the usual, obvious way, comparing the first element of A
|
|
to the first of B, and moving B[0] to the merge area if it's less than A[0],
|
|
else moving A[0] to the merge area. Call that the "one pair at a time"
|
|
mode. The only twist here is keeping track of how many times in a row "the
|
|
winner" comes from the same run.
|
|
|
|
If that count reaches MIN_GALLOP, we switch to "galloping mode". Here
|
|
we *search* B for where A[0] belongs, and move over all the B's before
|
|
that point in one chunk to the merge area, then move A[0] to the merge
|
|
area. Then we search A for where B[0] belongs, and similarly move a
|
|
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 (see note REGION OF UNCERTAINTY). 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 López-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
|
|
McIlroy 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() (see note LEFT OR 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 range(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.
|
|
|
|
|
|
NOTES
|
|
-----
|
|
|
|
BINSORT
|
|
A "binary insertion sort" is just like a textbook insertion sort, but instead
|
|
of locating the correct position of the next item via linear (one at a time)
|
|
search, an equivalent to Python's bisect.bisect_right is used to find the
|
|
correct position in logarithmic time. Most texts don't mention this
|
|
variation, and those that do usually say it's not worth the bother: insertion
|
|
sort remains quadratic (expected and worst cases) either way. Speeding the
|
|
search doesn't reduce the quadratic data movement costs.
|
|
|
|
But in CPython's case, comparisons are extraordinarily expensive compared to
|
|
moving data, and the details matter. Moving objects is just copying
|
|
pointers. Comparisons can be arbitrarily expensive (can invoke arbitrary
|
|
user-supplied Python code), but even in simple cases (like 3 < 4) _all_
|
|
decisions are made at runtime: what's the type of the left comparand? the
|
|
type of the right? do they need to be coerced to a common type? where's the
|
|
code to compare these types? And so on. Even the simplest Python comparison
|
|
triggers a large pile of C-level pointer dereferences, conditionals, and
|
|
function calls.
|
|
|
|
So cutting the number of compares is almost always measurably helpful in
|
|
CPython, and the savings swamp the quadratic-time data movement costs for
|
|
reasonable minrun values.
|
|
|
|
|
|
LEFT OR RIGHT
|
|
gallop_left() and gallop_right() are akin to the Python bisect module's
|
|
bisect_left() and bisect_right(): they're the same unless the slice they're
|
|
searching contains a (at least one) value equal to the value being searched
|
|
for. In that case, gallop_left() returns the position immediately before the
|
|
leftmost equal value, and gallop_right() the position immediately after the
|
|
rightmost equal value. The distinction is needed to preserve stability. In
|
|
general, when merging adjacent runs A and B, gallop_left is used to search
|
|
thru B for where an element from A belongs, and gallop_right to search thru A
|
|
for where an element from B belongs.
|
|
|
|
|
|
REGION OF UNCERTAINTY
|
|
Two kinds of confusion seem to be common about the claim that after finding
|
|
a k such that
|
|
|
|
B[2**(k-1) - 1] < A[0] <= B[2**k - 1]
|
|
|
|
then a binary search requires exactly k-1 tries to find A[0]'s proper
|
|
location. For concreteness, say k=3, so B[3] < A[0] <= B[7].
|
|
|
|
The first confusion takes the form "OK, then the region of uncertainty is at
|
|
indices 3, 4, 5, 6 and 7: that's 5 elements, not the claimed 2**(k-1) - 1 =
|
|
3"; or the region is viewed as a Python slice and the objection is "but that's
|
|
the slice B[3:7], so has 7-3 = 4 elements". Resolution: we've already
|
|
compared A[0] against B[3] and against B[7], so A[0]'s correct location is
|
|
already known wrt _both_ endpoints. What remains is to find A[0]'s correct
|
|
location wrt B[4], B[5] and B[6], which spans 3 elements. Or in general, the
|
|
slice (leaving off both endpoints) (2**(k-1)-1)+1 through (2**k-1)-1
|
|
inclusive = 2**(k-1) through (2**k-1)-1 inclusive, which has
|
|
(2**k-1)-1 - 2**(k-1) + 1 =
|
|
2**k-1 - 2**(k-1) =
|
|
2*2**(k-1)-1 - 2**(k-1) =
|
|
(2-1)*2**(k-1) - 1 =
|
|
2**(k-1) - 1
|
|
elements.
|
|
|
|
The second confusion: "k-1 = 2 binary searches can find the correct location
|
|
among 2**(k-1) = 4 elements, but you're only applying it to 3 elements: we
|
|
could make this more efficient by arranging for the region of uncertainty to
|
|
span 2**(k-1) elements." Resolution: that confuses "elements" with
|
|
"locations". In a slice with N elements, there are N+1 _locations_. In the
|
|
example, with the region of uncertainty B[4], B[5], B[6], there are 4
|
|
locations: before B[4], between B[4] and B[5], between B[5] and B[6], and
|
|
after B[6]. In general, across 2**(k-1)-1 elements, there are 2**(k-1)
|
|
locations. That's why k-1 binary searches are necessary and sufficient.
|
|
|
|
OPTIMIZATION OF INDIVIDUAL COMPARISONS
|
|
As noted above, even the simplest Python comparison triggers a large pile of
|
|
C-level pointer dereferences, conditionals, and function calls. This can be
|
|
partially mitigated by pre-scanning the data to determine whether the data is
|
|
homogeneous with respect to type. If so, it is sometimes possible to
|
|
substitute faster type-specific comparisons for the slower, generic
|
|
PyObject_RichCompareBool.
|