| Intro |
| ----- |
| This describes an adaptive, stable, natural mergesort, modestly called |
| timsort (hey, I earned it <wink>). It has supernatural performance on many |
| kinds of partially ordered arrays (less than lg(N!) comparisons needed, and |
| as few as N-1), yet as fast as Python's previous highly tuned samplesort |
| hybrid on random arrays. |
| |
| In a nutshell, the main routine marches over the array once, left to right, |
| alternately identifying the next run, then merging it into the previous |
| runs "intelligently". Everything else is complication for speed, and some |
| hard-won measure of memory efficiency. |
| |
| |
| Comparison with Python's Samplesort Hybrid |
| ------------------------------------------ |
| + timsort can require a temp array containing as many as N//2 pointers, |
| which means as many as 2*N extra bytes on 32-bit boxes. It can be |
| expected to require a temp array this large when sorting random data; on |
| data with significant structure, it may get away without using any extra |
| heap memory. This appears to be the strongest argument against it, but |
| compared to the size of an object, 2 temp bytes worst-case (also expected- |
| case for random data) doesn't scare me much. |
| |
| It turns out that Perl is moving to a stable mergesort, and the code for |
| that appears always to require a temp array with room for at least N |
| pointers. (Note that I wouldn't want to do that even if space weren't an |
| issue; I believe its efforts at memory frugality also save timsort |
| significant pointer-copying costs, and allow it to have a smaller working |
| set.) |
| |
| + Across about four hours of generating random arrays, and sorting them |
| under both methods, samplesort required about 1.5% more comparisons |
| (the program is at the end of this file). |
| |
| + In real life, this may be faster or slower on random arrays than |
| samplesort was, depending on platform quirks. Since it does fewer |
| comparisons on average, it can be expected to do better the more |
| expensive a comparison function is. OTOH, it does more data movement |
| (pointer copying) than samplesort, and that may negate its small |
| comparison advantage (depending on platform quirks) unless comparison |
| is very expensive. |
| |
| + On arrays with many kinds of pre-existing order, this blows samplesort out |
| of the water. It's significantly faster than samplesort even on some |
| cases samplesort was special-casing the snot out of. I believe that lists |
| very often do have exploitable partial order in real life, and this is the |
| strongest argument in favor of timsort (indeed, samplesort's special cases |
| for extreme partial order are appreciated by real users, and timsort goes |
| much deeper than those, in particular naturally covering every case where |
| someone has suggested "and it would be cool if list.sort() had a special |
| case for this too ... and for that ..."). |
| |
| + Here are exact comparison counts across all the tests in sortperf.py, |
| when run with arguments "15 20 1". |
| |
| Column Key: |
| *sort: random data |
| \sort: descending data |
| /sort: ascending data |
| 3sort: ascending, then 3 random exchanges |
| +sort: ascending, then 10 random at the end |
| %sort: ascending, then randomly replace 1% of elements w/ random values |
| ~sort: many duplicates |
| =sort: all equal |
| !sort: worst case scenario |
| |
| First the trivial cases, trivial for samplesort because it special-cased |
| them, and trivial for timsort because it naturally works on runs. Within |
| an "n" block, the first line gives the # of compares done by samplesort, |
| the second line by timsort, and the third line is the percentage by |
| which the samplesort count exceeds the timsort count: |
| |
| n \sort /sort =sort |
| ------- ------ ------ ------ |
| 32768 32768 32767 32767 samplesort |
| 32767 32767 32767 timsort |
| 0.00% 0.00% 0.00% (samplesort - timsort) / timsort |
| |
| 65536 65536 65535 65535 |
| 65535 65535 65535 |
| 0.00% 0.00% 0.00% |
| |
| 131072 131072 131071 131071 |
| 131071 131071 131071 |
| 0.00% 0.00% 0.00% |
| |
| 262144 262144 262143 262143 |
| 262143 262143 262143 |
| 0.00% 0.00% 0.00% |
| |
| 524288 524288 524287 524287 |
| 524287 524287 524287 |
| 0.00% 0.00% 0.00% |
| |
| 1048576 1048576 1048575 1048575 |
| 1048575 1048575 1048575 |
| 0.00% 0.00% 0.00% |
| |
| The algorithms are effectively identical in these cases, except that |
| timsort does one less compare in \sort. |
| |
| Now for the more interesting cases. lg(n!) is the information-theoretic |
| limit for the best any comparison-based sorting algorithm can do on |
| average (across all permutations). When a method gets significantly |
| below that, it's either astronomically lucky, or is finding exploitable |
| structure in the data. |
| |
| n lg(n!) *sort 3sort +sort %sort ~sort !sort |
| ------- ------- ------ ------- ------- ------ ------- -------- |
| 32768 444255 453096 453614 32908 452871 130491 469141 old |
| 448885 33016 33007 50426 182083 65534 new |
| 0.94% 1273.92% -0.30% 798.09% -28.33% 615.87% %ch from new |
| |
| 65536 954037 972699 981940 65686 973104 260029 1004607 |
| 962991 65821 65808 101667 364341 131070 |
| 1.01% 1391.83% -0.19% 857.15% -28.63% 666.47% |
| |
| 131072 2039137 2101881 2091491 131232 2092894 554790 2161379 |
| 2057533 131410 131361 206193 728871 262142 |
| 2.16% 1491.58% -0.10% 915.02% -23.88% 724.51% |
| |
| 262144 4340409 4464460 4403233 262314 4445884 1107842 4584560 |
| 4377402 262437 262459 416347 1457945 524286 |
| 1.99% 1577.82% -0.06% 967.83% -24.01% 774.44% |
| |
| 524288 9205096 9453356 9408463 524468 9441930 2218577 9692015 |
| 9278734 524580 524633 837947 2916107 1048574 |
| 1.88% 1693.52% -0.03% 1026.79% -23.92% 824.30% |
| |
| 1048576 19458756 19950272 19838588 1048766 19912134 4430649 20434212 |
| 19606028 1048958 1048941 1694896 5832445 2097150 |
| 1.76% 1791.27% -0.02% 1074.83% -24.03% 874.38% |
| |
| Discussion of cases: |
| |
| *sort: There's no structure in random data to exploit, so the theoretical |
| limit is lg(n!). Both methods get close to that, and timsort is hugging |
| it (indeed, in a *marginal* sense, it's a spectacular improvement -- |
| there's only about 1% left before hitting the wall, and timsort knows |
| darned well it's doing compares that won't pay on random data -- but so |
| does the samplesort hybrid). For contrast, Hoare's original random-pivot |
| quicksort does about 39% more compares than the limit, and the median-of-3 |
| variant about 19% more. |
| |
| 3sort, %sort, and !sort: No contest; there's structure in this data, but |
| not of the specific kinds samplesort special-cases. Note that structure |
| in !sort wasn't put there on purpose -- it was crafted as a worst case for |
| a previous quicksort implementation. That timsort nails it came as a |
| surprise to me (although it's obvious in retrospect). |
| |
| +sort: samplesort special-cases this data, and does a few less compares |
| than timsort. However, timsort runs this case significantly faster on all |
| boxes we have timings for, because timsort is in the business of merging |
| runs efficiently, while samplesort does much more data movement in this |
| (for it) special case. |
| |
| ~sort: samplesort's special cases for large masses of equal elements are |
| extremely effective on ~sort's specific data pattern, and timsort just |
| isn't going to get close to that, despite that it's clearly getting a |
| great deal of benefit out of the duplicates (the # of compares is much less |
| than lg(n!)). ~sort has a perfectly uniform distribution of just 4 |
| distinct values, and as the distribution gets more skewed, samplesort's |
| equal-element gimmicks become less effective, while timsort's adaptive |
| strategies find more to exploit; in a database supplied by Kevin Altis, a |
| sort on its highly skewed "on which stock exchange does this company's |
| stock trade?" field ran over twice as fast under timsort. |
| |
| However, despite that timsort does many more comparisons on ~sort, and |
| that on several platforms ~sort runs highly significantly slower under |
| timsort, on other platforms ~sort runs highly significantly faster under |
| timsort. No other kind of data has shown this wild x-platform behavior, |
| and we don't have an explanation for it. The only thing I can think of |
| that could transform what "should be" highly significant slowdowns into |
| highly significant speedups on some boxes are catastrophic cache effects |
| in samplesort. |
| |
| But timsort "should be" slower than samplesort on ~sort, so it's hard |
| to count that it isn't on some boxes as a strike against it <wink>. |
| |
| + Here's the highwater mark for the number of heap-based temp slots (4 |
| bytes each on this box) needed by each test, again with arguments |
| "15 20 1": |
| |
| 2**i *sort \sort /sort 3sort +sort %sort ~sort =sort !sort |
| 32768 16384 0 0 6256 0 10821 12288 0 16383 |
| 65536 32766 0 0 21652 0 31276 24576 0 32767 |
| 131072 65534 0 0 17258 0 58112 49152 0 65535 |
| 262144 131072 0 0 35660 0 123561 98304 0 131071 |
| 524288 262142 0 0 31302 0 212057 196608 0 262143 |
| 1048576 524286 0 0 312438 0 484942 393216 0 524287 |
| |
| Discussion: The tests that end up doing (close to) perfectly balanced |
| merges (*sort, !sort) need all N//2 temp slots (or almost all). ~sort |
| also ends up doing balanced merges, but systematically benefits a lot from |
| the preliminary pre-merge searches described under "Merge Memory" later. |
| %sort approaches having a balanced merge at the end because the random |
| selection of elements to replace is expected to produce an out-of-order |
| element near the midpoint. \sort, /sort, =sort are the trivial one-run |
| cases, needing no merging at all. +sort ends up having one very long run |
| and one very short, and so gets all the temp space it needs from the small |
| temparray member of the MergeState struct (note that the same would be |
| true if the new random elements were prefixed to the sorted list instead, |
| but not if they appeared "in the middle"). 3sort approaches N//3 temp |
| slots twice, but the run lengths that remain after 3 random exchanges |
| clearly has very high variance. |
| |
| |
| A detailed description of timsort follows. |
| |
| Runs |
| ---- |
| count_run() returns the # of elements in the next run. A run is either |
| "ascending", which means non-decreasing: |
| |
| a0 <= a1 <= a2 <= ... |
| |
| or "descending", which means strictly decreasing: |
| |
| a0 > a1 > a2 > ... |
| |
| Note that a run is always at least 2 long, unless we start at the array's |
| last element. |
| |
| The definition of descending is strict, because the main routine reverses |
| a descending run in-place, transforming a descending run into an ascending |
| run. Reversal is done via the obvious fast "swap elements starting at each |
| end, and converge at the middle" method, and that can violate stability if |
| the slice contains any equal elements. Using a strict definition of |
| descending ensures that a descending run contains distinct elements. |
| |
| If an array is random, it's very unlikely we'll see long runs. If a natural |
| run contains less than minrun elements (see next section), the main loop |
| artificially boosts it to minrun elements, via a stable binary insertion sort |
| applied to the right number of array elements following the short natural |
| run. In a random array, *all* runs are likely to be minrun long as a |
| result. This has two primary good effects: |
| |
| 1. Random data strongly tends then toward perfectly balanced (both runs have |
| the same length) merges, which is the most efficient way to proceed when |
| data is random. |
| |
| 2. Because runs are never very short, the rest of the code doesn't make |
| heroic efforts to shave a few cycles off per-merge overheads. For |
| example, reasonable use of function calls is made, rather than trying to |
| inline everything. Since there are no more than N/minrun runs to begin |
| with, a few "extra" function calls per merge is barely measurable. |
| |
| |
| Computing minrun |
| ---------------- |
| If N < 64, minrun is N. IOW, binary insertion sort is used for the whole |
| array then; it's hard to beat that given the overheads of trying something |
| fancier. |
| |
| When N is a power of 2, testing on random data showed that minrun values of |
| 16, 32, 64 and 128 worked about equally well. At 256 the data-movement cost |
| in binary insertion sort clearly hurt, and at 8 the increase in the number |
| of function calls clearly hurt. Picking *some* power of 2 is important |
| here, so that the merges end up perfectly balanced (see next section). We |
| pick 32 as a good value in the sweet range; picking a value at the low end |
| allows the adaptive gimmicks more opportunity to exploit shorter natural |
| runs. |
| |
| Because sortperf.py only tries powers of 2, it took a long time to notice |
| that 32 isn't a good choice for the general case! Consider N=2112: |
| |
| >>> divmod(2112, 32) |
| (66, 0) |
| >>> |
| |
| If the data is randomly ordered, we're very likely to end up with 66 runs |
| each of length 32. The first 64 of these trigger a sequence of perfectly |
| balanced merges (see next section), leaving runs of lengths 2048 and 64 to |
| merge at the end. The adaptive gimmicks can do that with fewer than 2048+64 |
| compares, but it's still more compares than necessary, and-- mergesort's |
| bugaboo relative to samplesort --a lot more data movement (O(N) copies just |
| to get 64 elements into place). |
| |
| If we take minrun=33 in this case, then we're very likely to end up with 64 |
| runs each of length 33, and then all merges are perfectly balanced. Better! |
| |
| What we want to avoid is picking minrun such that in |
| |
| q, r = divmod(N, minrun) |
| |
| q is a power of 2 and r>0 (then the last merge only gets r elements into |
| place, and r < minrun is small compared to N), or q a little larger than a |
| power of 2 regardless of r (then we've got a case similar to "2112", again |
| leaving too little work for the last merge to do). |
| |
| Instead we pick a minrun in range(32, 65) such that N/minrun is exactly a |
| power of 2, or if that isn't possible, is close to, but strictly less than, |
| a power of 2. This is easier to do than it may sound: take the first 6 |
| bits of N, and add 1 if any of the remaining bits are set. In fact, that |
| rule covers every case in this section, including small N and exact powers |
| of 2; merge_compute_minrun() is a deceptively simple function. |
| |
| |
| The Merge Pattern |
| ----------------- |
| In order to exploit regularities in the data, we're merging on natural |
| run lengths, and they can become wildly unbalanced. That's a Good Thing |
| for this sort! It means we have to find a way to manage an assortment of |
| potentially very different run lengths, though. |
| |
| Stability constrains permissible merging patterns. For example, if we have |
| 3 consecutive runs of lengths |
| |
| A:10000 B:20000 C:10000 |
| |
| we dare not merge A with C first, because if A, B and C happen to contain |
| a common element, it would get out of order wrt its occurrence(s) in B. The |
| merging must be done as (A+B)+C or A+(B+C) instead. |
| |
| So merging is always done on two consecutive runs at a time, and in-place, |
| although this may require some temp memory (more on that later). |
| |
| When a run is identified, its base address and length are pushed on a stack |
| in the MergeState struct. merge_collapse() is then called to see whether it |
| should merge it with preceding run(s). We would like to delay merging as |
| long as possible in order to exploit patterns that may come up later, but we |
| like even more to do merging as soon as possible to exploit that the run just |
| found is still high in the memory hierarchy. We also can't delay merging |
| "too long" because it consumes memory to remember the runs that are still |
| unmerged, and the stack has a fixed size. |
| |
| What turned out to be a good compromise maintains two invariants on the |
| stack entries, where A, B and C are the lengths of the three righmost not-yet |
| merged slices: |
| |
| 1. A > B+C |
| 2. B > C |
| |
| Note that, by induction, #2 implies the lengths of pending runs form a |
| decreasing sequence. #1 implies that, reading the lengths right to left, |
| the pending-run lengths grow at least as fast as the Fibonacci numbers. |
| Therefore the stack can never grow larger than about log_base_phi(N) entries, |
| where phi = (1+sqrt(5))/2 ~= 1.618. Thus a small # of stack slots suffice |
| for very large arrays. |
| |
| If A <= B+C, the smaller of A and C is merged with B (ties favor C, for the |
| freshness-in-cache reason), and the new run replaces the A,B or B,C entries; |
| e.g., if the last 3 entries are |
| |
| A:30 B:20 C:10 |
| |
| then B is merged with C, leaving |
| |
| A:30 BC:30 |
| |
| on the stack. Or if they were |
| |
| A:500 B:400: C:1000 |
| |
| then A is merged with B, leaving |
| |
| AB:900 C:1000 |
| |
| on the stack. |
| |
| In both examples, the stack configuration after the merge still violates |
| invariant #2, and merge_collapse() goes on to continue merging runs until |
| both invariants are satisfied. As an extreme case, suppose we didn't do the |
| minrun gimmick, and natural runs were of lengths 128, 64, 32, 16, 8, 4, 2, |
| and 2. Nothing would get merged until the final 2 was seen, and that would |
| trigger 7 perfectly balanced merges. |
| |
| The thrust of these rules when they trigger merging is to balance the run |
| lengths as closely as possible, while keeping a low bound on the number of |
| runs we have to remember. This is maximally effective for random data, |
| where all runs are likely to be of (artificially forced) length minrun, and |
| then we get a sequence of perfectly balanced merges (with, perhaps, some |
| oddballs at the end). |
| |
| OTOH, one reason this sort is so good for partly ordered data has to do |
| with wildly unbalanced run lengths. |
| |
| |
| Merge Memory |
| ------------ |
| Merging adjacent runs of lengths A and B in-place is very difficult. |
| Theoretical constructions are known that can do it, but they're too difficult |
| and slow for practical use. But if we have temp memory equal to min(A, B), |
| it's easy. |
| |
| 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. 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() 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. |