Messages in this thread Patch in this message | | | From | George Spelvin <> | Date | Thu, 21 Feb 2019 08:21:42 +0000 | Subject | [PATCH 2/5] lib/sort: Use more efficient bottom-up heapsort variant |
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This uses fewer comparisons than the previous code (61% as many for large random inputs), but produces identical results; it actually performs the exact same series of swap operations.
Standard heapsort, when sifting down, performs two comparisons per level: One to find the greater child, and a second to see if the current node should be exchanged with that child.
Bottom-up heapsort observes that it's better to postpone the second comparison and search for the leaf where -infinity would be sent to, then search back *up* for the current node's destination.
Since sifting down usually proceeds to the leaf level (that's where half the nodes are), this does many fewer second comparisons. That saves a lot of (expensive since Spectre) indirect function calls.
The one time it's worse than the previous code is if there are large numbers of duplicate keys, when the top-down algorithm is O(n) and bottom-up is O(n log n). For distinct keys, it's provably always better.
(The code is not significantly more complex. This patch also merges the heap-building and -extracting sift-down loops, resulting in a net code size savings.)
x86-64 code size 885 -> 770 bytes (-115)
(I see the checkpatch complaint about "else if (n -= size)". The alternative is significantly uglier.)
Signed-off-by: George Spelvin <lkml@sdf.org> --- lib/sort.c | 102 +++++++++++++++++++++++++++++++++++++++-------------- 1 file changed, 75 insertions(+), 27 deletions(-)
diff --git a/lib/sort.c b/lib/sort.c index dff2ab2e196e..2aef4631e7d3 100644 --- a/lib/sort.c +++ b/lib/sort.c @@ -117,6 +117,32 @@ static void generic_swap(void *a, void *b, int size) } while (n); } +/** + * parent - given the offset of the child, find the offset of the parent. + * @i: the offset of the heap element whose parent is sought. Non-zero. + * @lsbit: a precomputed 1-bit mask, equal to "size & -size" + * @size: size of each element + * + * In terms of array indexes, the parent of element j = i/size is simply + * (j-1)/2. But when working in byte offsets, we can't use implicit + * truncation of integer divides. + * + * Fortunately, we only need one bit of the quotient, not the full divide. + * size has a least significant bit. That bit will be clear if i is + * an even multiple of size, and set if it's an odd multiple. + * + * Logically, we're doing "if (i & lsbit) i -= size;", but since the + * branch is unpredictable, it's done with a bit of clever branch-free + * code instead. + */ +__attribute_const__ __always_inline +static size_t parent(size_t i, unsigned int lsbit, size_t size) +{ + i -= size; + i -= size & -(i & lsbit); + return i / 2; +} + /** * sort - sort an array of elements * @base: pointer to data to sort @@ -125,21 +151,26 @@ static void generic_swap(void *a, void *b, int size) * @cmp_func: pointer to comparison function * @swap_func: pointer to swap function or NULL * - * This function does a heapsort on the given array. You may provide a - * swap_func function optimized to your element type. + * This function does a heapsort on the given array. You may provide a + * swap_func function if you need to do something more than a memory copy + * (e.g. fix up pointers or auxiliary data), but the built-in swap isn't + * usually a bottleneck. * * Sorting time is O(n log n) both on average and worst-case. While * qsort is about 20% faster on average, it suffers from exploitable * O(n*n) worst-case behavior and extra memory requirements that make * it less suitable for kernel use. */ - void sort(void *base, size_t num, size_t size, int (*cmp_func)(const void *, const void *), void (*swap_func)(void *, void *, int size)) { /* pre-scale counters for performance */ - int i = (num/2 - 1) * size, n = num * size, c, r; + size_t n = num * size, a = (num/2) * size; + unsigned const lsbit = size & -size; /* Used to find parent */ + + if (!n) + return; if (!swap_func) { if (alignment_ok(base, size, 8)) @@ -150,30 +181,47 @@ void sort(void *base, size_t num, size_t size, swap_func = generic_swap; } - /* heapify */ - for ( ; i >= 0; i -= size) { - for (r = i; r * 2 + size < n; r = c) { - c = r * 2 + size; - if (c < n - size && - cmp_func(base + c, base + c + size) < 0) - c += size; - if (cmp_func(base + r, base + c) >= 0) - break; - swap_func(base + r, base + c, size); - } - } + /* + * Loop invariants: + * 1. elements [a,n) satisfy the heap property (compare greater than + * all of their children), + * 2. elements [n,num*size) are sorted, and + * 3. a <= b <= c <= d <= n (whenever they are valid). + */ + for (;;) { + size_t b, c, d; - /* sort */ - for (i = n - size; i > 0; i -= size) { - swap_func(base, base + i, size); - for (r = 0; r * 2 + size < i; r = c) { - c = r * 2 + size; - if (c < i - size && - cmp_func(base + c, base + c + size) < 0) - c += size; - if (cmp_func(base + r, base + c) >= 0) - break; - swap_func(base + r, base + c, size); + if (a) /* Building heap: sift down --a */ + a -= size; + else if (n -= size) /* Sorting: Extract root to --n */ + swap_func(base, base + n, size); + else /* Sort complete */ + break; + + /* + * Sift element at "a" down into heap. This is the + * "bottom-up" variant, which significantly reduces + * calls to cmp_func(): we find the sift-down path all + * the way to the leaves (one compare per level), then + * backtrack to find where to insert the target element. + * + * Because elements tend to sift down close to the leaves, + * this uses fewer compares than doing two per level + * on the way down. (A bit more than half as many on + * average, 3/4 worst-case.) + */ + for (b = a; c = 2*b + size, (d = c + size) < n;) + b = cmp_func(base + c, base + d) >= 0 ? c : d; + if (d == n) /* Special case last leaf with no sibling */ + b = c; + + /* Now backtrack from "b" to the correct location for "a" */ + while (b != a && cmp_func(base + a, base + b) >= 0) + b = parent(b, lsbit, size); + c = b; /* Where "a" belongs */ + while (b != a) { /* Shift it into place */ + b = parent(b, lsbit, size); + swap_func(base + b, base + c, size); } } } -- 2.20.1
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