Messages in this thread Patch in this message | | | From | George Spelvin <> | Date | Thu, 21 Feb 2019 08:21:42 +0000 | Subject | [PATCH v2 2/5] lib/sort: Use more efficient bottom-up heapsort variant |
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This uses fewer comparisons than the previous code (approaching half as many for large random inputs), but produces identical results; it actually performs the exact same series of swap operations.
Specifically, it reduces the average number of compares from 2*n*log2(n) - 3*n + o(n) to n*log2(n) + 0.37*n + o(n).
This is still 1.63*n worse than glibc qsort() which manages n*log2(n) - 1.26*n, but at least the leading coefficient is correct.
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 O(1) second comparisons rather than log2(n). 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, doing 1.5*n*log2(n) + O(n) in the worst case.
(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 -> 767 bytes (-118)
(I see the checkpatch complaint about "else if (n -= size)". The alternative is significantly uglier.)
Signed-off-by: George Spelvin <lkml@sdf.org> Acked-by: Andrey Abramov <st5pub@yandex.ru> --- lib/sort.c | 110 ++++++++++++++++++++++++++++++++++++++--------------- 1 file changed, 80 insertions(+), 30 deletions(-)
diff --git a/lib/sort.c b/lib/sort.c index ec79eac85e21..0d24d0c5c0fc 100644 --- a/lib/sort.c +++ b/lib/sort.c @@ -1,8 +1,13 @@ // SPDX-License-Identifier: GPL-2.0 /* - * A fast, small, non-recursive O(nlog n) sort for the Linux kernel + * A fast, small, non-recursive O(n log n) sort for the Linux kernel * - * Jan 23 2005 Matt Mackall <mpm@selenic.com> + * This performs n*log2(n) + 0.37*n + o(n) comparisons on average, + * and 1.5*n*log2(n) + O(n) in the (very contrived) worst case. + * + * Glibc qsort() manages n*log2(n) - 1.26*n for random inputs (1.63*n + * better) at the expense of stack usage and much larger code to avoid + * quicksort's O(n^2) worst case. */ #define pr_fmt(fmt) KBUILD_MODNAME ": " fmt @@ -15,7 +20,7 @@ * is_aligned - is this pointer & size okay for word-wide copying? * @base: pointer to data * @size: size of each element - * @align: required aignment (typically 4 or 8) + * @align: required alignment (typically 4 or 8) * * Returns true if elements can be copied using word loads and stores. * The size must be a multiple of the alignment, and the base address must @@ -115,6 +120,32 @@ static void swap_bytes(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 @@ -129,17 +160,20 @@ static void swap_bytes(void *a, void *b, int size) * 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 + * quicksort is slightly 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; + const unsigned int lsbit = size & -size; /* Used to find parent */ + + if (!a) /* num < 2 || size == 0 */ + return; if (!swap_func) { if (is_aligned(base, size, 8)) @@ -150,32 +184,48 @@ void sort(void *base, size_t num, size_t size, swap_func = swap_bytes; } - /* 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); } } } - EXPORT_SYMBOL(sort); -- 2.20.1
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