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Subject[PATCH v2 2/5] lib/sort: Use more efficient bottom-up heapsort variant
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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