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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.)

Link: http://lkml.kernel.org/r/2de8348635a1a421a72620677898c7fd5bd4b19d.1552704200.git.lkml@sdf.org
Signed-off-by: George Spelvin <lkml@sdf.org>
Acked-by: Andrey Abramov <st5pub@yandex.ru>
Acked-by: Rasmus Villemoes <linux@rasmusvillemoes.dk>
Reviewed-by: Andy Shevchenko <andriy.shevchenko@linux.intel.com>
Cc: Daniel Wagner <daniel.wagner@siemens.com>
Cc: Dave Chinner <dchinner@redhat.com>
Cc: Don Mullis <don.mullis@gmail.com>
Cc: Geert Uytterhoeven <geert@linux-m68k.org>
Signed-off-by: Andrew Morton <akpm@linux-foundation.org>
Signed-off-by: Linus Torvalds <torvalds@linux-foundation.org>
This commit is contained in:
George Spelvin 2019-05-14 15:42:52 -07:00 committed by Linus Torvalds
parent 37d0ec34d1
commit 22a241ccb2
1 changed files with 80 additions and 30 deletions
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110
lib/sort.c
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@ -2,7 +2,12 @@
/*
* 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)
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;
swap_func(base + r, base + c, size);
}
}
}
/*
* 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);