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SubjectRe: [PATCH] [rmap] operator-sparse Fibonacci hashing of waitqueues
Date
On February 17, 2002 10:01 am, William Lee Irwin III wrote:
> After distilling with hpa's help the results of some weeks-old
> numerological experiments^W^Wnumber crunching, I've devised a patch
> here for -rmap to make the waitqueue hashing somewhat more palatable
> for SPARC and several others.
>
> This patch uses some operator-sparse Fibonacci hashing primes in order
> to allow shift/add implementations of the hash function used for hashed
> waitqueues.
>
> Dan, Dave, could you take a look here and please comment?

Could you explain in very simple terms, suitable for Aunt Tillie (ok, not
*that* simple) how the continued fraction works, how it's notated, and how
the terms of the expansion relate to good performance as a hash?

--
Daniel

> Randy, any chance you could benchmark -rmap with this on top for
> comparison against standard -rmap to ensure there is no regression?
>
>
> Thanks,
> Bill
>
> P.S.: Dave: Sorry I took so long to get this thing out, but here it is.
>
>
> # This is a BitKeeper generated patch for the following project:
> # Project Name: Long-term Linux VM development
> # This patch format is intended for GNU patch command version 2.5 or higher.
> # This patch includes the following deltas:
> # ChangeSet 1.199 ->
> # include/linux/mm_inline.h 1.14 -> 1.15
> # mm/filemap.c 1.52 -> 1.53
> #
> # The following is the BitKeeper ChangeSet Log
> # --------------------------------------------
> # 02/02/17 wli@holomorphy.com 1.200
> # Use operator-sparse Fibonacci hashing primes for the hashed waitqueues so as to reduce the cost of
> # the hashing computation on several major RISC architectures and most 64-bit machines.
> #
> # Specifically the search for operator-sparse primes was documented around the definitions of the
> # GOLDEN_RATIO_PRIME values, where the number of terms in the sum of powers of 2 was chosen in
> # advance and the numbers generated this way were sorted by their continued fraction expansion
> # and then filtered for primality.
> #
> # In turn the definition of page_waitqueue() also needed to be altered so that for 64-bit machines,
> # whose compilers cannot perform the transformation of multiplication by a constant to shifts and
> # adds, the transformation is explcitly coded, and documentation is included for it describing how
> # the shifts and adds correspond to the signed sum of powers of two representation of the golden
> # ratio prime.
> # --------------------------------------------
> #
> diff -Nru a/include/linux/mm_inline.h b/include/linux/mm_inline.h
> --- a/include/linux/mm_inline.h Sun Feb 17 00:49:26 2002
> +++ b/include/linux/mm_inline.h Sun Feb 17 00:49:26 2002
> @@ -9,12 +9,43 @@
> * Chuck Lever verified the effectiveness of this technique for several
> * hash tables in his paper documenting the benchmark results:
> * http://www.citi.umich.edu/techreports/reports/citi-tr-00-1.pdf
> + *
> + * These prime numbers were determined by searching all numbers
> + * composed of sums of seven terms of the form 2^n with n decreasing
> + * with the index of the summand and sorting by the continued fraction
> + * expansion of p/2^BITS_PER_LONG, then filtering for primality.
> + * Verifying that the confidence tests on the chi^2 statistics passed
> + * is necessary because this process can produce "bad" factors, but on
> + * the other hand it often produces excellent factors, so it's a very
> + * useful process for generating hash functions.
> */
> +
> #if BITS_PER_LONG == 32
> -#define GOLDEN_RATIO_PRIME 2654435761UL
> +/*
> + * 2654404609 / 2^32 has the continued fraction expansion
> + * 0,1,1,1,1,1,1,1,1,1,1,1,1,18,7,1,3,7,18,1,1,1,1,1,1,1,1,1,1,2,0
> + * which is very close to phi.
> + * It is of the form 2^31 + 2^29 - 2^25 + 2^22 - 2^19 - 2^16 + 1
> + * which makes it suitable for compiler optimization to shift/add
> + * implementations on machines where multiplication is a slow operation.
> + * gcc is successfully able to optimize this on 32-bit SPARC and MIPS.
> + */
> +#define GOLDEN_RATIO_PRIME 0x9e370001UL
>
> #elif BITS_PER_LONG == 64
> -#define GOLDEN_RATIO_PRIME 11400714819323198549UL
> +/*
> + * 11400862456688148481 / 2^64 has the continued fraction expansion
> + * 0,1,1,1,1,1,1,1,1,1,1,1,2,1,14,1,1048579,15,1,2,1,1,3,9,1,1,8,3,1,7,
> + * 1,1,9,1,2,1,1,1,1,2,0
> + * which is very close to phi = (sqrt(5)-1)/2 = 0,1,1,1,....
> + *
> + * It is of the form 2^63 + 2^61 - 2^57 + 2^54 - 2^51 - 2^18 + 1
> + * which makes it suitable for shift/add implementations of the hash
> + * function. 64-bit architectures typically have slow multiplies (or
> + * no hardware multiply) and also gcc is unable to optimize 64-bit
> + * multiplies for these bit patterns so explicit expansion is used.
> + */
> +#define GOLDEN_RATIO_PRIME 0x9e37fffffffc0001UL
>
> #else
> #error Define GOLDEN_RATIO_PRIME in mm_inline.h for your wordsize.
> diff -Nru a/mm/filemap.c b/mm/filemap.c
> --- a/mm/filemap.c Sun Feb 17 00:49:26 2002
> +++ b/mm/filemap.c Sun Feb 17 00:49:26 2002
> @@ -787,6 +787,9 @@
> * collisions. This cost is great enough that effective hashing
> * is necessary to maintain performance.
> */
> +
> +#if BITS_PER_LONG == 32
> +
> static inline wait_queue_head_t *page_waitqueue(struct page *page)
> {
> const zone_t *zone = page_zone(page);
> @@ -798,6 +801,50 @@
>
> return &wait[hash];
> }
> +
> +#elif BITS_PER_LONG == 64
> +
> +static inline wait_queue_head_t *page_waitqueue(struct page *page)
> +{
> + const zone_t *zone = page_zone(page);
> + wait_queue_head_t *wait = zone->wait_table;
> + unsigned long hash = (unsigned long)page;
> + unsigned long n;
> +
> + /*
> + * The bit-sparse GOLDEN_RATIO_PRIME is a sum of powers of 2
> + * with varying signs: 2^63 + 2^61 - 2^57 + 2^54 - 2^51 - 2^18 + 1
> + * The shifts in the code below correspond to the differences
> + * in the exponents above.
> + * The primary reason why the work of expanding the multiplication
> + * this way is not given to the compiler is because 64-bit code
> + * generation does not seem to be capable of doing it. So the code
> + * here manually expands it.
> + * The differences are used in order to both reduce the number of
> + * variables used and also to reduce the size of the immediates
> + * needed for the shift instructions, whose precision is limited
> + * on some architectures.
> + */
> +
> + n = hash;
> + n <<= 18;
> + hash -= n;
> + n <<= 33;
> + hash -= n;
> + n <<= 3;
> + hash += n;
> + n <<= 3;
> + hash -= n;
> + n <<= 4;
> + hash += n;
> + n <<= 2;
> + hash += n;
> +
> + hash >>= zone->wait_table_shift;
> + return &wait[hash];
> +}
> +
> +#endif /* BITS_PER_LONG == 64 */
>
>
> /*
> -
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>

--
Daniel
-
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