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Adopted alternative implementation for lfs_ctz_index

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Same runtime cost, however reduces the logic and avoids one of
the two big branches. See the DESIGN.md for more info.

Now uses these equations instead of the messy guess and correct method:
n = (N - w/8(popcount(N/(B-2w/8)) + 2)) / (B-2w/8)
off = N - (B-w2/8)n - w/8popcount(n)
This commit is contained in:
Christopher Haster
2017-10-18 00:33:59 -05:00
parent 46e22b2a38
commit 0825d34f3d
2 changed files with 18 additions and 43 deletions
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45
DESIGN.md
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@@ -396,42 +396,23 @@ for a file size:
Unfortunately, we're not quite done. The popcount function is non-injective, Unfortunately, we're not quite done. The popcount function is non-injective,
so we can only find the file size from the block index, not the other way so we can only find the file size from the block index, not the other way
around. However, we can guess and correct. Consider an n' block index that around. However, we can solve for an n' block index that is greater than n
is greater than n, we can find one pretty easily: with an error bounded by the range of the popcount function. We can then
repeatedly substitute this n' into the original equation until the error
is smaller than the integer division. As it turns out, we only need to
perform this substitution once. Now we directly calculate our block index:
![summation3step1](https://latex.codecogs.com/svg.latex?n%27%20%3D%20%5Cleft%5Clfloor%5Cfrac%7BN%7D%7BB-2%5Cfrac%7Bw%7D%7B8%7D%7D%5Cright%5Crfloor) ![formulaforn](https://latex.codecogs.com/svg.latex?n%20%3D%20%5Cleft%5Clfloor%5Cfrac%7BN-%5Cfrac%7Bw%7D%7B8%7D%5Cleft%28%5Ctext%7Bpopcount%7D%5Cleft%28%5Cfrac%7BN%7D%7BB-2%5Cfrac%7Bw%7D%7B8%7D%7D-1%5Cright%29+2%5Cright%29%7D%7BB-2%5Cfrac%7Bw%7D%7B8%7D%7D%5Cright%5Crfloor)
where: Now that we have our block index n, we can just plug it back into the above
n' >= n equation to find the offset. However, we do need to rearrange the equation
a bit to avoid integer overflow:
We can plug n' back into our popcount equation to find an N' file size that ![formulaforoff](https://latex.codecogs.com/svg.latex?%5Cmathit%7Boff%7D%20%3D%20N%20-%20%5Cleft%28B-2%5Cfrac%7Bw%7D%7B8%7D%5Cright%29n%20-%20%5Cfrac%7Bw%7D%7B8%7D%5Ctext%7Bpopcount%7D%28n%29)
is greater than N. However, we need to rearrange our terms a bit to avoid
integer overflow:
![summation3step2](https://latex.codecogs.com/svg.latex?N%27%20%3D%20%28B-2%5Cfrac%7Bw%7D%7B8%7D%29n%27+%5Cfrac%7Bw%7D%7B8%7D%5Ctext%7Bpopcount%7D%28n%27%29) The solution involves quite a bit of math, but computers are very good at math.
We can now solve for the block index + offset while only needed to store the
where: file size in O(1).
N' >= N
Now that we have N', we can find our block offset:
![summation3step3](https://latex.codecogs.com/svg.latex?%5Cmathit%7Boff%7D%27%20%3D%20N%20-%20N%27)
where:
off' >= off, our byte offset in the block
Now we're getting somewhere. N' is greater than or equal to N, and as long as
the number of pointers per block is bounded by the block size, it can only be
different by at most one block. So we have two cases that can be determined by
the sign of off'. If off' is negative, we correct n' and add a block to off'.
Note that we also need to incorporate the overhead of the last block to get
the right offset.
![summation3step4](https://latex.codecogs.com/svg.latex?n%2C%20%5Cmathit%7Boff%7D%20%3D%20%5Cbegin%7Bcases%7D%20n%27-1%2C%20%5Cmathit%7Boff%7D%27+B%20%26%20%5Cmathit%7Boff%7D%27%20%3C%200%20%5C%5C%20n%27%2C%20%5Cmathit%7Boff%7D%27+%5Cfrac%7Bw%7D%7B8%7D%5Cleft%5B%5Ctext%7Bctz%7D%28n%27%29+1%5Cright%5D%20%26%20%5Cmathit%7Boff%7D%27%20%5Cgeq%200%20%5Cend%7Bcases%7D)
It's a lot of math, but computers are very good at math. With these equations
we can solve for the block index + offset while only needed to store the file
size in O(1).
Here is what it might look like to update a file stored with a CTZ skip-list: Here is what it might look like to update a file stored with a CTZ skip-list:
``` ```

14
lfs.c
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@@ -1005,21 +1005,15 @@ int lfs_dir_rewind(lfs_t *lfs, lfs_dir_t *dir) {
/// File index list operations /// /// File index list operations ///
static int lfs_ctz_index(lfs_t *lfs, lfs_off_t *off) { static int lfs_ctz_index(lfs_t *lfs, lfs_off_t *off) {
lfs_off_t size = *off; lfs_off_t size = *off;
lfs_off_t i = size / (lfs->cfg->block_size-2*4); lfs_off_t b = lfs->cfg->block_size - 2*4;
lfs_off_t i = size / b;
if (i == 0) { if (i == 0) {
return 0; return 0;
} }
lfs_off_t nsize = (lfs->cfg->block_size-2*4)*i + 4*lfs_popc(i-1) + 2*4; i = (size - 4*(lfs_popc(i-1)+2)) / b;
lfs_soff_t noff = size - nsize; *off = size - b*i - 4*lfs_popc(i);
if (noff < 0) {
*off = noff + lfs->cfg->block_size;
return i-1;
} else {
*off = noff + 4*(lfs_ctz(i) + 1);
return i; return i;
}
} }
static int lfs_ctz_find(lfs_t *lfs, static int lfs_ctz_find(lfs_t *lfs,