## Friday, December 14, 2018

### LSM math - size of search space for LSM tree configuration

I have written before and will write again about using 3-tuples to explain the shape of an LSM tree. This makes it easier to explain the configurations supported today and configurations we might want to support tomorrow in addition to traditional tiered and leveled compaction. The summary is that n LSM tree has N levels labeled from L1 to Ln and Lmax is another name for L1. There is one 3-tuple per level and the components of the 3-tuple are (type, fanout, runs) for Lk (level k) where:
• type is Tiered or Leveled and explains compaction into that level
• fanout is the size of a sorted run in Lk relative to a sorted run from Lk-1, a real and >= 1
• runs is the number of sorted runs in that level, an integer and >= 1
Given the above how many valid configurations exist for an LSM tree? There are additional constraints that can be imposed on the 3-tuple but I will ignore most of them except for limiting fanout and runs to be <= 20. The answer is easy - there are an infinite number of configurations because fanout is a real.

The question is more interesting when fanout is limited to an integer and the number of levels is limited to between 1 and 10. I am doing this to explain the size of the search space but I don't think that fanout should be limited to an integer.

There are approximately 2^11 configurations only considering compaction type, which has 2 values, and 1 to 10 levels because there are 2^N configurations of compaction types for a tree with N levels and the sum of 2^1 + 2^2 + ... + 2^9 + 2^10 = 2^11 - 1

But when type, fanout and runs are considered then there are 2 x 20 x 20 = 800 choices per level and 800^N combinations for an LSM tree with N levels. Considering LSM trees with 1 to 10 levels then the number of valid configurations is the sum 800^1 + 800^2 + ... + 800^9 + 800^10. That is a large number of configurations if exhaustive search were to be used to find the best configuration. Note that I don't think exhaustive search should be used.

## Thursday, December 13, 2018

### LSM math - how many levels minimizes write amplification?

How do you configure an LSM tree with leveled compaction to minimize write amplification? For a given number of levels write-amp is minimal when the same fanout (growth factor) is used between all levels, but that does not explain the number of levels to use. In this post I answer that question.
1. The number of levels that minimizes write-amp is one of ceil(ln(T)) or floor(ln(T)) where T is the total fanout -- sizeof(database) / sizeof(memtable)
2. When #1 is done then the per-level fanout is e when the number of levels is ln(t) and a value close to e when the number of levels is an integer.
Introduction

I don't recall reading this result elsewhere, but I am happy to update this post with a link to such a result. I was encouraged to answer this after a discussion with the RocksDB team and thank Siying Dong for stating #2 above while leaving the math to me. I assume the original LSM paper didn't address this problem because that system used a fixed number of levels.

One result from the original LSM paper and updated by me is that write-amp is minimized when the per-level growth factor is constant. Sometimes I use fanout or per-level fanout rather than per-level growth factor. In RocksDB the option name is max_bytes_for_level_multiplier. Yes, this can be confusing. The default fanout in RocksDB is 10.

Math

I solve this for pure-leveled compaction which differs from what RocksDB calls leveled. In pure-leveled all levels used leveled compaction. In RocksDB leveled the first level, L0, uses tiered and the other levels used leveled. I started to explain this here where I claim that RocksDB leveled is really tiered+leveled. But I am not asking for them to change the name.

Assumptions:
• LSM tree uses pure-leveled compaction and compaction from memtable flushes into the first level of the LSM tree uses leveled compaction
• total fanout is T and is size(Lmax) / size(memtable) where Lmax is the max level of the LSM tree
• workload is update-only so the number of keys in the database is fixed
• workload has no write skew and all keys are equally likely to be updated
• per-level write-amp == per-level growth factor. In practice and in theory the per-level write-amp tends to be less than the per-level growth factor.
• total write-amp is the sum of per-level write-amp. I ignore write-amp from the WAL.

Specify function for write-amp and determine critical points

# wa is the total write-amp
# n is the number of levels
# per-level fanout is the nth root of the total fanout

# per-level fanout = per-level write-amp
# therefore wa = number of levels * per-level fanout

wa = n * t^(1/n)

# given the function for write-amp as wa = a * b
# ... then below is a' * b + a * b'
a = n, b = t^(1/n)

wa' = t^(1/n) + n * ln(t) * t^(1/n) * (-1) * (1/n^2)

# which simplifies to

wa' = t^(1/n) - (1/n) * ln(t) * t^(1/n)

# critical point for this occurs when wa' = 0
t^(1/n) - (1/n) * ln(t) * t^(1/n) = 0

t^(1/n) = (1/n) * ln(t) * t^(1/n)
1 = (1/n) * ln(t)

n = ln(t)

When t = 1024 then n = ln(1024) ~= 6.93. In this case write-amp is minimized when 7 levels are used although 6 isn't a bad choice.

Assuming the cost function is convex (see below) the critical point is the minimum for write-amp. However, n must be an integer so the number of levels that minimizes write-amp is one of: ceil(ln(t)) or floor(ln(t)).

The graph for wa when t=1024 can be viewed thanks to Desmos. The function looks convex and I show below that it is.

Determine whether critical point is a min or max

The critical point found above is a minimum for wa if wa is convex so we must show that the second derivative is positive.

wa = n * t ^ (1/n)
wa' = t^(1/n) - (1/n) * ln(t) * t^(1/n)
wa' = t^(1/n) * (1 - (1/n) * ln(t))

# assuming wa' is a * b then wa'' is a' * b + a * b'

a  = t^(1/n)
a' = ln(t) * t^(1/n) * -1 * (1/n^2)
a' = - ln(t) * t^(1/n) * (1/n^2)

b  = 1 - (1/n) * ln(t)
b' = (1/n^2) * ln(t)

# a' * b
- ln(t) * t^(1/n) * (1/n^2)         --> called x below
+ ln(t) * ln(t) * (1/n^3) * t^(1/n) --> called y below

# b' * a
t^(1/n) * (1/n^2) * ln(t)           --> called z below

# therefore wa'' = x + y + z
# note that x, y and z all contain: t^(1/n), 1/n and ln(t)
wa'' = t^(1/n) * (1/n) * ln(t) * (-(1/n) + (ln(t) * 1/n^2) + (1/n))
wa'' = t^(1/n) * (1/n) * ln(t) * ( ln(t) * 1/n^2 )'
wa'' = t^(1/n) * 1/n^3 * ln(t)^2

Therefore wa'' is positive, wa is convex and the critical point is a minimum value for wa

Solve for per-level fanout

The next step is to determine the value of the per-level fanout when write-amp is minimized. If the number of levels doesn't have to be an integer then this occurs when ln(t) levels are used and below I show that the per-level fanout is e in that case. When the number of levels is limited to an integer then the per-level fanout that minimizes write-amp is a value that is close to e.

# total write-amp is number of levels * per-level fanout
wa = n * t^(1/n)

# The per-level fanout is t^(1/n) and wa is minimized when n = ln(t)

# Therefore we show that t^(1/n) = e when n = ln(t)
Assume t^(1 / ln(t)) = e
ln (t^(1 / ln(t))) = ln e
(1 / ln(t)) * ln(t) = 1
1=1

When the t=1024 then ln(t) ~= 6.93. With 7 levels the per-level fanout is t^(1/7) ~= 2.69 while e ~= 2.72.

## Saturday, December 1, 2018

### Pixelbook review

This has nothing to do with databases. This is a review of a Pixelbook (Chromebook laptop) that I got on sale last month. This one has a core i5, 8gb RAM and 128gb storage. It runs Linux too but I haven't done much with that. I expected a lot from this given that my 2013 Nexus 7 tablet is still awesome. I have been mostly happy with the laptop but if you care about keyboards and don't like the new Macs thanks to the butterfly keyboard then this might not be the laptop for you. My 3 complaints:

1. keyboard is hard to read. It is grey on grey and too hard to read when there is light on my back even with the backlight (backlit?) turned all the way up. I don't get it -- grey on grey. So this is a great device for using in a dark room or for improving your touch typing skills.
2. touchpad control is too coarse grained so it is either too fast or too slow. The settings has 5 values via a slider (1=slowest, 5=fastest). I have been using it at 3 which is a bit too fast for me while 2 is a bit too slow. I might go back to 2 but that means picking up my finger more frequently when moving a pointer across the screen.
3. no iMessage - my family uses Apple devices and I can't run that here as I can on a Mac laptop

## Monday, November 19, 2018

This is review of TRIAD which was published in USENIX ATC 2017. It explains how to reduce write amplification for RocksDB leveled compaction although the ideas are useful for many LSM implementations. I share a review here because the paper has good ideas. It isn't easy to keep up with all of the LSM research, even when limiting the search to papers that reference RocksDB, and I didn't notice this paper until recently.

TRIAD reduces write amplification for an LSM with leveled compaction and with a variety of workloads gets up to 193% more throughput, up to 4X less write amplification and spends up to 77% less time doing compaction and flush. Per the RUM Conjecture improvements usually come at a cost and the cost in this case is more cache amplification (more memory overhead/key) and possibly more read amplification. I assume this is a good tradeoff in many cases.

The paper explains the improvements via 3 components -- TRIAD-MEM, TRIAD-DISK and TRIAD-LOG -- that combine to reduce write amplification.

TRIAD-MEM reduces write-amp by keeping frequently updated keys (hot keys) in the memtable. It divides keys into the memtable into two classes: hot and cold. On flush the cold keys are written into a new L0 SST while the hot keys are copied over to the new memtable. The hot keys must be written again to the new WAL so that the old WAL can be dropped. TRIAD-MEM tries to keep the K hottest keys in the memtable and there is work in progress to figure out a good value for K without being told by the DBA.

An extra 4-bytes/key is used for the memtable to track write frequency and identify hot keys. Note that RocksDB already 8 bytes/key for metadata. So TRIAD-MEM has a cost in cache-amp but I don't think that is a big deal.

Assuming the per-level write-amp is 1 from the memtable flush this reduces it to 0 in the best case where all keys are hot.

TRIAD-DISK reduces write-amp by delaying L0:L1 compaction until there is sufficient overlap between keys to be compacted. TRIAD continues to use an L0:L1 compaction trigger based on the number of files in the L0 but can trigger compaction earlier when there is probably sufficient overlap between the L0 and L1 SSTs.

Overlap is estimated via Hyperloglog (HLL) which requires 4kb/SST and is estimated as the following where file-i is the i-th SST under consideration, UniqueKeys is the estimated number of distinct keys across all of the SSTs and Keys(file-i) is the number of keys in the i-th SST. The paper states that both UniqueKeys and Keys are approximated using HLL. But I assume that per-SST metadata already has an estimate or exact value for the number of keys in the SST. The formula for overlap is:
UniqueKeys(file-1, file-2, ... file-n) / sum( Keys( file-i))

The benefit from early L0:L1 compaction is less read-amp, because there will be fewer sorted runs to search on a query. The cost from always doing early compaction is more per-level write-amp which is etimated by size(L1 input) / size(L0 input). TRIAD-DISK provides the benefit with less cost.

In RocksDB today you can manually schedule early compaction by setting the trigger to 1 or 2 files, or you can always schedule it to be less early with a trigger set to 8 or more files. But this setting is static. TRIAD-DISK uses a cost-based approach to do early compaction when it won't hurt the per-level write-amp. This is an interesting idea.

TRIAD-LOG explains improvements to memtable flush that reduce write-amp. Data in an L0 SST has recently been written to the WAL. So they use the WAL in place of writing the L0 SST. But something extra, an index into the WAL, is written on memtable flush because everything in the L0 must have an index. The WAL in the SST (called the CL-SST for commit log SST) will be deleted when it is compacted into the L1.

There is cache-amp from TRIAD-LOG. Each key in the CL-SST (L0) and maybe in the memtable needs 8 extra bytes -- 4 bytes for CL-SST ID, 4 bytes for the WAL offset.

Assuming the per-level write-amp is one from the memtable flush for cold keys this reduces that to 0.

Reducing write amplification

The total write-amp for an LSM tree with leveled compaction is the sum of:
• writing the WAL = 1
• memtable flush = 1
• L0:L1 compaction ~= size(L1) / size(L0)
• Ln compaction for n>1 ~= fanout, the per-level growth factor, usually 8 or 10. Note that this paper explains why it is usually a bit less than fanout.
TRIAD avoids the write-amp from memtable flush thanks to TRIAD-MEM for hot keys and TRIAD-LOG for cold keys. I will wave my hands and suggest that TRIAD-DISK reduces write-amp for L0:L1 from 3 to 1 based on the typical LSM configuration I use. So TRIAD reduces the total write-amp by 1+2 or 3.

Reducing total write-amp by 3 is a big deal when the total write-amp for the LSM tree is small, for example <= 10. But that only happens when there are few levels beyond the L1. Assuming you accept my estimate for total write-amp above then per-level write-amp is ~8 for both L1:L2 and L2:L3. The total write-amp for an LSM tree without TRIAD would be 1+1+3+8 = 13 if the max level is L2 and 1+1+3+8+8 = 21 if the max level is L3. And then TRIAD reduces that from 13 to 10 or from 21 to 18.

But my write-amp estimate above is more true for workloads without skew and less true for workloads with skew. Many of the workloads tested in the paper have a large amount of skew. So while I have some questions about the paper I am not claiming they are doing it wrong. What I am claiming is that the benefit from TRIAD is significant when total write-amp is small and less significant otherwise. Whether this matters is workload dependent. It would help to know more about the LSM tree from each benchmark. How many levels were in the LSM tree per benchmark? What is the per-level write-amp with and without TRIAD? Most of this can be observed from compaction statistics provided by RocksDB. The paper has some details on the workloads but that isn't sufficient to answer the questions above.

Questions

The paper documents the memory overhead, but limits the definition of read amplification to IO and measured none. I am interested in IO and CPU and suspect there might be some CPU read-amp from using the commit-log SST in the L0 both for searches and during compaction as logically adjacent data is no longer physically adjacent in the commit-log SST.
impact of more levels?

Another question is how far down the LSM compaction occurs. For example if the write working set fits in the L2, should compaction stop at the L2. It might with some values of compaction priority in RocksDB but it doesn't for all.  When the workload has significant write skew then the write working set is likely to fit into one of the smaller levels of the LSM tree.

An interesting variant on this is a workload with N streams of inserts that are each appending (right growing). When N=1 there is an optimization in RocksDB that limits write-amp to 2 (one for WAL, one for SST). I am not aware of optimizations in RocksDB for N>2 but am curious if we could do something better.

## Friday, November 2, 2018

### Converting an LSM to a B-Tree and back again

I wonder if it is possible to convert an LSM to a B-Tree. The goal is to do it online and in-place -- so I don't want two copies of the database while the conversion is in progress. I am interested in data structures for data management that adapt dynamically to improve performance and efficiency for a given workload.

Workloads change in the short and long term. I hope that data structures can be adapt to the change and converting between an LSM and a B-Tree is one way to adapt. This is more likely to be useful when the data structure supports some kind of partitioning in the hope that different workloads can be isolated to different partitions -- and then some can use an LSM while others use a B-Tree.

LSM to B-Tree

A B-Tree is one tree. An LSM is a sequence of trees. Each sorted run in the LSM is a tree. With leveled compaction in RocksDB there are a few sorted runs in level 0 (L0) and then one sorted run in each of L1, L2 up to the max level (Lmax).

A B-Tree persists changes by writing back pages -- either in-place or copy-on-write (UiP or CoW). An LSM persists changes by writing and then re-writing rows. I assume that page write back is required if you want to limit the database to one tree and row write back implies there will be more than one tree.

There are two things that must be done online and in-place:
1. Convert the LSM from many trees to one tree
2. Convert from row write back to page write back
Note that my goal has slightly changed. I want to move from an LSM to a data structure with one tree. For the one-tree solution a B-Tree is preferred but not required.

The outline of a solution:
1. Reconfigure the LSM to use 2 levels -- L0 and L1 -- and 3 trees -- memtable, L0, L1.
2. Disable the L0. At this point the LSM has two trees -- memtable and L1.
3. Flush the memtable and merge it into the L1. Now there is one tree.
4. After the flush disable the memtable and switch to a page cache. Changes now require a copy of the L1 block in the page cache that eventually get written back via UiP or CoW.
The outline above doesn't explain how to maintain indexes for the L1. Note that after step 2 there is one tree on disk and the layout isn't that different from the leaf level of a B-Tree. The interior levels of the B-Tree could be created by reading/rewriting the block indexes stored in the SSTs.

B-Tree to LSM

The conversion can also be done in the opposite direction (B-Tree to LSM)
1. Treat the current B-Tree as the max level of the LSM tree. While it might help to flush the page cache I don't think that is required. This is easier to do when your LSM uses a B-Tree per level, as done by WiredTiger.
2. Record new changes for insert, update, delete in a memtable rather than a page cache.
3. When the memtable is full then flush it to create a new tree (sorted run, SST) on disk.
4. Eventually start to do compaction.

## Friday, October 19, 2018

### Combining tiered and leveled compaction

There are simple optimization problems for LSM tuning. For example use leveled compaction to minimize space amplification and use tiered to minimize write amplification. But there are interesting problems that are harder to solve:
1. maximize throughput given a constraint on write and/or space amplification
2. minimize space and/or write amplification given a constraint on read amplification
To solve the first problem use leveled compaction if it can satisfy the write amp constraint, else use tiered compaction if it can satisfy the space amp constraint, otherwise there is no solution. The lack of a solution might mean the constraints are unreasonable but it can also mean we need to enhance LSM implementations to support more diversity in LSM tree shapes. Even when there is a solution using leveled or tiered compaction there are solutions that would do much better were an LSM to support more varieties of tiered+leveled and leveled-N.

When I mention solved above I leave out that there is more work to find a solution even when tiered or leveled compaction is used. For both there are decisions about the number of levels and per-level fanout. If minimizing write amp is the goal then that is a solved problem. But there are usually more things to consider.

Tiered+leveled

I defined tiered+leveled and leveled-N in a previous post. They occupy the middle ground between tiered and leveled compaction with better read efficiency than tiered and better write efficiency than leveled. They are not supported today by popular LSM implementations but I think they can and should be supported.

While we tend to explain compaction as a property of an LSM tree (all tiered or all leveled) it is really a property of a level of an LSM tree and RocksDB already supports hybrids, combinations of tiered and leveled. For tiered compaction in RocksDB all levels except the largest use tiered. The largest level is usually configured to use leveled to reduce space amp. For leveled compaction in RocksDB all levels except the smallest use leveled and the smallest (L0) uses tiered.

So tiered+leveled isn't new but I think we need more flexibility. When a string of T and L is created from the per-level compaction choices then the regex for the strings that RocksDB supports is T+L or TL+. I want to support T+L+. I don't want to support cases where leveled is used for a smaller level and tiered for a larger level. So I like TTLL but not LTTL. My reasons for not supporting LTTL are:
1. The benefit from tiered is less write amp and is independent of the level on which it is used. The reduction in write amp is the same whether tiered is used for L1, L2 or L3.
2. The cost from tiered is more read and space amp and that is dependent on the level on which it is used. The cost is larger for larger levels. When space amp is 2 more space is wasted on larger levels than smaller levels. More IO read amp is worse for larger levels because they have a lower hit rate than smaller levels and more IO will be done. More IO implies more CPU cost from decompression and the CPU overhead of performing IO.
From above the benefit from using T is the same for all levels but the cost increases for larger levels so when T and L are both used then T (tiered) should be used on the smaller levels and L (leveled) on the larger levels.

Leveled-N

I defined leveled-N in a previous post. Since then a co-worker, Maysam Yabandeh, explained to me that a level that uses leveled-N can also be described as two levels where the smaller uses leveled and the larger uses tiered. So leveled-N might be syntactic sugar in the LSM tree configuration language.

For example with an LSM defined using the triple syntax from here as (compaction type, fanout, runs-per-level) then this is valid: (T,1,8) (T,8,2) (L,8,2) (L,8,1) and has total fanout of 512 (8 * 8 * 8). The third level (L,8,2) uses leveled-N with N=2. Assuming we allow LSM trees where T follows L then the leveled-N level can be replaced with two levels: (L,8,1) (T,1,8). Then the LSM tree is defined as (T,1,8) (T,8,2) (L,8,1) (T,1,8) (L,8,1). These LSM trees have the same total fanout and total read/write/space amp. Compaction from (L,8,1) to (T,1,8) is special. It has zero write amp because it is done by a file move rather than merging/writing data so all that must be updated is LSM metadata to record the move.

So in general I don't support T after L but I do support it in the special case. Of course we can pretend the special case doesn't exist if we use the syntactic sugar provided by leveled-N. But I appreciate that Maysam discovered this.

## Wednesday, October 3, 2018

### Minimizing write amplification in an LSM

Write-amplification for an LSM with leveled compaction is minimized when the per-level growth factor (fanout) is the same between all levels. This is a result for an LSM tree using a given number of levels. To find the minimal write-amplification for any number of levels this result can be repeated for 2, 3, 4, ... up to a large value. You might find that a large number of levels is needed to get the least write-amp and that comes at price of more read-amp, as the RUM Conjecture predicts.

In all cases below I assume that compaction into the smallest level (from a write buffer flush) has no write-amp. This is done to reduce the size of this blog post.

tl;dr - for an LSM with L1, L2, L3 and L4 what values for per-level fanout minimizes write-amp when the total fanout is 1000?
• (10, 10, 10) for leveled
• (6.3, 12.6, 12.6) for leveled-N assuming two of the levels have 2 sorted runs
• (>1, >1, >1) for tiered

Minimizing write-amp for leveled compaction

For an LSM with 4 levels (L1, L2, L3, L4) there is a per-level fanout between L1:L2, L2:L3 and L3:L4. Assume this uses classic leveled compaction so the total fanout is size(L4) / size(L1). The product of the per-level fanouts must equal the total fanout. The total write-amp is the sum of the per-level write-amp. I assume that the per-level write amp is the same as the per-level fanout although in practice and in theory it isn't that simple. Lets use a, b and c as the variables for the per-level fanout (write-amp) then the math problem is:
1. minimize a+b+c
2. such that a*b*c=k and a, b, c > 1
While I have been working on my math skills this year they aren't great and corrections are welcome. This is a constrained optimization problem that can be solved using Lagrange Multipliers. From above #1 is the sum of per-level write-amp and #2 means that the product of per-level fanout must equal the total fanout. The last constraint is that a, b and c must (or should) all be > 1.

This result uses Lagrange Multipliers for an LSM tree with 4 levels do there are 3 variables: a, b, c. But the math holds for an LSM tree with fewer levels or with more levels. If there are N levels then there are N-1 variables.

L(a, b, c) = a + b + c - lambda * (a*b*c - k)
dL/da = 1 - lambda * bc
dL/db = 1 - lambda * ac
dL/dc = 1 - lambda * ab
then
lambda = 1/bc = 1/ac = 1/ab
bc == ac == ab
and a == b == c to minimize the sum in #1

I wrote a Python script to discover the (almost) best values and the results match the math above.

Minimizing write-amp for tiered compaction

Assuming you can reason about tiered compaction using the notion of levels then the math changes a bit because the per-level write-amp with tiered equals 1 regardless of the per-level fanout. For tiered with 4 levels and 3 variables the problem is:
1. minimize 1+1+1
2. such that a*b*c = k and a, b, c > 1
Any values for a, b and c are sufficient as long they satisfy the constraints in #2. But it still helps to minimize a+b+c if that is predicts read-amp because a, b and c are also the number of sorted runs in L2, L3 and L4. So my advice is to use a == b == c in most cases.

Minimizing write-amp for leveled-N compaction
I explain leveled-N compaction here and here. It is like leveled compaction but allows a level to have more than one sorted run. This reduces the per-level write-amp at the cost of more read-amp. Sometimes that is a good trade.

The math above can also be used to determine how to configure per-level fanout to minimize write-amp for leveled-N. Assume an LSM tree with 4 levels (L1, L2, L3, L4) and 2 sorted runs in L2 and L3. The problem is:
1. minimize a + b/2 + c/2
2. such that a*b*c = k and a, b, c > 1
For leveled compaction I assume that per-level write-amp is all-size(Ln+1) / all-size(Ln) for compaction from Ln into Ln+1. For leveled-N I assume it is run-size(Ln+1) / all-size(Ln) where all-size is the size of all sorted runs on that level and run-size is the size of one sorted run. The astute reader might notice that all-size(Ln) == run-size(Ln) for traditional leveled. For leveled-N I assume that fanout continues to be run-size(Ln+1) / run-size(Ln).

Therefore with leveled-N the per-level write-amp is b/2 for L2 to L3 and c/2 for L3 to L4 because there are 2 sorted runs in the compaction input (twice as much data) in those cases. Were there 3 sorted runs then the values would be b/3 and c/3.

Lagrange Multipliers can be used to solve this assuming we want to minimize a + b/2 + c/2.

L(a, b, c) = a + b/2 + c/2 - lambda * (a*b*c - k)
dL/da = 1   - lambda * bc
dL/db = 1/2 - lambda * ac
dL/dc = 1/2 - lambda * ab
then
lambda = 1/bc = 1/2ac = 1/2ab
bc == 2ac -> b == 2a
bc == 2ab -> c == 2a
2ac == 2ab -> c == b
and 2a == b == c to minimize the sum

If the total fanout is 1000 then the per-level fanout values that minimize write-amp are 10, 10, 10 for leveled and 6.30, 12.60, 12.60 for this example with leveled-N and can be computed by "bc -l"
# for leveled-N
e(l(1000/4)/3)
6.29960524947436582381

e(l(1000/4)/3) * 2
12.59921049894873164762

# and for leveled
e(l(1000)/3)

9.99999999999999999992

One way to think of this result is that with leveled compaction the goal is to use the same per-level fanout between levels. This also uses the same per-level write-amp between levels because per-level write-amp == the per-level fanout for leveled.

But with leveled-N compaction we need to adjust the per-level fanout for levels to continue to get the same per-level write-amp between levels.

## Tuesday, October 2, 2018

### Describing tiered and leveled compaction

This is another attempt by me to define the shape of an LSM tree with more formality and this builds on previous posts here and here. My key point is that compaction is the property of a level in an LSM tree rather than the LSM tree. Some levels can use tiered and others can use leveled. This combination of tiered and leveled is already done in popular LSM implementations but it hasn't been called out as a feature.

Stepped Merge

The Stepped Merge paper might have been the first description of tiered compaction. It is a way to improve B-Tree insert performance. It looked like an LSM tree with a few sorted runs at each level. When a level was full the sorted runs at that level were merged to create a larger sorted run in the next level. The per-level write-amplification was 1 because compaction into level N+1 merged runs from level N but did not read/rewrite a run already on level N+1.

This looks like tiered compaction. However it allows for N sorted runs on the max level which means that space-amplification will be >= N. I assume that is too much for most users of tiered compaction in Cassandra, RocksDB and HBase. But this isn't a problem for Stepped Merge because it is an algorithm for buffering changes to a B-Tree, not for storing the entire database and it doesn't lead to a large space-amp for that workload. Note that the InnoDB change buffer is a B-Tree that buffers changes to other B-Trees for a similar reason.

Compaction per level

I prefer to define compaction as a property of a level in an LSM tree rather than a property of the LSM tree. Unfortunately this can hamper discussion because it takes more time and text to explain compaction per level.

1. When a level is full then compaction is done from it to the next larger level. For now I ignore compaction across many levels, but that is a thing (see "major compaction" in HBase).
2. A sorted run is a sequence of key-value pairs stored in key order. It is stored using 1+ files.
3. A level is tiered when compaction into it doesn't read/rewrite sorted runs already in that level.
4. A level is leveled when compaction into that level reads/rewrites sorted runs already in that level.
5. Levels are full when they have a configurable number of sorted runs. In classic leveled compaction a level has one sorted run. A tiered level is full when it has X sorted runs where X is some value >= 2.
6. leveled-N uses leveled compaction which reads/rewrites an existing sorted run, but it allows N sorted runs (full when runs == N) rather than 1.
7. The per level fanout is size(sorted-run in level N) / size(sorted-run in level N-1)
8. The total fanout is the product of the per level fanouts. When the write buffer is 1G and the database is 1000G then the total fanout must be 1000.
9. The runs-per-level is the number of sorted runs in a level when it is full.
10. The per level write-amplification is the work done to compact from Ln to Ln+1. It is 1 for tiered, all-size(Ln+1) / all-size(Ln) for leveled and run-size(Ln+1) / all-size(Ln) for leveled-N where run-size is the size of a sorted run and all-size is the sum of the sizes of all sorted runs on a level.
A level can be described by a 3-tuple (c, f, r) where c is the type of compaction (T or L for tiered or leveled), f is the fanout and r is the runs-per-level. Unfortunately, now we have made the description of an LSM tree even more complex because there is a 3-tuple per level. For now I don't use 3-tuples to describe the write buffer (memory component). That is a topic for another post. Example 3-tuples include:
• T:1:4 - this is tiered with fanout=1 and runs-per-level=4. It is a common configuration for the RocksDB level 0 (L0) where the fanout is 1 because the compaction input is a write buffer flush so the size of a sorted run in L0 is similar to the size of a full write buffer. For now I ignore that RocksDB can merge write buffers on a flush.
• T:8:8 - this is tiered with fanout=8 and runs-per-level=8. When Ln and Ln+1 both use tiered then runs-per-level in Ln == fanout in Ln+1.
• T:8:4 - this is tiered with fanout=8 and runs-per-level=4. It might be used when the next larger level uses leveled and runs-per-level on this level can be smaller than fanout to reduce read-amp.
• L:10:1 - this is common in RocksDB with leveled compaction, fanout=10 and runs-per-level=1
• L:10:2 - this is leveled-N with runs-per-level=2

Compaction per LSM tree

An LSM tree can be described using the per level 3-tuples from the previous section. The following are examples for popular algorithms.

Classic LSM with total fanout = 1000 is:
• C0 is the write buffer
• C1, C2 and C3 are L:10:1
RocksDB leveled with total fanout = 1000 is:
• L0 is T:1:4
• L1 is L:1:1
• L2, L3, L4 are L:10:1
Stepped Merge with total fanout = 1000 is:
• L1 is T:1:10
• L2, L3, L4 are T:10:10
Tiered in HBase and Cassandra with total fanout = 1000 might be:
• L1 is T:1:10
• L2, L3 are T:10:10
• L4 is L:10:1

Tiered+leveled

Note that some smaller levels using tiered and some larger levels using leveled is done by both RocksDB leveled and Cassandra/HBase tiered. I think both of these are examples of an LSM variant that I call tiered+leveled but I won't ask any of the projects update their docs. My definition of tiered+leveled is the smallest (1 or more) levels use tiered compaction, then 0 or more levels use leveled-N, then the remaining levels use leveled.  When tiered=T, leveled=L and leveled-N=N then the regex for this is T+N*L+.

I don't think it is a good idea for leveled levels to precede tiered levels in tiered+leveled (TTL is OK, LTL is not) but that is a topic for another post.

The largest level should use leveled compaction with runs-per-level=1 to avoid too much space amplification.

LSM trees with tiered+leveled can be described using 3-tuples and the previous section does that but here I provide one for a tree that uses leveled-N for L1 and L2 with total fanout = 1000:
• L0 is T:1:4
• L1 is L:1:2
• L2 is L:10:2
• L3, L4 are L:10:1

And another example to show that tiered levels don't have to use the same fanout or runs-per-level, but runs-per-level for Ln == fanout for Ln+1:
• L0 is T:1:20
• L1 is T:20:10
• L2 is T:10:2
• L3 is L:5:1

Leveled-N

Leveled-N can reduce the per level write-amp at the cost of increasing the per level read-amp.

The regex for an LSM tree that uses leveled-N is N+L+ (see the previous section). The largest level should use leveled compaction with runs-per-level=1 to avoid too much space amplification. An example 3-tuple for leveled-N with fanout=1000 that has runs-per-level=2 for L1 and L2 is:
• L1 is L:10:2
• L2 is L:10:2
• L3 is L:10:1

## Monday, October 1, 2018

### Transaction Processing in NewSQL

This is a list of references for transaction processing in NewSQL systems. The work is exciting. I don't have much to add and wrote this to avoid losing interesting links. My focus is on OLTP, but some of these systems support more than that.

By NewSQL I mean the following. I am not trying to define "NewSQL" for the world:
1. Support for multiple nodes because the storage/compute on one node isn't sufficient.
2. Support for SQL with ACID transactions. If there are shards then cross-shard operations can be consistent and isolated.
3. Replication does not prevent properties listed above when you are wiling to pay the price in commit overhead. Alas synchronous geo-replication is slow and too-slow commit is another form of downtime. I hope NewSQL systems make this less of a problem (async geo-replication for some or all commits, commutative operations). Contention and conflict are common in OLTP and it is important to understand the minimal time between commits to a single row or the max number of commits/second to a single row.
NewSQL Systems
• MySQL Cluster - this was NewSQL before NewSQL was a thing. There is a nice book that explains the internals. There is a company that uses it to make HDFS better. Cluster seems to be more popular for uses other than web-scale workloads.
• VoltDB - another early NewSQL system that is still getting better. It was after MySQL Cluster but years before Spanner and came out of the H-Store research effort.
• Spanner - XA across-shards, Paxos across replicas, special hardware to reduce clock drift between nodes. Sounds amazing, but this is Google so it just works. See the papers that explain the system and support for SQL. This got the NewSQL movement going.
• CockroachDB - the answer to implementing Spanner without GPS and atomic clocks. From that URL they explain it as "while Spanner always waits after writes, CockroachDB sometimes waits before reads". It uses RocksDB and they help make it better.
• FaunaDB - FaunaDB is inspired by Calvin and Daniel Abadi explains the difference between it and Spanner -- here and here. Abadi is great at explaining distributed systems, see his work on PACELC (and the pdf). A key part of Calvin is that "Calvin uses preprocessing to order transactions. All transactions are inserted into a distributed, replicated log before being processed." This approach might limit the peak TPS on a large cluster, but I assume that doesn't matter for a large fraction of the market.
• YugaByte - another user of RocksDB. There is much discussion about it in the recent Abadi post. Their docs are amazing -- slides, transaction IO path, single-shard write IO path, distributed ACID and single-row ACID.
• TiDB - I don't know much about it but they are growing fast and are part of the MySQL community. It uses RocksDB (I shouldn't have forgotten that).
Other relevant systems

## Wednesday, September 19, 2018

### Durability debt

I define durability debt to be the amount of work that can be done to persist changes that have been applied to a database. Dirty pages must be written back for a b-tree. Compaction must be done for an LSM. Durability debt has IO and CPU components. The common IO overhead is from writing something back to the database. The common CPU overhead is from computing a checksum and optionally from compressing data.

From an incremental perspective (pending work per modified row) an LSM usually has less IO and more CPU durability debt than a B-Tree. From an absolute perspective the maximum durability debt can be much larger for an LSM than a B-Tree which is one reason why tuning can be more challenging for an LSM than a B-Tree.

In this post by LSM I mean LSM with leveled compaction.

B-Tree

The maximum durability debt for a B-Tree is limited by the size of the buffer pool. If the buffer pool has N pages then there will be at most N dirty pages to write back. If the buffer pool is 100G then there will be at most 100G to write back. The IO is more random or less random depending on whether the B-Tree is update-in-place, copy-on-write random or copy-on-write sequential. I prefer to describe this as small writes (page at a time) or large writes (many pages grouped into a larger block) rather than random or sequential. InnoDB uses small writes and WiredTiger uses larger writes. The distinction between small writes and large writes is more important with disks than with SSD.

There is a small CPU overhead from computing the per-page checksum prior to write back. There can be a larger CPU overhead from compressing the page. Compression isn't popular with InnoDB but is popular with WiredTiger.

There can be an additional IO overhead when torn-write protection is enabled as provided by the InnoDB double write buffer.

LSM

The durability debt for an LSM is the work required to compact all data into the max level (Lmax). A byte in the write buffer causes more debt than a byte in the L1 because more work is needed to move the byte from the write buffer to Lmax than from L1 to Lmax.

The maximum durability debt for an LSM is limited by the size of the storage device. Users can configure RocksDB such that the level 0 (L0) is huge. Assume that the database needs 1T of storage were it compacted into one sorted run and the write-amplification to move data from the L0 to the max level (Lmax) is 30. Then the maximum durability debt is 30 * sizeof(L0). The L0 is usually configured to be <= 1G in which case the durability debt from the L0 is <= 30G. But were the L0 configured to be <= 1T then the debt from it could grow to 30T.

I use the notion of per-level write-amp to explain durability debt in an LSM. Per-level write-amp is defined in the next section. Per-level write-amp is a proxy for all of the work done by compaction, not just the data to be written. When the per-level write-amp is X then for compaction from Ln to Ln+1 for every key-value pair from Ln there are ~X key-value pairs from Ln+1 for which work is done including:
• Read from Ln+1. If Ln is a small level then the data is likely to be in the LSM block cache or OS page cache. Otherwise it is read from storage. Some reads will be cached, all writes go to storage. So the write rate to storage is > the read rate from storage.
• The key-value pairs are decompressed if the level is compressed for each block not in the LSM block cache.
• The key-value pairs from Ln+1 are merged with Ln. Note that this is a merge, not a merge sort because the inputs are ordered. The number of comparisons might be less than you expect because one iterator is ~X times larger than the other and there are optimizations for that.
The output from the merge is then compressed and written back to Ln+1. Some of the work above (reads, decompression) are also done for Ln but most of the work comes from Ln+1 because it is many times larger than Ln. I stated above that an LSM usually has more IO and less CPU durability debt per modified row. The extra CPU overheads come from decompression and the merge. I am not sure whether to count the compression overhead as extra.

Assuming the per-level growth factor is 10 and f is 0.7 (see below) then the per-level write-amp is 7 for L1 and larger levels. If sizeof(L1) == sizeof(L0) then the per-level write-amp is 2 for the L0. And the per-level write-amp is always 1 for the write buffer.

From this we can estimate the pending write-amp for data at any level in the LSM tree.
1. Key-value pairs in the write buffer have the most pending write-amp. Key-value pairs in the max level (L5 in this case) have none. Key-value pairs in the write buffer are further from the max level.
2. Starting with the L2 there is more durability debt from a full Ln+1 than a full Ln -- while there is more pending write-amp for Ln, there is more data in Ln+1.
3. Were I given the choice of L1, L2, L3 and L4 when first placing a write in the LSM tree then I would choose L4 as that has the least pending write-amp.
4. Were I to choose to make one level 10% larger then I prefer to do that for a smaller level given the values in the rel size X pend column.

legend:
w-amp per-lvl   : per-level write-amp
w-amp pend      : write-amp to move byte to Lmax from this level
rel size        : size of level relative to write buffer
rel size X pend : write-amp to move all data from that level to Lmax

w-amp   w-amp   rel     rel size
level   per-lvl pend    size    X pend
-----   ------- -----   -----   --------
wbuf    1       31          1      31
L0      2       30          4     120
L1      7       28          4     112
L2      7       21         40     840
L3      7       14        400    5600
L4      7       7        4000   28000
L5      0       0       40000       0

Per-level write-amp in an LSM

The per-level write-amplification is the work required to move data between adjacent levels. The per-level write-amp for the write buffer is 1 because a write buffer flush creates a new SST in L0 without reading/re-writing an SST already in L0.

I assume that any key in Ln is already in Ln+1 so that merging Ln into Ln+1 does not make Ln+1 larger. This isn't true in real life, but this is a model.

The per-level write-amp for Ln is approximately sizeof(Ln+1) / sizeof(Ln). For n=0 this is 2 with a typical RocksDB configuration. For n>0 this is the per-level growth factor and the default is 10 in RocksDB. Assume that the per-level growth factor is equal to X, in reality the per-level write-amp is f*X rather than X where f ~= 0.7. See this excellent paper or examine the compaction IO stats from a production RocksDB instance. Too many excellent conference papers assume it is X rather than f*X in practice.

The per-level write-amp for Lmax is 0 because compaction stops at Lmax.

## Tuesday, September 18, 2018

### Bloom filter and cuckoo filter

The multi-level cuckoo filter (MLCF) in SlimDB builds on the cuckoo filter (CF) so I read the cuckoo filter paper. The big deal about the cuckoo filter is that it supports delete and a bloom filter does not. As far as I know the MLCF is updated when sorted runs arrive and depart a level -- so delete is required. A bloom filter in an LSM is per sorted run and delete is not required because the filter is created when the sorted run is written and dropped when the sorted run is unlinked.

I learned of the blocked bloom filter from the cuckoo filter paper (see here or here). RocksDB uses this but I didn't know it had a name. The benefit of it is to reduce the number of cache misses per probe. I was curious about the cost and while the math is complicated, the paper estimates a 10% increase on the false positive rate for a bloom filter with 8 bits/key and a 512-bit block which is similar to a typical setup for RocksDB.

Space Efficiency

I am always interested in things that use less space for filters and block indexes with an LSM so I spent time reading the paper. It is a great paper and I hope that more people read it. The cuckoo filter (CF) paper claims better space-efficiency than a bloom filter and the claim is repeated in the SlimDB paper as:
However, by selecting an appropriate fingerprint size f and bucket size b, it can be shown that the cuckoo filter is more space-efficient than the Bloom filter when the target false positive rate is smaller than 3%
The tl;dr for me is that the space savings from a cuckoo filter is significant when the false positive rate (FPR) is sufficiently small. But when the target FPR is 1% then a cuckoo filter uses about the same amount of space as a bloom filter.

The paper has a lot of interesting math that I was able to follow. It provides formulas for the number of bits/key for a bloom filter, cuckoo filter and semisorted cuckoo filter. The semisorted filter uses 1 less bit/key than a regular cuckoo filter. The formulas assuming E is the target false positive rate, b=4, and A is the load factor:
• bloom filter: ceil(1.44 * log2(1/E))
• cuckoo filter: ceil(log2(1/E) + log2(2b)) / A == (log2(1/E) + 3) / A
• semisorted cuckoo filter: ceil(log2(1/E) + 2) / A

The target load factor is 0.95 (A = 0.95) and that comes at a cost in CPU overhead when creating the CF. Assuming A=0.95 then a bloom filter uses 10 bits/key, a cuckoo filter uses 10.53 and a semisorted cuckoo filter uses 9.47. So the cuckoo filter uses either 5% more or 5% less space than a bloom filter when the target FPR is 1% which is a different perspective from the quote I listed above. Perhaps my math is wrong and I am happy for an astute reader to explain that.

When the target FPR rate is 0.1% then a bloom filter uses 15 bits/key, a cuckoo filter uses 13.7 and a semisorted cuckoo filter uses 12.7. The savings from a cuckoo filter are larger here but the common configuration for a bloom filter in an LSM has been to target a 1% FPR. I won't claim that we have proven that FPR=1% is the best rate and recent research on Monkey has shown that we can do better when allocating space to bloom filters.

The first graph shows the number of bits/key as a function of the FPR for a bloom filter (BF) and cuckoo filter (CF). The second graph shows the ratio for bits/key from BF versus bits/key from CF. The results for semisorted CF, which uses 1 less bit/key, are not included.  For the second graph a CF uses less space than a BF when the value is greater than one. The graph covers FPR from 0.00001 to 0.09 which is 0.001% to 9%. R code to generate the graphs is here.

CPU Efficiency

From the paper there is more detail on CPU efficiency in table 3, figure 5 and figure 7. Table 3 has the speed to create a filter, but the filter is much larger (192MB) than a typical per-run filter with an LSM and there will be more memory system stalls in that case. Regardless the blocked bloom filter has the least CPU overhead during construction.

Figure 5 shows the lookup performance as a function of the hit rate. Fortunately performance doesn't vary much with the hit rate. The cuckoo filter is faster than the blocked bloom filter and the block bloom filter is faster than the semisorted cuckoo filter.

Figure 7 shows the insert performance as a function of the cuckoo filter load factor. The CPU overhead per insert grows significantly when the load factor exceeds 80%.

## Thursday, September 13, 2018

### Review of SlimDB from VLDB 2018

SlimDB is a paper worth reading from VLDB 2018. The highlights from the paper are that it shows:
1. How to use less memory for filters and indexes with an LSM
2. How to reduce the CPU penalty for queries with tiered compaction
3. The benefit of more diversity in LSM tree shapes
Overview

Cache amplification has become more important as database:RAM ratios increase. With SSD it is possible to attach many TB of usable data to a server for OLTP. By usable I mean that the SSD has enough IOPs to access the data. But it isn't possible to grow the amount of RAM per server at that rate. Many of the early RocksDB workloads used database:RAM ratios that were about 10:1 and everything but the max level (Lmax) of the LSM tree was in memory. As the ratio grows that won't be possible unless filters and block indexes use less memory. SlimDB does that via three-level block indexes and multi-level cuckoo-filters.

Tiered compaction uses more CPU and IO for point and range queries because there are more places to check for data when compared to level compaction. The multi-level cuckoo filter in SlimDB reduces the CPU overhead for point queries as there is only one filter to check per level rather than one per sorted run per level.

The SlimDB paper shows the value of hybrid LSM tree shapes, combinations of tiered and leveled, and then how to choose the best combination based on IO costs. Prior to this year, hybrid didn't get much discussion -- the choices were usually tiered or leveled. While RocksDB and LevelDB with the L0 have always been hybrids of tiered (L0) and leveled (L1 to Lmax), we rarely discuss that. But more diversity in LSM tree shape means more complexity in tuning and the SlimDB solution is to make a cost-based decision (cost == IO overhead) subject to a constraint on the amount of memory to use.

This has been a great two years for storage engine efficiency. First we had several papers from Harvard DASLab that have begun to explain cost-based algorithm design and engine configuration and SlimDB continues in that tradition. I have much more reading to do starting with The Periodic Table of Data Structures.

Below I review the paper. Included with that is some criticism. Papers can be great without being perfect. This paper is a major contribution and worth reading.

Semi-sorted

The paper starts by explaining the principle of semi-sorted data. When the primary key can be split into two parts -- prefix and suffix -- there are some workloads that don't need data ordered over the entire primary key (prefix + suffix). Semi-sorted supports queries that fetch all data that matches the prefix of the PK while still enforcing uniqueness for the entire PK. The PK can be on (a,b,c,d) and (a,b) is prefix and queries are like "a=X and b=Y" without predicates on (c,d) that require index ordering. SlimDB takes advantage of this to use less space for the block index.

There are many use cases for this, but the paper cites Linkbench which isn't correct. See the Linkbench and Tao papers for queries that do an exact match on the prefix but only want the top-N rows in the result. So ordering on the suffix is required to satisfy query response time goals when the total number of rows that match the prefix is much larger than N. I assume this issue with top-N is important for other social graph workloads because some graph nodes are popular. Alas, things have changed with the social graph workload since those papers were published and I hope the changes are explained one day.

Note that MyRocks can use a prefix bloom filter to support some range queries with composite indexes. Assume the index is on (a,b,c) and the query has a=X and b=Y order by c limit 10. A prefix bloom on (a,b) can be used for such a query.

Stepped Merge

The paper implements tiered compaction but calls it stepped merge. I didn't know about the stepped merge paper prior to reading the SlimDB paper. I assume that people who chose the name tiered might also have missed that paper.

LSM compaction algorithms haven't been formally defined. I tried to advance the definitions in a previous post. One of the open issues for tiered is whether it requires only one sorted run at the max level or allows for N runs at the max level. With N runs at the max level the space-amplification is at least N which is too much for many workloads. With 1 run at the max level compaction into the max level is always leveled rather than tiered -- the max level is read/rewritten and the per-level write-amplification from that is larger than 1 (while the per-level write-amp from tiered == 1). With N runs at the max level many of the compaction steps into the max level can be tiered, but some will be leveled -- when the max level is full (has N runs) then something must be done to reduce the number of runs.

3-level block index

Read the paper. It is complex and a summary by me here won't add value. It uses an Entropy Coded Trie (ECT) that builds on ideas from SILT -- another great paper from CMU.

ECT uses ~2 bits/key versus at least 8 bits/key for LevelDB for the workloads they considered. This is a great result. ECT also uses 5X to 7X more CPU per lookup than LevelDB which means you might limit the use of it to the largest levels of the LSM tree -- because those use the most memory and the place where we are willing to spend CPU to save memory.

Multi-level cuckoo filter

SlimDB can use a cuckoo filter for leveled levels of the LSM tree and a multi-level cuckoo filter for tiered levels. Note that leveled levels have one sorted run and tiered levels have N sorted runs. SlimDB and the Stepped Merge paper use the term sub-levels, but I prefer N sorted runs.

The cuckoo filter is used in place of a bloom filter to save space given target false positive rates of less than 3%. The paper has examples where the cuckoo filter uses 13 bits/key (see Table 1) and a bloom filter with 10 bits/key (RocksDB default) has a false positive rate of much less than 3%. It is obvious that I need to read another interesting CMU paper cited by SlimDB -- Cuckoo Filter Practically Better than Bloom.

The multi-level cuckoo filter (MLCF) extends the cuckoo filter by using a few bits/entry to name the sub-level (sorted run) in the level that might contain the search key. With tiered and a bloom filter per sub-level (sorted run) a point query must search a bloom filter per sorted run. With the MLCF there is only one search per level (if I read the paper correctly).

The MLCF might go a long way to reduce the point-query CPU overhead when using many sub-levels which is a big deal. While a filter can't be used for general range queries, SlimDB doesn't support general range queries. Assuming the PK is on (a,b,c,d) and the prefix is (a,b) then SlimDB supports range queries like fetch all rows where a=X and b=Y. It wasn't clear to me whether the MLCF could be used in that case. But many sub-levels can create more work for range queries as iterators must be positioned in each sub-level in the worst case and that is more work.

This statement from the end of the paper is tricky. SlimDB allows for an LSM tree to use leveled compaction on all levels, tiered on all levels or a hybrid. When all levels are leveled, then performance should be similar to RocksDB with leveled, when all or some levels are tiered then write-amplification will be reduced at the cost of read performance and the paper shows that range queries are slower when some levels are tiered. Lunch isn't free as the RUM Conjecture asserts.
In contrast, with the support of dynamic use of a stepped merge algorithm and optimized in-memory indexes, SlimDB minimizes write amplification without sacrificing read performance.
The memory overhead for MLCF is ~2 bits. I am not sure this was explained by the paper but that might be to name the sub-level, in which case there can be at most 4 sub-levels per level and the cost would be larger with more sub-levels.

The paper didn't explain how the MLCF is maintained. With a bloom filter per sorted run the bloom filter is created when SST files are created during compaction and memtable flush. This is an offline or batch computation. But the MLCF covers all the sub-levels (sorted runs) in a level. And the sub-levels in a level arrive and depart one at a time, not at the same time. They  arrive as output from compaction and depart when they were compaction input. The arrival or departure of a new sub-level requires incremental changes to the MLCF.

LSM tree shapes

For too long there has not been much diversity in LSM tree shapes. The usual choice was all tiered or all leveled. RocksDB leveled is really a hybrid -- tiered for L0, leveled for L1 to Lmax. But the SlimDB paper makes the case for more diversity. It explains that some levels (smaller ones) can be tiered while the larger levels can be leveled. And the use of multi-level cuckoo filters, three-level indexes and cuckoo filters is also a decision to make per-level.

Even more interesting is the use of a cost-model to choose the best configuration subject to a constraint -- the memory budget. They enumerate a large number of LSM tree configurations, generate estimated IO-costs per operation (write-amp, IO per point query that returns a row, IO per point query that doesn't return a row, memory overhead) and then the total IO cost is computed for for a workload -- where a workload specifies the frequency of each operation (for example - 30% writes, 40% point hits, 30% point misses).

The Dostoevsky paper also makes the case for more diversity and uses rigorous models to show how to choose the best LSM tree shape.

I think this work is a big step in the right direction. Although cost models must be expanded to include CPU overheads and constraints expanded to include the maximum write and space amplification that can be tolerated.

I disagree with a statement from the related work section. We can already navigate some of the read, write and space amplification space but I hope there is more flexibility in the future. RocksDB tuning is complex in part to support this via changing the number of levels (or growth factor per level), enabling/disabling the bloom filter, using different compression (or none) on different levels, changing the max space amplification allowed, changing the max number of sorted runs in the L0 or max number of write buffers, changing the L0:L1 size ratio, changing the number of bloom filter bits/key. Of course I want more flexibility in the future while also making RocksDB easier to tune.

Existing LSM-tree based key-value stores do not allow trading among read cost, write cost and main memory footprint.

Performance Results

Figuring out why X was faster than Y in academic papers is not my favorite task. I realize that space constraints are a common reason for the lack of details but I am wary of results that have not been explained and I know that mistakes can be made (note: don't use serializable with InnoDB). I make many mistakes myself. I am willing to provide advice for MyRocks, MySQL and RocksDB. AFAIK most authors who hack on RocksDB or compare with it for research are not reaching out to us. We are happy to help in private.

SlimDB was faster than RocksDB on their evaluation except for range queries. There were few details about the configurations used, so I will guess. First I assume that SlimDB used stepped merge with MLCF for most levels. I am not sure why point queries were faster with SlimDB than RocksDB. Maybe RocksDB wasn't configured to use bloom filters. Writes were about 4X faster with SlimDB because stepped merge (tiered) compaction was used, write-amplification was 4X less and when IO is the bottleneck then an approach that has less write-amp will go faster.

## Wednesday, September 5, 2018

### 5 things to set when configuring RocksDB and MyRock

The 5 options to set for RocksDB and MyRocks are:
1. block cache size
3. compaction priority
4. dynamic leveled compaction
5. bloom filters
I have always wanted to do a "10 things" posts but prefer to keep this list small. It is unlikely that RocksDB can provide a great default for the block cache size and number of background threads because they depend on the amount of RAM and number of CPU cores in a server. But I hope RocksDB or MyRocks are changed to get better defaults for the other three which would shrink this list from 5 to 2.

Options

My advice on setting the size of the RocksDB block cache has not changed assuming it is configured to use buffered IO (the default). With MyRocks this option is rocksdb_block_cache_size and with RocksDB you will write a few lines of code to setup the LRU.

The number of background threads for flushing memtables and doing compaction is set by the option rocksdb_max_background_jobs in MyRocks and max_background_jobs in RocksDB. There used to be two options for this. While RocksDB can use async read-ahead and write-behind during compaction, it still uses synchronous reads and a potentially slow fsync/fdatasync call. Using more than 1 background job helps to overlap CPU and IO. A common configuration for me is number-of-CPU-cores / 4. With too few threads there will be more stalls from throttling. With too many threads there the threads handling user queries might suffer.

There are several strategies for choosing the next data to compact with leveled compaction in RocksDB. The strategy is selected via the compaction_pri option in RocksDB. This is harder to set for MyRocks -- see compaction_pri in rocksdb_default_cf_options. The default value is kByCompensatedSize but the better choice is kMinOverlappingRatio. With MyRocks the default is 0 and the better value is 3 (3 == kMinOverlappingRatio). I first wrote about compaction_pri prior to the arrival of kMinOverlappingRatio. Throughput is better and write amplification is reduced with kMinOverlappingRatio. An awesome paper by Hyeontaek Lim et al explains this.

Leveled compaction in RocksDB limits the amount of data per level of the LSM tree. A great description of this is here. There is a target size per level and this is enforced top down (smaller to larger levels) or bottom up (larger to smaller levels). With the bottom up approach the largest level has ~10X (or whatever the fanout is set to) more data than the next to last level. With the top down approach the largest level frequently has less data than the next to last level. I strongly prefer the bottom up approach to reduce space amplification. This is enabled via the level_compaction_dynamic_level_bytes option in RocksDB. It is harder to set for MyRocks -- see rocksdb_default_cf_options.

Bloom filters are disabled by default for MyRocks and RocksDB. I prefer to use a bloom filer on all but the largest level. This is set via rocksdb_default_cf_options with MyRocks. The reason for not using it with the max level is to consume less memory (reduce cache amplification). The bloom filter is skipped for the largest level in MyRocks via the optimize_filter_for_hits option. The example at the end of this post has more information on enabling bloom filters. All of this is set via rocksdb_default_cf_options.

Examples

A previous post recently explained how to set rocksdb_default_cf_options for compression with MyRocks. Below I share an example my.cnf for MyRocks to set the 5 options I listed above. I set transaction isolation because read committed is a better choice for MyRocks today. Repatable read will be a great choice after gap locks are added to match InnoDB semantics. In rocksdb_default_cf_options block_based_table_factory is used to enable the bloom filter, level_compaction_dynamic_level_bytes enables bottom up management of level sizes, optimize_filters_for_hits disables the bloom filter for the largest level of the LSM tree and compaction_pri sets the compaction priority.

default-storage-engine=rocksdb
rocksdb

rocksdb_default_cf_options=block_based_table_factory={filter_policy=bloomfilter:10:false};level_compaction_dynamic_level_bytes=true;optimize_filters_for_hits=true;compaction_pri=kMinOverlappingRatio
rocksdb_block_cache_size=2g
rocksdb_max_background_jobs=4

## Thursday, August 30, 2018

### Name that compaction algorithm

First there was leveled compaction and it was a great paper. Then tiered compaction arrived in BigTable, HBase and Cassandra. Eventually LevelDB arrived with leveled compaction and RocksDB emerged from that. Along the way a few interesting optimizations have been added including support for time series data. My summary is missing a few details because it is a summary.

Compaction algorithms constrain the LSM tree shape. They determine which sorted runs can be merged by it and which sorted runs need to be accessed for a read operation. I am not sure whether they have been formally defined, but I hope there can be agreement on the basics. I will try to do that now for a few - leveled, tiered, tiered+leveled, leveled-N and time-series. There are two new names on this list -- tiered+leveled and leveled-N.

LSM tree used to imply leveled compaction. I prefer to expand the LSM tree definition to include leveled, tiered and more.

I reference several papers below. All of them are awesome, even when not perfect -- they are major contributions to write-optimized databases and worth reading. One of the best things about my job is getting time to read papers like this and then speak with the authors.

There are many interesting details in academic papers and existing systems (RocksDB, Cassandra, HBase, ScyllaDB) that I ignore. I don't want to get lost in the details.

Leveled

Leveled compaction minimizes space amplification at the cost of read and write amplification.

The LSM tree is a sequence of levels. Each level is one sorted run that can be range partitioned into many files. Each level is many times larger than the previous level. The size ratio of adjacent levels is sometimes called the fanout and write amplification is minimized when the same fanout is used between all levels. Compaction into level N (Ln) merges data from Ln-1 into Ln. Compaction into Ln rewrites data that was previously merged into Ln. The per-level write amplification is equal to the fanout in the worst case, but it tends to be less than the fanout in practice as explained in this paper by Hyeontaek Lim et al. Compaction in the original LSM paper was all-to-all -- all data from Ln-1 is merged with all data from Ln. It is some-to-some for LevelDB and RocksDB -- some data from Ln-1 is merged with some (the overlapping) data in Ln.

While write amplification is usually worse with leveled than with tiered there are a few cases where leveled is competitive. The first is key-order inserts and a RocksDB optimization greatly reduces write-amp in that case. The second one is skewed writes where only a small fraction of the keys are likely to be updated. With the right value for compaction priority in RocksDB compaction should stop at the smallest level that is large enough to capture the write working set -- it won't go all the way to the max level. When leveled compaction is some-to-some then compaction is only done for the slices of the LSM tree that overlap the written keys, which can generate less write amplification than all-to-all compaction.

Tiered

Tiered compaction minimizes write amplification at the cost of read and space amplification.

The LSM tree can still be viewed as a sequence of levels as explained in the Dostoevsky paper by Niv Dayan and Stratos Idreos. Each level has N sorted runs. Each sorted run in Ln is ~N times larger than a sorted run in Ln-1. Compaction merges all sorted runs in one level to create a new sorted run in the next level. N in this case is similar to fanout for leveled compaction. Compaction does not read/rewrite sorted runs in Ln when merging into Ln. The per-level write amplification is 1 which is much less than for leveled where it was fanout.

Most implementations of tiered compaction don't behave exactly as described in the previous paragraph. I hope they are close enough, because the model above makes it easy to reason about performance and estimate the worst-case write amplification. A common approach for tiered is to merge sorted runs of similar size, without having the notion of levels (which imply a target for the number of sorted runs of specific sizes). Most include some notion of major compaction that includes the largest sorted run and conditions that trigger major and non-major compaction. Too many files and too many bytes are typical conditions.

The stepped merge paper is the earliest reference I found for tiered compaction. It reduces random IO for b-tree changes by buffering them in an LSM tree that uses tiered compaction. While the stepped merge algorithm is presented as different from an LSM, it is tiered compaction. The MaSM paper is similar but the SM in MaSM stands for sort merge. The paper uses an external sort rather than an LSM to reduce write amplification. It assumes that LSM implies leveled compaction but an external sort looks a lot like tiered compaction. The InnoDB change buffer has a similar goal of reducing random IO for changes to a b-tree but doesn't use an LSM. In what year did the InnoDB change buffer get designed or implemented?

I prefer that tiered not require N sorted runs at the max level because that means N copies of the database which is too much space amplification. I define it to allow K copies at the max level where K is between 2 and N. But it still does tiered compaction at the max level and when the max level is full (has K sorted runs) then the K runs are merged and the output (1 sorted run) replaces the K runs in the max level. One day I hope to learn whether HBase or Cassandra support 1, a few or N sorted runs at the max level -- although this can be confusing because they don't enforce the notion of levels. Tiered compaction in RocksDB has a configuration option to limit the worst-case space amplification which should prevent too many full copies (too many sorted runs at the max level) but I don't have much experience with tiered in RocksDB. I hope the RocksDB wiki gets updated to explain this.

There are a few challenges with tiered compaction:
• Transient space amplification is large when compaction includes a sorted run from the max level.
• The block index and bloom filter for large sorted runs will be large. Splitting them into smaller parts is a good idea.
• Compaction for large sorted runs takes a long time. Multi-threading would help.
• Compaction is all-to-all. When there is skew and most of the keys don't get updates, large sorted runs might get rewritten because compaction is all-to-all. In a traditional tiered algorithm there is no way to rewrite a subset of a large sorted run.
For tiered compaction the notion of levels are usually a concept to reason about the shape of the LSM tree and estimate write amplification. With RocksDB they are also an implementation detail. The levels of the LSM tree beyond L0 can be used to store the larger sorted runs. The benefit from this is to partition large sorted runs into smaller SSTs. This reduces the size of the largest bloom filter and block index chunks -- which is friendlier to the block cache -- and was a big deal before partitioned index/filter was supported. With subcompactions this enables multi-threaded compaction of the largest sorted runs. Note that RocksDB used the name universal rather than tiered. More docs on this are here.

Tiered+Leveled

Tiered+Leveled has less write amplification than leveled and less space amplification than tiered.

The tiered+leveled approach is a hybrid that uses tiered for the smaller levels and leveled for the larger levels. It is flexible about the level at which the LSM tree switches from tiered to leveled. For now I assume that if Ln is leveled then all levels that follow (Ln+1, Ln+2, ...) must be leveled.

SlimDB from VLDB 2018 is an example of tiered+leveled although it might allow Lk to be tiered when Ln is leveled for k > n. Fluid LSM is described as tiered+leveled but I think it is leveled-N.

Leveled compaction in RocksDB is also tiered+leveled, but we didn't explain it that way until now. There can be N sorted runs at the memtable level courtesy of the max_write_buffer_number option -- only one is active for writes, the rest are read-only waiting to be flushed. A memtable flush is similar to tiered compaction -- the memtable output creates a new sorted run in L0 and doesn't read/rewrite existing sorted runs in L0. There can be N sorted runs in level 0 (L0) courtesy of level0_file_num_compaction_trigger. So the L0 is tiered. Compaction isn't done into the memtable level so it doesn't have to be labeled as tiered or leveled. Subcompactions in the RocksDB L0 makes this even more interesting, but that is a topic for another post. I hope we get more docs on this interesting feature from Andrew Kryczka.

Leveled-N

Leveled-N compaction is like leveled compaction but with less write and more read amplification. It allows more than one sorted run per level. Compaction merges all sorted runs from Ln-1 into one sorted run from Ln, which is leveled. And then "-N" is added to the name to indicate there can be n sorted runs per level.

The Dostoevsky paper defined a compaction algorithm named Fluid LSM in which the max level has 1 sorted run but the non-max levels can have more than 1 sorted run. Leveled compaction is done into the max level. The paper states that tiered compaction is done into the smaller levels when they have more than 1 sorted run. But from my reading of the paper it uses leveled-N for the non-max levels.

In Fluid LSM each level is T times larger than the previous level (T == fanout), the max level has Z sorted runs and the non-max levels have K sorted runs. When Z=1 and K=1 then this is leveled compaction. When Z=1 and K>1 or Z>1 and K>1 then I claim this uses leveled-N.

Assuming K>1 for Ln-1 then compaction with Fluid LSM into Ln merges K runs from Ln-1 with 1 run from Ln. This doesn't match my definition of tiered compaction because compaction into Ln reads & rewrites a sorted run from Ln and per-level write amplification is likely to be larger than 1. Regardless I like the idea.

Examples of write amplification with Fluid LSM for compaction from Ln-1 to Ln:
• T==K - there are T (or K) sorted runs in each of Ln-1 and Ln. When each run in Ln-1 has size 1, then each run in Ln has size T. Compaction into Ln merges T runs from Ln-1 with 1 run from Ln to create a new run in Ln. This reads T bytes from Ln-1 and T bytes from Ln and the new run has a size between T and 2T -- size T when all keys in Ln-1 are duplicates of keys in the run from Ln and size > T otherwise. When the new run has size 2T the per-level write amp is 2 because 2T bytes were written to move T bytes from Ln-1. When the new run has size T the per-level write amp is 1. Otherwise the per-level write-amp is between 1 and 2.
• T > K - there are K sorted runs in each of Ln-1 and Ln. Each run in Ln-1 has size T/K and each run in Ln has size T^2/K. K runs in Ln-1 have size T. Compaction reads T bytes from Ln-1, T^2/K bytes from Ln and writes a new run in Ln that has a size between T^2/K and (T^2/K + T). The per-level write-amp is as small as T^2/K / T, which reduces to T/K, when all keys in Ln-1 are duplicates with the run in Ln. It can be as large as (T^2/K + T) / T, which reduces to T/K + 1, when there is no overlap. Otherwise it is between T/K and T/K + 1.
When K=2 and T=10 then the per-level write-amp is ~5 which is about half of the per-level write-amp from leveled compaction.

Time Series

There are compaction algorithms optimized for time series workloads. I have no experience with them but they are worth mentioning. Cassandra had DTCS and has TWCS. InfluxDB has or had TSM and TSI. I hope we eventually do something interesting for time series with RocksDB.

Other

There are other interesting LSM engines:
• Tarantool - Sphia begat Vinyl and I lost track of it. But I have high hopes.
• WiredTiger - has an LSM but they are busy making the CoW b-tree better
• Kudu - didn't use RocksDB and I like the reasons for not using it
My summary of Sphia and Tarantool probably has bugs. My memory is that Sophia was a great design assuming the database : RAM ratio wasn't too large. It had a memtable and a sorted run on disk -- both were partitioned (not sure if range or hash). When a memtable partition became full then leveled compaction was done between it and its disk partition. Vinyl has changed enough from this design that I won't try to summarize it here. It has clever ideas for managing the partitions.