From e595ef179f706ee5d22736307dc58b92d376f575 Mon Sep 17 00:00:00 2001 From: Yiding Cui Date: Tue, 4 Sep 2018 17:19:00 +0800 Subject: [PATCH 01/11] proposal: add proposal for maintaining histograms in plan. --- docs/design/2018-09-04-histograms-in-plan.md | 115 +++++++++++++++++++ 1 file changed, 115 insertions(+) create mode 100644 docs/design/2018-09-04-histograms-in-plan.md diff --git a/docs/design/2018-09-04-histograms-in-plan.md b/docs/design/2018-09-04-histograms-in-plan.md new file mode 100644 index 0000000000000..331d9d7fb809a --- /dev/null +++ b/docs/design/2018-09-04-histograms-in-plan.md @@ -0,0 +1,115 @@ +# Proposal: Maintain statistics in `Plan` + +- Author: Yiding CUI +- Last updated: 2018-09-04 + +## Abstract + +This proposal proposes to maintain the histogram in `Plan`’s statistics information. + + +## Background + +Currently, TiDB only uses statistics when deciding which physical scan method a table should use. And TiDB only stores simple statistics in the plan structure. But when deciding the join order and considering some other optimization rules, we need more detailed statistics. +So we need to maintain the statistics in the plan structure to get sufficient statistics information to do optimizations. + +## Proposal + +The new `statsInfo` of `plan` should be something like the following structure: + +``` +type statsInfo struct { + // This two attributes are already used in current statsInfo. + count float64 + // The number of distinct value of each column. + ndv []float64 + + // HistColl collects histograms of columns. + histColl statistics.HistColl + // rangesOfXXX This won’t be maintained in the first version. + rangesOfColumn map[int][]*ranger.Range + rangesOfIndex map[int][]*ranger.Range + // max/minValues is a simplified version of rangesOfXXX. Since maintaining rangesOfXXXX consumes a lot of time and memory, we can first check whether this can meet our needs. + maxValues map[int][]types.Datum + minValues map[int][]types.Datum +} +``` + +This struct will be maintained when we call `deriveStats`. + +Currently we don't change the histogram itself during planning. Because it will consume a lot of time and memory space. I’ll try to maintain ranges slice or the max/min value to improve the accuracy of row count estimation instead. + +Before we maintain `rangesOfXXX` or `max and min values`: + +We maintain the histogram in `Projection`, `Selection`, `Join`, `Aggregation`, `Sort`, `Limit` and `DataSource` operators. + +For `Sort`, we can just copy children's `statsInfo` without doing any change. + +For `Limit`, we can just copy children's `statsInfo` or ignore the histogram information. As you know, its execution logic is based on randomization. Hard to maintain the statistics information after it. But we may use the information before it to do some estimation in some scenarios. + +For `Projection`, change the reflection of the map we maintained. + +For `Aggregation`, use the histogram to estimate the ndv of group-by items. If one index cannot cover the group-by item, we’ll multiply the ndv of each group-by column. If the output of `Aggregation` includes group-by columns, we’ll maintain the histogram of them for future use. + +For `Join`, there’re joins as follows: + +- Inner join: use histograms to do the row count estimation with the join key condition. Since it won’t have one side filter, we only need to consider the composite filters after considering the join key. We can simply multiply `selectionFactor` if there are other composite filters in our first version of implementation. Since `Selectivity` cannot calculate selectivity of expression that containing multiple column. + +- One side outer join: It depends on the join keys’ NDV. And we can just use histograms to estimate it if there’re non-join-key filters. + +- Semi join: It’s something similar to inner join. But no data expanding occurs. When we maintain the range information. We can get a nearly accurate answer of its row count. + +- Anti semi join: Same with semi join. + +For `Selection`, just use it to calculate the selectivity. + +For `DataSource`, if it’s a non-partitioned table, we just maintain the map. If it’s a partitioned table, we now only store the statistics of each partition So we need to merge them. We’ll need a cache or something else to ensure that we won’t merge them each time we need it, which will consume tooooo much time and memory space. + +For other plan operators or `DataSource` which do not contain any histogram, we just use the original way to maintain `statsInfo`. We won’t maintain histograms for them. Taking `Union` for an example, if we maintain histogram for it, we need to merge the histograms which is really an expensive operation. But we can consider maintaining this if a hint is given. + +When we maintain `rangesOfXXX` or `max and min values`: + +Most things are the same with above. But there will be something more to do. + +For `Selection` and `Join`, we need to cut off the things which are not in ranges when doing estimation. And update the ranges information after the estimation. Also the NDV of column can be estimated more accurately. + +For `Aggregation`, we only need to cut off the things which are not in ranges when doing estimation. There is no need to update the ranges information. + +For `TopN`, we now have alibity to maintain histograms of the order-by items. + + +## Rationale + +### How other systems solve the same issue? + +I’ve looked into Spark. They did nearly the same thing with what I said. They only maintain the max and min values, no `ranges` information. And they don’t have index, so they only maintain the column’s max/min value which make problem far more much easier. + +As for Orca and Calcite, I haven’t discovered where they maintain this information. But there’s something about statistics in Orca’s paper. According to the paper, I think they construct new histogram during planning and cache it for not building to often. + +### What is the disadvantage of this design? + +This may have side effects on OLTP performance. But I’ll try to reduce it. + +For now, only join reorder and the position `after logical optimize before physical optimize` will trigger this mechanism and it’s controlled by `tidb_optimizer_selectivity_level` which is disabled by default. This may not have much more side effects on simple point queries. + +And the `expectedCount` we used in physical plan is something same with `Limit`. So the row count modification during physical plan won’t be affected. + +After we switch to the cascade like planner. The rule that needs cost to make decision is still a small set of all. And the existence of `Group` can also help us. If we lazily construct the `statsInfo`, this may not be the bottleneck. + +### What is the impact of not doing this? + +Many cases reported by our customer already prove that we need more accurate statistics to choose a better join order and a proper join algorithm. Only maintaining a number about row count and a slice about ndv is not enough for making that decision. + +If we don’t do this, we cannot fix some cases except we use hints. + +## Compatibility + +There’s no compatibility issue. + +## Implementation + +First maintain the histogram in `DataSource`. In this step, there will be some changes in the `statistics` package to make it work. It may take a little long time to do this. [PR#7385](https://github.com/pingcap/tidb/pull/7385) + +Then, all changes will be things inside `statsInfo`, which won’t affect other parts. It may not take too much time to implement this part. Like [PR#7595](https://github.com/pingcap/tidb/pull/7595) + +The future work will depend on the benchmark result of the join reorder algorithm. From b55c9244fc9ceff0b5aeb3f50394c77f8ed3c6ee Mon Sep 17 00:00:00 2001 From: Yiding Cui Date: Wed, 5 Sep 2018 15:48:04 +0800 Subject: [PATCH 02/11] address comments. --- docs/design/2018-09-04-histograms-in-plan.md | 28 ++++++++++---------- 1 file changed, 14 insertions(+), 14 deletions(-) diff --git a/docs/design/2018-09-04-histograms-in-plan.md b/docs/design/2018-09-04-histograms-in-plan.md index 331d9d7fb809a..4c3ffd27b69c7 100644 --- a/docs/design/2018-09-04-histograms-in-plan.md +++ b/docs/design/2018-09-04-histograms-in-plan.md @@ -11,7 +11,7 @@ This proposal proposes to maintain the histogram in `Plan`’s statistics inform ## Background Currently, TiDB only uses statistics when deciding which physical scan method a table should use. And TiDB only stores simple statistics in the plan structure. But when deciding the join order and considering some other optimization rules, we need more detailed statistics. -So we need to maintain the statistics in the plan structure to get sufficient statistics information to do optimizations. +So we need to maintain the statistics in the plan structure to get sufficient statistics information to do optimizations. ## Proposal @@ -45,46 +45,46 @@ We maintain the histogram in `Projection`, `Selection`, `Join`, `Aggregation`, ` For `Sort`, we can just copy children's `statsInfo` without doing any change. -For `Limit`, we can just copy children's `statsInfo` or ignore the histogram information. As you know, its execution logic is based on randomization. Hard to maintain the statistics information after it. But we may use the information before it to do some estimation in some scenarios. +For `Limit`, we can just copy children's `statsInfo` or ignore the histogram information. As you know, its execution logic is based on randomization. It's hard to maintain the statistics information after `Limit`. But we may use the information before it to do some estimation in some scenarios. For `Projection`, change the reflection of the map we maintained. -For `Aggregation`, use the histogram to estimate the ndv of group-by items. If one index cannot cover the group-by item, we’ll multiply the ndv of each group-by column. If the output of `Aggregation` includes group-by columns, we’ll maintain the histogram of them for future use. +For `Aggregation`, use the histogram to estimate the NDV of group-by items. If one index cannot cover the group-by item, we’ll multiply the NDV of each group-by column. If the output of `Aggregation` includes group-by columns, we’ll maintain the histogram of them for future use. For `Join`, there’re joins as follows: -- Inner join: use histograms to do the row count estimation with the join key condition. Since it won’t have one side filter, we only need to consider the composite filters after considering the join key. We can simply multiply `selectionFactor` if there are other composite filters in our first version of implementation. Since `Selectivity` cannot calculate selectivity of expression that containing multiple column. +- Inner join: use histograms to do the row count estimation with the join key condition. Since it won’t have one side filter, we only need to consider the composite filters after considering the join key. We can simply multiply `selectionFactor` if there are other composite filters in our first version of implementation. Since `Selectivity` cannot calculate selectivity of an expression containing multiple columns. - One side outer join: It depends on the join keys’ NDV. And we can just use histograms to estimate it if there’re non-join-key filters. -- Semi join: It’s something similar to inner join. But no data expanding occurs. When we maintain the range information. We can get a nearly accurate answer of its row count. +- Semi join: It’s something similar to inner join. But no data expanding occurs. When we maintain the range information, we can get a nearly accurate answer of its row count. - Anti semi join: Same with semi join. For `Selection`, just use it to calculate the selectivity. -For `DataSource`, if it’s a non-partitioned table, we just maintain the map. If it’s a partitioned table, we now only store the statistics of each partition So we need to merge them. We’ll need a cache or something else to ensure that we won’t merge them each time we need it, which will consume tooooo much time and memory space. +For `DataSource`, if it’s a non-partitioned table, we just maintain the map. If it’s a partitioned table, we now only store the statistics of each partition. So we need to merge them. We need a cache or something else to ensure that we won’t merge them each time we need it, which will consume tooooo much time and memory space. -For other plan operators or `DataSource` which do not contain any histogram, we just use the original way to maintain `statsInfo`. We won’t maintain histograms for them. Taking `Union` for an example, if we maintain histogram for it, we need to merge the histograms which is really an expensive operation. But we can consider maintaining this if a hint is given. +For other plan operators or `DataSource` which do not contain any histogram, we just use the original way to maintain `statsInfo`. We won’t maintain histograms for them. Taking `Union` for an example, if we maintain histograms for it, we need to merge the histograms, which is really an expensive operation. But we can consider maintaining this if a hint is given. When we maintain `rangesOfXXX` or `max and min values`: Most things are the same with above. But there will be something more to do. -For `Selection` and `Join`, we need to cut off the things which are not in ranges when doing estimation. And update the ranges information after the estimation. Also the NDV of column can be estimated more accurately. +For `Selection` and `Join`, we need to cut off the things which are not in ranges when doing estimation, and update the ranges information after the estimation. Also the NDV of column can be estimated more accurately. For `Aggregation`, we only need to cut off the things which are not in ranges when doing estimation. There is no need to update the ranges information. -For `TopN`, we now have alibity to maintain histograms of the order-by items. +For `TopN`, we now have the alibity to maintain histograms of the order-by items. ## Rationale ### How other systems solve the same issue? -I’ve looked into Spark. They did nearly the same thing with what I said. They only maintain the max and min values, no `ranges` information. And they don’t have index, so they only maintain the column’s max/min value which make problem far more much easier. +I’ve looked into Spark. They did nearly the same thing with what I said. They only maintain the max and min values, rather than the `ranges` information. And they don’t have the index, so they only maintain the column’s max/min value which make problem much easier to solve. -As for Orca and Calcite, I haven’t discovered where they maintain this information. But there’s something about statistics in Orca’s paper. According to the paper, I think they construct new histogram during planning and cache it for not building to often. +As for Orca and Calcite, I haven’t discovered where they maintain this information. But there’s something about statistics in Orca’s paper. According to the paper, I think they construct new histogram during planning and cache it to avoid building too many times. ### What is the disadvantage of this design? @@ -94,11 +94,11 @@ For now, only join reorder and the position `after logical optimize before physi And the `expectedCount` we used in physical plan is something same with `Limit`. So the row count modification during physical plan won’t be affected. -After we switch to the cascade like planner. The rule that needs cost to make decision is still a small set of all. And the existence of `Group` can also help us. If we lazily construct the `statsInfo`, this may not be the bottleneck. +After we switch to the cascade-like planner. The rule that needs cost to make decision is still a small set of all. And the existence of `Group` can also help us. If we lazily construct the `statsInfo`, this may not be the bottleneck. ### What is the impact of not doing this? -Many cases reported by our customer already prove that we need more accurate statistics to choose a better join order and a proper join algorithm. Only maintaining a number about row count and a slice about ndv is not enough for making that decision. +Many cases reported by our customer already prove that we need more accurate statistics to choose a better join order and a proper join algorithm. Only maintaining a number about row count and a slice about NDV is not enough for making that decision. If we don’t do this, we cannot fix some cases except we use hints. @@ -108,7 +108,7 @@ There’s no compatibility issue. ## Implementation -First maintain the histogram in `DataSource`. In this step, there will be some changes in the `statistics` package to make it work. It may take a little long time to do this. [PR#7385](https://github.com/pingcap/tidb/pull/7385) +First, maintain the histogram in `DataSource`. In this step, there will be some changes in the `statistics` package to make it work. It may take a little long time to do this. [PR#7385](https://github.com/pingcap/tidb/pull/7385) Then, all changes will be things inside `statsInfo`, which won’t affect other parts. It may not take too much time to implement this part. Like [PR#7595](https://github.com/pingcap/tidb/pull/7595) From e693210eefd8ecfd1ad603970e0131166ecfabc8 Mon Sep 17 00:00:00 2001 From: Yiding Cui Date: Thu, 6 Sep 2018 16:53:47 +0800 Subject: [PATCH 03/11] address comments --- docs/design/2018-09-04-histograms-in-plan.md | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/docs/design/2018-09-04-histograms-in-plan.md b/docs/design/2018-09-04-histograms-in-plan.md index 4c3ffd27b69c7..5167ca627e85d 100644 --- a/docs/design/2018-09-04-histograms-in-plan.md +++ b/docs/design/2018-09-04-histograms-in-plan.md @@ -1,6 +1,6 @@ # Proposal: Maintain statistics in `Plan` -- Author: Yiding CUI +- Author: [Yiding CUI](https://github.com/winoros) - Last updated: 2018-09-04 ## Abstract @@ -11,6 +11,7 @@ This proposal proposes to maintain the histogram in `Plan`’s statistics inform ## Background Currently, TiDB only uses statistics when deciding which physical scan method a table should use. And TiDB only stores simple statistics in the plan structure. But when deciding the join order and considering some other optimization rules, we need more detailed statistics. + So we need to maintain the statistics in the plan structure to get sufficient statistics information to do optimizations. ## Proposal @@ -75,7 +76,7 @@ For `Selection` and `Join`, we need to cut off the things which are not in range For `Aggregation`, we only need to cut off the things which are not in ranges when doing estimation. There is no need to update the ranges information. -For `TopN`, we now have the alibity to maintain histograms of the order-by items. +For `TopN`, we now have the ability to maintain histograms of the order-by items. ## Rationale @@ -84,7 +85,7 @@ For `TopN`, we now have the alibity to maintain histograms of the order-by items I’ve looked into Spark. They did nearly the same thing with what I said. They only maintain the max and min values, rather than the `ranges` information. And they don’t have the index, so they only maintain the column’s max/min value which make problem much easier to solve. -As for Orca and Calcite, I haven’t discovered where they maintain this information. But there’s something about statistics in Orca’s paper. According to the paper, I think they construct new histogram during planning and cache it to avoid building too many times. +As for Orca and Calcite, I haven’t discovered where they maintain this information. But there’s something about statistics in Orca’s paper. According to the paper, I think they construct new histograms during planning and cache it to avoid building too many times. ### What is the disadvantage of this design? @@ -94,7 +95,7 @@ For now, only join reorder and the position `after logical optimize before physi And the `expectedCount` we used in physical plan is something same with `Limit`. So the row count modification during physical plan won’t be affected. -After we switch to the cascade-like planner. The rule that needs cost to make decision is still a small set of all. And the existence of `Group` can also help us. If we lazily construct the `statsInfo`, this may not be the bottleneck. +After we switch to the cascade-like planner, the rule that needs cost to make decision is still a small set of all. And the existence of `Group` can also help us. If we lazily construct the `statsInfo`, this may not be the bottleneck. ### What is the impact of not doing this? From 8e012e6b9cec1c736a0e19f020be3c8f01c1ad8d Mon Sep 17 00:00:00 2001 From: Yiding Cui Date: Thu, 13 Sep 2018 17:40:19 +0800 Subject: [PATCH 04/11] Change structure. --- docs/design/2018-09-04-histograms-in-plan.md | 49 +++++++++++++------- 1 file changed, 32 insertions(+), 17 deletions(-) diff --git a/docs/design/2018-09-04-histograms-in-plan.md b/docs/design/2018-09-04-histograms-in-plan.md index 5167ca627e85d..9ea37a90f3002 100644 --- a/docs/design/2018-09-04-histograms-in-plan.md +++ b/docs/design/2018-09-04-histograms-in-plan.md @@ -18,7 +18,7 @@ So we need to maintain the statistics in the plan structure to get sufficient st The new `statsInfo` of `plan` should be something like the following structure: -``` +```go type statsInfo struct { // This two attributes are already used in current statsInfo. count float64 @@ -40,43 +40,58 @@ This struct will be maintained when we call `deriveStats`. Currently we don't change the histogram itself during planning. Because it will consume a lot of time and memory space. I’ll try to maintain ranges slice or the max/min value to improve the accuracy of row count estimation instead. -Before we maintain `rangesOfXXX` or `max and min values`: +###Before we maintain range information We maintain the histogram in `Projection`, `Selection`, `Join`, `Aggregation`, `Sort`, `Limit` and `DataSource` operators. -For `Sort`, we can just copy children's `statsInfo` without doing any change. +####`Sort` +We can just copy children's `statsInfo` without doing any change. -For `Limit`, we can just copy children's `statsInfo` or ignore the histogram information. As you know, its execution logic is based on randomization. It's hard to maintain the statistics information after `Limit`. But we may use the information before it to do some estimation in some scenarios. +####`Limit` +we can just copy children's `statsInfo` or ignore the histogram information. As you know, its execution logic is based on randomization. It's hard to maintain the statistics information after `Limit`. But we may use the information before it to do some estimation in some scenarios. -For `Projection`, change the reflection of the map we maintained. +####`Projection` +Change the reflection of the map we maintained. -For `Aggregation`, use the histogram to estimate the NDV of group-by items. If one index cannot cover the group-by item, we’ll multiply the NDV of each group-by column. If the output of `Aggregation` includes group-by columns, we’ll maintain the histogram of them for future use. +####`Aggregation` +Use the histogram to estimate the NDV of group-by items. If one index cannot cover the group-by item, we’ll multiply the NDV of each group-by column. If the output of `Aggregation` includes group-by columns, we’ll maintain the histogram of them for future use. -For `Join`, there’re joins as follows: +####`Join` +There're several kinds of joins. -- Inner join: use histograms to do the row count estimation with the join key condition. Since it won’t have one side filter, we only need to consider the composite filters after considering the join key. We can simply multiply `selectionFactor` if there are other composite filters in our first version of implementation. Since `Selectivity` cannot calculate selectivity of an expression containing multiple columns. +#####Inner join +Use histograms to do the row count estimation with the join key condition. Since it won’t have one side filter, we only need to consider the composite filters after considering the join key. We can simply multiply `selectionFactor` if there are other composite filters in our first version of implementation. Since `Selectivity` cannot calculate selectivity of an expression containing multiple columns. -- One side outer join: It depends on the join keys’ NDV. And we can just use histograms to estimate it if there’re non-join-key filters. +#####One side outer join +It depends on the join keys’ NDV. And we can just use histograms to estimate it if there’re non-join-key filters. -- Semi join: It’s something similar to inner join. But no data expanding occurs. When we maintain the range information, we can get a nearly accurate answer of its row count. +#####Semi join +It’s something similar to inner join. But no data expanding occurs. When we maintain the range information, we can get a nearly accurate answer of its row count. -- Anti semi join: Same with semi join. +#####Anti semi join +Same with semi join. -For `Selection`, just use it to calculate the selectivity. +####`Selection` +Just use it to calculate the selectivity. -For `DataSource`, if it’s a non-partitioned table, we just maintain the map. If it’s a partitioned table, we now only store the statistics of each partition. So we need to merge them. We need a cache or something else to ensure that we won’t merge them each time we need it, which will consume tooooo much time and memory space. +####`DataSource` +If it’s a non-partitioned table, we just maintain the map. If it’s a partitioned table, we now only store the statistics of each partition. So we need to merge them. We need a cache or something else to ensure that we won’t merge them each time we need it, which will consume tooooo much time and memory space. +####Others For other plan operators or `DataSource` which do not contain any histogram, we just use the original way to maintain `statsInfo`. We won’t maintain histograms for them. Taking `Union` for an example, if we maintain histograms for it, we need to merge the histograms, which is really an expensive operation. But we can consider maintaining this if a hint is given. -When we maintain `rangesOfXXX` or `max and min values`: +###After we maintain range information Most things are the same with above. But there will be something more to do. -For `Selection` and `Join`, we need to cut off the things which are not in ranges when doing estimation, and update the ranges information after the estimation. Also the NDV of column can be estimated more accurately. +####`Selection` and `Join` +We need to cut off the things which are not in ranges when doing estimation, and update the ranges information after the estimation. Also the NDV of column can be estimated more accurately. -For `Aggregation`, we only need to cut off the things which are not in ranges when doing estimation. There is no need to update the ranges information. +####`Aggregation` +We only need to cut off the things which are not in ranges when doing estimation. There is no need to update the ranges information. -For `TopN`, we now have the ability to maintain histograms of the order-by items. +####`TopN` +We now have the ability to maintain histograms of the order-by items. ## Rationale From bdd26fa1aa90a5a373aed764f13f45943e072d88 Mon Sep 17 00:00:00 2001 From: Yiding Cui Date: Fri, 14 Sep 2018 16:08:19 +0800 Subject: [PATCH 05/11] add formula. --- docs/design/2018-09-04-histograms-in-plan.md | 49 +++-- .../svgs/305e3aca4fcdd416e505e59b337ec67f.png | Bin 0 -> 25489 bytes .../svgs/305e3aca4fcdd416e505e59b337ec67f.svg | 174 ++++++++++++++++++ .../svgs/30df1c648fa9fe43985776847c8dbe60.png | Bin 0 -> 8171 bytes .../svgs/30df1c648fa9fe43985776847c8dbe60.svg | 51 +++++ .../svgs/9b88d7d105ad2b263532733169dd94fe.png | Bin 0 -> 17005 bytes .../svgs/9b88d7d105ad2b263532733169dd94fe.svg | 110 +++++++++++ .../svgs/caab8c4da85732f108d5e1b1cfe5e285.png | Bin 0 -> 15534 bytes .../svgs/caab8c4da85732f108d5e1b1cfe5e285.svg | 86 +++++++++ 9 files changed, 452 insertions(+), 18 deletions(-) create mode 100644 docs/design/svgs/305e3aca4fcdd416e505e59b337ec67f.png create mode 100644 docs/design/svgs/305e3aca4fcdd416e505e59b337ec67f.svg create mode 100644 docs/design/svgs/30df1c648fa9fe43985776847c8dbe60.png create mode 100644 docs/design/svgs/30df1c648fa9fe43985776847c8dbe60.svg create mode 100644 docs/design/svgs/9b88d7d105ad2b263532733169dd94fe.png create mode 100644 docs/design/svgs/9b88d7d105ad2b263532733169dd94fe.svg create mode 100644 docs/design/svgs/caab8c4da85732f108d5e1b1cfe5e285.png create mode 100644 docs/design/svgs/caab8c4da85732f108d5e1b1cfe5e285.svg diff --git a/docs/design/2018-09-04-histograms-in-plan.md b/docs/design/2018-09-04-histograms-in-plan.md index 9ea37a90f3002..666af169996bb 100644 --- a/docs/design/2018-09-04-histograms-in-plan.md +++ b/docs/design/2018-09-04-histograms-in-plan.md @@ -40,57 +40,70 @@ This struct will be maintained when we call `deriveStats`. Currently we don't change the histogram itself during planning. Because it will consume a lot of time and memory space. I’ll try to maintain ranges slice or the max/min value to improve the accuracy of row count estimation instead. -###Before we maintain range information +### Before we maintain range information We maintain the histogram in `Projection`, `Selection`, `Join`, `Aggregation`, `Sort`, `Limit` and `DataSource` operators. -####`Sort` +#### `Sort` We can just copy children's `statsInfo` without doing any change. -####`Limit` +#### `Limit` we can just copy children's `statsInfo` or ignore the histogram information. As you know, its execution logic is based on randomization. It's hard to maintain the statistics information after `Limit`. But we may use the information before it to do some estimation in some scenarios. -####`Projection` +#### `Projection` Change the reflection of the map we maintained. -####`Aggregation` +#### `Aggregation` Use the histogram to estimate the NDV of group-by items. If one index cannot cover the group-by item, we’ll multiply the NDV of each group-by column. If the output of `Aggregation` includes group-by columns, we’ll maintain the histogram of them for future use. -####`Join` +#### `Join` There're several kinds of joins. -#####Inner join -Use histograms to do the row count estimation with the join key condition. Since it won’t have one side filter, we only need to consider the composite filters after considering the join key. We can simply multiply `selectionFactor` if there are other composite filters in our first version of implementation. Since `Selectivity` cannot calculate selectivity of an expression containing multiple columns. +##### Inner join +Use histograms to do the row count estimation with the join key condition. -#####One side outer join +It can be calculated as this formula $selecivity=joinKeySelectivity*leftRowCount*rightRowCount$ + +Where $selectivity = \frac{rowCount(t1) - nulls(t1.key)}{rowCount(t1)*NDV(t1.key)}*\frac{rowCount(t2) - nulls(t2.key)}{rowCount(t2)*NDV(t2.key)}*ndvAfterJoin$. + +The `ndvAfterJoin` can be $min(NDV(t1.key), NDV(t2.key))$ or a more detailed one if we can caculate it. + + +Since it won’t have one side filter, we only need to consider the composite filters after considering the join key. We can simply multiply `selectionFactor` if there are other composite filters in our first version of implementation. Since `Selectivity` cannot calculate selectivity of an expression containing multiple columns. + +##### One side outer join It depends on the join keys’ NDV. And we can just use histograms to estimate it if there’re non-join-key filters. -#####Semi join +##### Semi join It’s something similar to inner join. But no data expanding occurs. When we maintain the range information, we can get a nearly accurate answer of its row count. -#####Anti semi join +##### Anti semi join Same with semi join. -####`Selection` +#### `Selection` Just use it to calculate the selectivity. -####`DataSource` +And to maintain `NDV`, we can use $NDV_{a} = NDV_{a} * (1 - (1 - selectivity_a)^{cntPerVal})*(1 - (1 - selectivity_{others})^{cntPerVal})$ to estimate it. + +If we have NDV of every bucket of histogram, we can make this formula more accurate. + +#### `DataSource` If it’s a non-partitioned table, we just maintain the map. If it’s a partitioned table, we now only store the statistics of each partition. So we need to merge them. We need a cache or something else to ensure that we won’t merge them each time we need it, which will consume tooooo much time and memory space. -####Others +#### Others For other plan operators or `DataSource` which do not contain any histogram, we just use the original way to maintain `statsInfo`. We won’t maintain histograms for them. Taking `Union` for an example, if we maintain histograms for it, we need to merge the histograms, which is really an expensive operation. But we can consider maintaining this if a hint is given. -###After we maintain range information +### After we maintain range information Most things are the same with above. But there will be something more to do. -####`Selection` and `Join` +#### `Selection` and `Join` We need to cut off the things which are not in ranges when doing estimation, and update the ranges information after the estimation. Also the NDV of column can be estimated more accurately. -####`Aggregation` +#### `Aggregation` We only need to cut off the things which are not in ranges when doing estimation. 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Date: Mon, 8 Oct 2018 17:02:55 +0800 Subject: [PATCH 06/11] modify proposal. --- docs/design/2018-09-04-histograms-in-plan.md | 92 ++++--- docs/design/histogram-1.png | Bin 0 -> 57800 bytes docs/design/histogram-2.png | Bin 0 -> 66736 bytes docs/design/histogram-3.png | Bin 0 -> 37320 bytes .../svgs/291c9eb6e8db885402c716ffc3e17a65.png | Bin 0 -> 14388 bytes ...g => 291c9eb6e8db885402c716ffc3e17a65.svg} | 259 ++++++------------ .../svgs/305e3aca4fcdd416e505e59b337ec67f.png | Bin 25489 -> 0 bytes .../svgs/35fa60f709be6b9ab8aa9036bd5e7f7f.png | Bin 0 -> 15502 bytes ...g => 35fa60f709be6b9ab8aa9036bd5e7f7f.svg} | 61 ++--- .../svgs/9b88d7d105ad2b263532733169dd94fe.png | Bin 17005 -> 0 bytes .../svgs/9b88d7d105ad2b263532733169dd94fe.svg | 110 -------- .../svgs/caab8c4da85732f108d5e1b1cfe5e285.png | Bin 15534 -> 0 bytes 12 files changed, 168 insertions(+), 354 deletions(-) create mode 100644 docs/design/histogram-1.png create mode 100644 docs/design/histogram-2.png create mode 100644 docs/design/histogram-3.png create mode 100644 docs/design/svgs/291c9eb6e8db885402c716ffc3e17a65.png rename docs/design/svgs/{305e3aca4fcdd416e505e59b337ec67f.svg => 291c9eb6e8db885402c716ffc3e17a65.svg} (57%) delete mode 100644 docs/design/svgs/305e3aca4fcdd416e505e59b337ec67f.png create mode 100644 docs/design/svgs/35fa60f709be6b9ab8aa9036bd5e7f7f.png rename docs/design/svgs/{caab8c4da85732f108d5e1b1cfe5e285.svg => 35fa60f709be6b9ab8aa9036bd5e7f7f.svg} (79%) delete mode 100644 docs/design/svgs/9b88d7d105ad2b263532733169dd94fe.png delete mode 100644 docs/design/svgs/9b88d7d105ad2b263532733169dd94fe.svg delete mode 100644 docs/design/svgs/caab8c4da85732f108d5e1b1cfe5e285.png diff --git a/docs/design/2018-09-04-histograms-in-plan.md b/docs/design/2018-09-04-histograms-in-plan.md index 666af169996bb..a7e1844b8502b 100644 --- a/docs/design/2018-09-04-histograms-in-plan.md +++ b/docs/design/2018-09-04-histograms-in-plan.md @@ -27,12 +27,6 @@ type statsInfo struct { // HistColl collects histograms of columns. histColl statistics.HistColl - // rangesOfXXX This won’t be maintained in the first version. - rangesOfColumn map[int][]*ranger.Range - rangesOfIndex map[int][]*ranger.Range - // max/minValues is a simplified version of rangesOfXXX. Since maintaining rangesOfXXXX consumes a lot of time and memory, we can first check whether this can meet our needs. - maxValues map[int][]types.Datum - minValues map[int][]types.Datum } ``` @@ -40,91 +34,99 @@ This struct will be maintained when we call `deriveStats`. Currently we don't change the histogram itself during planning. Because it will consume a lot of time and memory space. I’ll try to maintain ranges slice or the max/min value to improve the accuracy of row count estimation instead. -### Before we maintain range information +### How to maintain it. We maintain the histogram in `Projection`, `Selection`, `Join`, `Aggregation`, `Sort`, `Limit` and `DataSource` operators. -#### `Sort` -We can just copy children's `statsInfo` without doing any change. +#### `Select` +Just use it to calculate the selectivity. -#### `Limit` -we can just copy children's `statsInfo` or ignore the histogram information. As you know, its execution logic is based on randomization. It's hard to maintain the statistics information after `Limit`. But we may use the information before it to do some estimation in some scenarios. +And we can get the range information of each column and index involved in filters. Then we can use the range information to cut the histogram bucket of these columns. + +For column/index not matched in the filter. The `NDV` and `totalCnt` of each bucket can just mutiply the total selectivity of filters(based on independent assumption). #### `Projection` Change the reflection of the map we maintained. -#### `Aggregation` -Use the histogram to estimate the NDV of group-by items. If one index cannot cover the group-by item, we’ll multiply the NDV of each group-by column. If the output of `Aggregation` includes group-by columns, we’ll maintain the histogram of them for future use. - #### `Join` There're several kinds of joins. ##### Inner join -Use histograms to do the row count estimation with the join key condition. +Use histograms to do the row count estimation with the join key condition. It can be described as the following procedures: + +- Align the histogram buckets so that each bucket boundary in one histogram can be matched in the other histogram. For example(the following figure), bucket `b1` and `c1`'s left boundary is the same. But the right boundary is different. So we can split `c1` into two buckets. The first one's boundary is totally the same with `b1`. The second one starts from where the first one ends. +- Since the boundary of the bucket have been aligned. Then we can estimate the join's size bucket by bucket by the well-know way. + +Original: -It can be calculated as this formula $selecivity=joinKeySelectivity*leftRowCount*rightRowCount$ +original histogram -Where $selectivity = \frac{rowCount(t1) - nulls(t1.key)}{rowCount(t1)*NDV(t1.key)}*\frac{rowCount(t2) - nulls(t2.key)}{rowCount(t2)*NDV(t2.key)}*ndvAfterJoin$. +Step 1: + +Step 1 + +Step 2: + +
+Step 2 +
+ +The calculation inside the bucket can be calculated as this formula $selecivity=joinKeySelectivity*RowCount(t1)*RowCount(t2)$ + +Where $joinKeySelectivity = \frac{1}{NDV(t1.key)}*\frac{1}{NDV(t2.key)}*ndvAfterJoin$. The `ndvAfterJoin` can be $min(NDV(t1.key), NDV(t2.key))$ or a more detailed one if we can caculate it. -Since it won’t have one side filter, we only need to consider the composite filters after considering the join key. We can simply multiply `selectionFactor` if there are other composite filters in our first version of implementation. Since `Selectivity` cannot calculate selectivity of an expression containing multiple columns. +Since it won’t have one side filter, we only need to consider the composite filters after considering the join key. We can simply multiply `selectionFactor` if there are other composite filters since `Selectivity` cannot calculate selectivity of an expression containing multiple columns. ##### One side outer join -It depends on the join keys’ NDV. And we can just use histograms to estimate it if there’re non-join-key filters. +It's almost the same as inner join's behavior. But we need to consider two more thing: + +- The unmatched row will be filled as `NULL`. This should be calculated in the new histogram. The null count can be caculated when we estimate the matched count bucket by bucket. +- There will be one side filters of the outer table. If the filter is about join key and can be converted to range information, it's can be easily involved when we do the caculation bucket by bucket. Otherwise it's a little hard to deal with it. Don't consider this case currently. ##### Semi join -It’s something similar to inner join. But no data expanding occurs. When we maintain the range information, we can get a nearly accurate answer of its row count. +It’s something similar to inner join. But no data expanding occurs. So the `total` can be adjusted easily. ##### Anti semi join Same with semi join. -#### `Selection` -Just use it to calculate the selectivity. +#### `Aggregate` +Just read the NDV information from the `statsInfo` to dicide the row count after aggregate. If there's index can fully match the group-by items. We just use its NDV. Otherwise we multiply the ndv of each column(or index that can match part of the group-by item). + +If some of the group-by items are also in the select field. We will create new histograms modify the `totalCnt` of each bucket(set it the same with `NDV`). -And to maintain `NDV`, we can use $NDV_{a} = NDV_{a} * (1 - (1 - selectivity_a)^{cntPerVal})*(1 - (1 - selectivity_{others})^{cntPerVal})$ to estimate it. +#### `Sort` +We can just copy children's `statsInfo` without doing any change. Since the data distribution is not changed. -If we have NDV of every bucket of histogram, we can make this formula more accurate. +#### `Limit` +Currently we won't maintain hitogram information for it. But it can be considered in the future. #### `DataSource` -If it’s a non-partitioned table, we just maintain the map. If it’s a partitioned table, we now only store the statistics of each partition. So we need to merge them. We need a cache or something else to ensure that we won’t merge them each time we need it, which will consume tooooo much time and memory space. +Our `DataSource` is actually `source of data` + `select`. So if it’s a non-partitioned table, we just maintain the map and do what `Select` does. If it’s a partitioned table, we now only store the statistics of each partition. So we need to merge them into one histogram so the upper plan can use it. We need a cache or something else to ensure that we won’t merge them each time we need it. #### Others -For other plan operators or `DataSource` which do not contain any histogram, we just use the original way to maintain `statsInfo`. We won’t maintain histograms for them. Taking `Union` for an example, if we maintain histograms for it, we need to merge the histograms, which is really an expensive operation. But we can consider maintaining this if a hint is given. - -### After we maintain range information - -Most things are the same with above. But there will be something more to do. - -#### `Selection` and `Join` -We need to cut off the things which are not in ranges when doing estimation, and update the ranges information after the estimation. Also the NDV of column can be estimated more accurately. - -#### `Aggregation` -We only need to cut off the things which are not in ranges when doing estimation. There is no need to update the ranges information. - -#### `TopN` -We now have the ability to maintain histograms of the order-by items. - +For other plan operators or `DataSource` which do not contain any histogram, we just use the original way to maintain `statsInfo`. We won’t maintain histograms for them. ## Rationale ### How other systems solve the same issue? -I’ve looked into Spark. They did nearly the same thing with what I said. They only maintain the max and min values, rather than the `ranges` information. And they don’t have the index, so they only maintain the column’s max/min value which make problem much easier to solve. +I’ve looked into Spark. They only maintain the max and min values, won't create new histogram after `Select`/`Join`/`Aggregate`. And they don’t have the index, so maintaining the column’s max/min value is much easier to solve comparing with us. As for Orca and Calcite, I haven’t discovered where they maintain this information. But there’s something about statistics in Orca’s paper. According to the paper, I think they construct new histograms during planning and cache it to avoid building too many times. ### What is the disadvantage of this design? -This may have side effects on OLTP performance. But I’ll try to reduce it. +This may have side effects on OLTP performance. (Currently totally changing a histogram whose number of bucket is 256 and content is int value costs 7000ns. So we can create 143 histograms at 1ms currently) -For now, only join reorder and the position `after logical optimize before physical optimize` will trigger this mechanism and it’s controlled by `tidb_optimizer_selectivity_level` which is disabled by default. This may not have much more side effects on simple point queries. - -And the `expectedCount` we used in physical plan is something same with `Limit`. So the row count modification during physical plan won’t be affected. +For now, only join reorder and the position `after logical optimize before physical optimize` will trigger this mechanism and it’s controlled by `tidb_optimizer_selectivity_level` which is disabled by default. It will opened by default after we test it carefully. After we switch to the cascade-like planner, the rule that needs cost to make decision is still a small set of all. And the existence of `Group` can also help us. If we lazily construct the `statsInfo`, this may not be the bottleneck. +And thanks to `predicate push down` and some other optimization rules. The single column(column directly from the DataSource not one the generated by `Project`'s complex `Expr`) filters(Only this type of expression can be calculated to range information) can be all pushed down to the `DataSource` or the `Selection` on the `DataSource`. So simple querys won't create many new histograms. + ### What is the impact of not doing this? Many cases reported by our customer already prove that we need more accurate statistics to choose a better join order and a proper join algorithm. 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b/docs/design/2018-09-04-histograms-in-plan.md index a7e1844b8502b..2ead7a6fc9180 100644 --- a/docs/design/2018-09-04-histograms-in-plan.md +++ b/docs/design/2018-09-04-histograms-in-plan.md @@ -32,18 +32,16 @@ type statsInfo struct { This struct will be maintained when we call `deriveStats`. -Currently we don't change the histogram itself during planning. Because it will consume a lot of time and memory space. I’ll try to maintain ranges slice or the max/min value to improve the accuracy of row count estimation instead. - ### How to maintain it. We maintain the histogram in `Projection`, `Selection`, `Join`, `Aggregation`, `Sort`, `Limit` and `DataSource` operators. -#### `Select` +#### `Selection` Just use it to calculate the selectivity. And we can get the range information of each column and index involved in filters. Then we can use the range information to cut the histogram bucket of these columns. -For column/index not matched in the filter. The `NDV` and `totalCnt` of each bucket can just mutiply the total selectivity of filters(based on independent assumption). +For column/index not matched in the filter. The `NDV` and `totalCnt` of each bucket can just multiply the total selectivity of filters(based on independent assumption). #### `Projection` Change the reflection of the map we maintained. @@ -75,7 +73,7 @@ The calculation inside the bucket can be calculated as this formula $s
 
 Where <img alt=. -The `ndvAfterJoin` can be $min(NDV(t1.key), NDV(t2.key))$ or a more detailed one if we can caculate it. +The `ndvAfterJoin` can be $min(NDV(t1.key), NDV(t2.key))$ or a more detailed one if we can calculate it. Since it won’t have one side filter, we only need to consider the composite filters after considering the join key. We can simply multiply `selectionFactor` if there are other composite filters since `Selectivity` cannot calculate selectivity of an expression containing multiple columns. @@ -83,8 +81,8 @@ Since it won’t have one side filter, we only need to consider the composite fi ##### One side outer join It's almost the same as inner join's behavior. But we need to consider two more thing: -- The unmatched row will be filled as `NULL`. This should be calculated in the new histogram. The null count can be caculated when we estimate the matched count bucket by bucket. -- There will be one side filters of the outer table. If the filter is about join key and can be converted to range information, it's can be easily involved when we do the caculation bucket by bucket. Otherwise it's a little hard to deal with it. Don't consider this case currently. +- The unmatched row will be filled as `NULL`. This should be calculated in the new histogram. The null count can be calculated when we estimate the matched count bucket by bucket. +- There will be one side filters of the outer table. If the filter is about join key and can be converted to range information, it's can be easily involved when we do the calculation bucket by bucket. Otherwise it's a little hard to deal with it. Don't consider this case currently. ##### Semi join It’s something similar to inner join. But no data expanding occurs. So the `total` can be adjusted easily. @@ -93,15 +91,15 @@ It’s something similar to inner join. But no data expanding occurs. So the `to Same with semi join. #### `Aggregate` -Just read the NDV information from the `statsInfo` to dicide the row count after aggregate. If there's index can fully match the group-by items. We just use its NDV. Otherwise we multiply the ndv of each column(or index that can match part of the group-by item). +Just read the NDV information from the `statsInfo` to decide the row count after aggregate. If there's index can fully match the group-by items, we will just use its NDV. Otherwise we will multiply the ndv of each column(or index that can match part of the group-by item). -If some of the group-by items are also in the select field. We will create new histograms modify the `totalCnt` of each bucket(set it the same with `NDV`). +If some of the group-by items are also in the select field, we will create new histograms by modifying the `totalCnt` of each bucket(set it the same with `NDV`). #### `Sort` -We can just copy children's `statsInfo` without doing any change. Since the data distribution is not changed. +We can just copy children's `statsInfo` without doing any change, since the data distribution is not changed. #### `Limit` -Currently we won't maintain hitogram information for it. But it can be considered in the future. +Currently we won't maintain histogram information for it. But it can be considered in the future. #### `DataSource` Our `DataSource` is actually `source of data` + `select`. So if it’s a non-partitioned table, we just maintain the map and do what `Select` does. If it’s a partitioned table, we now only store the statistics of each partition. So we need to merge them into one histogram so the upper plan can use it. We need a cache or something else to ensure that we won’t merge them each time we need it. From 3f932a1f167b0761fa030a0c8011c2f761b8880b Mon Sep 17 00:00:00 2001 From: Yiding Cui Date: Wed, 24 Oct 2018 17:13:45 +0800 Subject: [PATCH 08/11] address comments --- docs/design/2018-09-04-histograms-in-plan.md | 12 +-- docs/design/histogram-1.png | Bin 57800 -> 0 bytes docs/design/histogram-2.png | Bin 66736 -> 0 bytes docs/design/histogram-3.png | Bin 37320 -> 0 bytes .../svgs/291c9eb6e8db885402c716ffc3e17a65.png | Bin 14388 -> 0 bytes .../svgs/291c9eb6e8db885402c716ffc3e17a65.svg | 97 ------------------ .../svgs/30df1c648fa9fe43985776847c8dbe60.png | Bin 8171 -> 0 bytes .../svgs/30df1c648fa9fe43985776847c8dbe60.svg | 51 --------- .../svgs/35fa60f709be6b9ab8aa9036bd5e7f7f.png | Bin 15502 -> 0 bytes .../svgs/35fa60f709be6b9ab8aa9036bd5e7f7f.svg | 85 --------------- 10 files changed, 6 insertions(+), 239 deletions(-) delete mode 100644 docs/design/histogram-1.png delete mode 100644 docs/design/histogram-2.png delete mode 100644 docs/design/histogram-3.png delete mode 100644 docs/design/svgs/291c9eb6e8db885402c716ffc3e17a65.png delete mode 100644 docs/design/svgs/291c9eb6e8db885402c716ffc3e17a65.svg delete mode 100644 docs/design/svgs/30df1c648fa9fe43985776847c8dbe60.png delete mode 100644 docs/design/svgs/30df1c648fa9fe43985776847c8dbe60.svg delete mode 100644 docs/design/svgs/35fa60f709be6b9ab8aa9036bd5e7f7f.png delete mode 100644 docs/design/svgs/35fa60f709be6b9ab8aa9036bd5e7f7f.svg diff --git a/docs/design/2018-09-04-histograms-in-plan.md b/docs/design/2018-09-04-histograms-in-plan.md index 2ead7a6fc9180..bba90f9af5124 100644 --- a/docs/design/2018-09-04-histograms-in-plan.md +++ b/docs/design/2018-09-04-histograms-in-plan.md @@ -57,23 +57,23 @@ Use histograms to do the row count estimation with the join key condition. It ca Original: -original histogram +original histogram Step 1: -Step 1 +Step 1 Step 2:
-Step 2 +Step 2
-The calculation inside the bucket can be calculated as this formula $selecivity=joinKeySelectivity*RowCount(t1)*RowCount(t2)$ +The calculation inside the bucket can be calculated as this formula $selectivity=joinKeySelectivity*RowCount(t1)*RowCount(t2)$ -Where $joinKeySelectivity = \frac{1}{NDV(t1.key)}*\frac{1}{NDV(t2.key)}*ndvAfterJoin$. +Where $joinKeySelectivity = \frac{1}{NDV(t1.key)}*\frac{1}{NDV(t2.key)}*ndvAfterJoin$. -The `ndvAfterJoin` can be $min(NDV(t1.key), NDV(t2.key))$ or a more detailed one if we can calculate it. +The `ndvAfterJoin` can be $min(NDV(t1.key), NDV(t2.key))$ or a more detailed one if we can calculate it. Since it won’t have one side filter, we only need to consider the composite filters after considering the join key. 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zqy*i_H2YQExWqXAcbhMS-GYM3&kWojBlBJUGUz`H}Fu*zAlxD@$>u5EXgzJ%%%* zpWGX+i<=#eqj3 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - \ No newline at end of file From 3f5358b5400279f9e7deabc637aaa188c5036d8b Mon Sep 17 00:00:00 2001 From: Yiding Cui Date: Wed, 24 Oct 2018 17:16:06 +0800 Subject: [PATCH 09/11] add to doc/design's readme --- docs/design/README.md | 1 + 1 file changed, 1 insertion(+) diff --git a/docs/design/README.md b/docs/design/README.md index 4318dbfeca3a3..ab79d40dd4717 100644 --- a/docs/design/README.md +++ b/docs/design/README.md @@ -12,6 +12,7 @@ A proposal template: [TEMPLATE.md](./TEMPLATE.md) - [Proposal: Enhance constraint propagation in TiDB logical plan](./2018-07-22-enhance-propagations.md) - [Proposal: A SQL Planner based on the Volcano/Cascades model](./2018-08-29-new-planner.md) - [Proposal: Implement Radix Hash Join](./2018-09-21-radix-hashjoin.md) +- [Proposal: Maintaining histograms in plan](./2018-09-04-histograms-in-plan.md) ## Completed From 089a5020cec0b787252af96ad5ae7cf5cc43f086 Mon Sep 17 00:00:00 2001 From: Yiding Cui Date: Wed, 24 Oct 2018 17:19:10 +0800 Subject: [PATCH 10/11] add imgs back --- docs/design/imgs/.DS_Store | Bin 0 -> 6148 bytes docs/design/imgs/histogram-1.jpeg | Bin 0 -> 58549 bytes docs/design/imgs/histogram-2.jpeg | Bin 0 -> 64945 bytes docs/design/imgs/histogram-3.jpeg | Bin 0 -> 39915 bytes docs/design/imgs/histogram-3.png | Bin 0 -> 29861 bytes docs/design/imgs/histogram-formula-1.png | Bin 0 -> 15755 bytes docs/design/imgs/histogram-formula-2.png | Bin 0 -> 14388 bytes docs/design/imgs/histogram-formula-3.png | Bin 0 -> 8171 bytes 8 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 docs/design/imgs/.DS_Store create mode 100644 docs/design/imgs/histogram-1.jpeg create mode 100644 docs/design/imgs/histogram-2.jpeg create mode 100644 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