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SELECT count(distinct ws1.ws_order_number) as order_count,
             sum(ws1.ws_ext_ship_cost) as total_shipping_cost,
             sum(ws1.ws_net_profit) as total_net_profit
FROM web_sales ws1
JOIN /*MJ1*/ customer_address ca ON (ws1.ws_ship_addr_sk = ca.ca_address_sk)
JOIN /*MJ2*/ web_site s ON (ws1.ws_web_site_sk = s.web_site_sk)
JOIN /*MJ3*/ date_dim d ON (ws1.ws_ship_date_sk = d.d_date_sk)
LEFT SEMI JOIN /*JOIN4*/ (SELECT ws2.ws_order_number as ws_order_number
               	          FROM web_sales ws2 JOIN /*JOIN1*/ web_sales ws3
               	          ON (ws2.ws_order_number = ws3.ws_order_number)
               	          WHERE ws2.ws_warehouse_sk <> ws3.ws_warehouse_sk) ws_wh1
ON (ws1.ws_order_number = ws_wh1.ws_order_number)
LEFT SEMI JOIN /*JOIN4*/ (SELECT wr_order_number
               	          FROM web_returns wr
               	          JOIN /*JOIN3*/ (SELECT ws4.ws_order_number as ws_order_number
                                          FROM web_sales ws4 JOIN /*JOIN2*/ web_sales ws5
                                          ON (ws4.ws_order_number = ws5.ws_order_number)
                                          WHERE ws4.ws_warehouse_sk <> ws5.ws_warehouse_sk) ws_wh2
                          ON (wr.wr_order_number = ws_wh2.ws_order_number)) tmp1
ON (ws1.ws_order_number = tmp1.wr_order_number)
WHERE d.d_date >= '2001-05-01' and
      d.d_date <= '2001-06-30' and
      ca.ca_state = 'NC' and
      s.web_company_name = 'pri';

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1. Input Correlation: A input table is used by multiple MapReduce tasks in the original operator tree.
2. Job Flow Correlation: Two dependent MapReduce tasks shuffle the data in the same way.

4. Correlation Detection

At the optimization side, Correlation Optimizer is located in the class of CorrelationOptimizer and it is a part of the package of org.apache.hadoop.hive.ql.optimizer.correlation. It works on the operator tree before this tree is cut to multiple MapReduce tasks. This optimizer detects correlations and transforms the operator tree accordingly. In this section, we first go through the part of correlation detection. In the next section, we will introduce how an operator tree is transformed based on detected correlations.

To detect correlations, we start to walk the tree from the FileSinkOperator (using DefaultGraphWalker). We stop by at every ReduceSinkOperator. Then, from this ReduceSinkOperator and its peer ReduceSinkOperators (in the case of handling a JoinOperator), we start to find correlated ReduceSinkOperators along the upstream direction (the direction of parent operators) in a layer by layer way. These ReduceSinkOperator which the search starts from are called topReduceSinkOperators. The search from topReduceSinkOperators will return all ReduceSinkOperators at the lowest layers we can reach as a list called bottomReduceSinkOperators. Finally, the optimizer will evaluate if we have found a sub-tree with correlations by comparing topReduceSinkOperators and bottomReduceSinkOperators.If topReduceSinkOperators and bottomReduceSinkOperators are not the same, we consider that we have found job flow correlations. If we found correlations, we will mark those ReduceSinkOperator belonging to the sub-tree with correlations, so the tree walker will not visit these ReduceSinkOperators again. Finally, the optimizer continues to walk the tree. It is worth noting that if hive.auto.convert.join=true, we will first check if any JoinOperator will be automatically converted to MapJoinOperator later by CommonJoinResolver. Then in the process of correlation detection, we will stop searching a branch if we reach a such kind of JoinOperator.

For example, in Figure 1 (we also show it below), the process of correlation detection is described as follows.
Image Added

1. Wiki Markup
   The tree walker visits {{RS4}}. We set {{topReduceSinkOperators=\[RS4\]}}.

2. From RS4, we track sorting columns and partitioning columns of RS4 backward until we reach RS2 (because tmp1.key is from the left table of JOIN1).
   1. Check if RS4 and RS2 are using the same sorting columns, sorting orders, and same partitioning columns. Also, we check if RS4 and RS2 do not have any conflict on the number of reducers. In this example, all of these checks pass.
   2. Because RS4 and RS2 are correlated and the child of RS2 is a JoinOperator, we analyze if we can consider RS3 as a correlated ReduceSinkOperator of RS4. In this example, JOIN1 is an inner join operation. So, RS4 and RS3 are also correlated. Because both parents of the JOIN1 are correlated ReduceSinkOperators, we can continue to search ReduceSinkOperators from both RS2 and RS3.
3. From RS2, we track sorting columns and partitioning columns of RS2 backward until we reach RS1.
   1. Check if RS2 and RS1 are using the same sorting columns, sorting orders, and same partitioning columns. Also, we check if RS2 and RS1 do not have any conflict on the number of reducers. In this example, all of these checks pass. So, RS2 and RS1 are correlated.
4. Because there is no ReduceSinkOperator we can track backward from RS1, we add RS1 to bottomReduceSinkOperators.
5. Because there is no ReduceSinkOperator we can track backward from RS3, we add RS3 to bottomReduceSinkOperators.

6. Wiki Markup
   We have {{topReduceSinkOperators=\[RS4\]}} and {{bottomReduceSinkOperators=\[RS1, RS3\]}}. Because {{topReduceSinkOperators}} and {{bottomReduceSinkOperators}} are not the same, we have found a sub-tree with correlations. This sub-tree starts from {{RS1}} and {{RS3}}, and all {{ReduceSinkOperators}} in this sub-tree are {{RS1}}, {{RS2}}, {{RS3}}, and {{RS4}}.

7. There is no ReduceSinkOperator which needs to be visited. The process of correlation detection stops.

In the process of searching correlated ReduceSinkOperators, if the child of a correlated ReduceSinkOperator is a JoinOperator, we analyze if other ReduceSinkOperators of this JoinOperator can be also considered as correlated ReduceSinkOperators in the following way. In a JoinOperator, there are multiple join conditions (joinConds). For a join condition, it has a left table and a right table. For a correlated ReduceSinkOperator, if it is the left table of a join condition, we consider that the ReduceSinkOperator corresponding to the right table is also correlated when the join type is either INNER_JOIN, LEFT_OUTER_JOIN, or LEFT_SEMI_JOIN. If a correlated ReduceSinkOperator is the right table of a join condition, we consider that the ReduceSinkOperator corresponding to the left table is also correlated when the join type is either INNER_JOIN, or RIGHT_OUTER_JOIN. Because a JoinOperator can have multiple join conditions, we recursively search all join conditions until we either have searched all join conditions or there is no more correlated ReduceSinkOperators. After this analysis, if all parent ReduceSinkOperators of the JoinOperator are correlated, we will continue to search ReduceSinkOperators at this branch. Otherwise, we will stop searching this branch and consider none of parent ReduceSinkOperators of the JoinOperator is correlated.

Right now, the process of correlation detection has a few limitations. We should improve these in our future work.

1. Conditions on checking if two ReduceSinkOperators are correlated are very restrict. Two ReduceSinkOperators are considered correlated if they have the same sorting columns, sorting orders, partitioning columns, and they do not have conflict on the number of reducers.
2. Input correlations are not explicitly detected. Right now, we only explicitly detect job flow correlations. If a sub-tree has job flow correlations, because we use a single MapReduce job to evaluate this sub-tree, input correlations in this sub-tree can be automatically exploited. However, there are cases which only have input correlations. Right now, these cases are not optimized.
3. If the input operator tree has multiple FileSinkOperators, we do not optimize this tree.
4. If the input operator tree already has MapJoinOperator, we do not optimize this tree.
5. In the process of searching ReduceSinkOperators, if we find a GroupByOperator with grouping sets or a PTFOperator in a branch, we stop searching this branch.

5. Operator Tree Transformation

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