### What changes were proposed in this pull request?
Reimplement `org.apache.spark.sql.catalyst.util.QuantileSummaries#merge` and add a test-case showing the previous bug.
### Why are the changes needed?
The original Greenwald-Khanna paper, from which the algorithm behind `approxQuantile` was taken, does not cover how to merge the result of multiple parallel QuantileSummaries. The current implementation violates some invariants and therefore the effective error can be larger than the specified.
### Does this PR introduce any user-facing change?
Yes, for same cases, the results from `approxQuantile` (`percentile_approx` in SQL) will now be within the expected error margin. For example:
```scala
var values = (1 to 100).toArray
val all_quantiles = values.indices.map(i => (i+1).toDouble / values.length).toArray
for (n <- 0 until 5) {
var df = spark.sparkContext.makeRDD(values).toDF("value").repartition(5)
val all_answers = df.stat.approxQuantile("value", all_quantiles, 0.1)
val all_answered_ranks = all_answers.map(ans => values.indexOf(ans)).toArray
val error = all_answered_ranks.zipWithIndex.map({ case (answer, expected) => Math.abs(expected - answer) }).toArray
val max_error = error.max
print(max_error + "\n")
}
```
In the current build it returns:
```
16
12
10
11
17
```
I couldn't run the code with this patch applied to double check the implementation. Can someone please confirm it now outputs at most `10`, please?
### How was this patch tested?
A new unit test was added to uncover the previous bug.
Closes#26029 from sitegui/SPARK-29336.
Authored-by: Guilherme <sitegui@sitegui.com.br>
Signed-off-by: Sean Owen <sean.owen@databricks.com>
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