In `reduce` and `aggregate`, the driver node spends linear time on the number of partitions. It becomes a bottleneck when there are many partitions and the data from each partition is big.

SPARK-1485 (#506) tracks the progress of implementing AllReduce on Spark. I did several implementations including butterfly, reduce + broadcast, and treeReduce + broadcast. treeReduce + BT broadcast seems to be right way to go for Spark. Using binary tree may introduce some overhead in communication, because the driver still need to coordinate on data shuffling. In my experiments, n -> sqrt(n) -> 1 gives the best performance in general, which is why I set "depth = 2" in MLlib algorithms. But it certainly needs more testing.

I left `treeReduce` and `treeAggregate` public for easy testing. Some numbers from a test on 32-node m3.2xlarge cluster.

code:

~~~
import breeze.linalg._
import org.apache.log4j._

Logger.getRootLogger.setLevel(Level.OFF)

for (n <- Seq(1, 10, 100, 1000, 10000, 100000, 1000000)) {
  val vv = sc.parallelize(0 until 1024, 1024).map(i => DenseVector.zeros[Double](n))
  var start = System.nanoTime(); vv.treeReduce(_ + _, 2); println((System.nanoTime() - start) / 1e9)
  start = System.nanoTime(); vv.reduce(_ + _); println((System.nanoTime() - start) / 1e9)
}
~~~

out:

| n | treeReduce(,2) | reduce |
|---|---------------------|-----------|
| 10 | 0.215538731 | 0.204206899 |
| 100 | 0.278405907 | 0.205732582 |
| 1000 | 0.208972182 | 0.214298272 |
| 10000 | 0.194792071 | 0.349353687 |
| 100000 | 0.347683285 | 6.086671892 |
| 1000000 | 2.589350682 | 66.572906702 |

CC: @pwendell

This is clearly more scalable than the default implementation. My question is whether we should use this implementation in `reduce` and `aggregate` or put them as separate methods. The concern is that users may use `reduce` and `aggregate` as collect, where having multiple stages doesn't reduce the data size. However, in this case, `collect` is more appropriate.

Author: Xiangrui Meng <meng@databricks.com>

Closes #1110 from mengxr/tree and squashes the following commits:

c6cd267 [Xiangrui Meng] make depth default to 2
b04b96a [Xiangrui Meng] address comments
9bcc5d3 [Xiangrui Meng] add depth for readability
7495681 [Xiangrui Meng] fix compile error
142a857 [Xiangrui Meng] merge master
d58a087 [Xiangrui Meng] move treeReduce and treeAggregate to mllib
8a2a59c [Xiangrui Meng] Merge branch 'master' into tree
be6a88a [Xiangrui Meng] use treeAggregate in mllib
0f94490 [Xiangrui Meng] add docs
eb71c33 [Xiangrui Meng] add treeReduce
fe42a5e [Xiangrui Meng] add treeAggregate