Problem:
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The Number of Inversions

For a given sequence $A = \{a_0, a_1, ... a_{n-1}\}$, the number of pairs $(i, j)$ where $a_i > a_j$ and $i < j$, is called the number of inversions. The number of inversions is equal to the number of swaps of Bubble Sort defined in the following program:

bubbleSort(A)
cnt = 0 // the number of inversions
for i = 0 to A.length-1
for j = A.length-1 downto i+1
if A[j] < A[j-1]
swap(A[j], A[j-1])
cnt++

return cnt

For the given sequence $A$, print the number of inversions of $A$. Note that you should not use the above program, which brings Time Limit Exceeded.

Input

In the first line, an integer $n$, the number of elements in $A$, is given. In the second line, the elements $a_i$ ($i = 0, 1, .. n-1$) are given separated by space characters.

output

Print the number of inversions in a line.

Constraints

$ 1 \leq n \leq 200,000$

$ 0 \leq a_i \leq 10^9$

$a_i$ are all different

Sample Input 1

5
3 5 2 1 4

Sample Output 1

6

Sample Input 2

3
3 1 2

Sample Output 2

2

Solution in Scala (judge verdict: Accepted)

import scala.io.StdIn.{readLine,readInt}
import scala.language.postfixOps

object Main {
  var cnt:Long = 0
  val l = new Array[Int](100002)
  val r = new Array[Int](100002)

  def merge(a:Array[Int],left:Int,mid:Int,right:Int) = {
    val n1 = mid-left
    val n2 = right-mid
    Array.copy(a,left,l,0,n1)
    l(n1) = Int.MaxValue
    Array.copy(a,mid,r,0,n2)
    r(n2) = Int.MaxValue
    var i=0
    var j=0
    for(k<-left until right) {
      if(l(i)<=r(j)){
        a(k) = l(i)
        i += 1
      } else  {
        a(k)=r(j)
        cnt = cnt+(n1-i).toLong
        j += 1
      }
    }
  }

  def scan(h:Int,l:List[IndexedSeq[(Int,Int)]]):List[IndexedSeq[(Int,Int)]] =
    if(h<=0) l
    else scan(h-1,(for(i<-l.head;j<-List((i._1,(i._2+i._1)/2),((i._2+i._1)/2,i._2))) yield j)::l)

  def getSortIndex(n:Int):Iterator[IndexedSeq[(Int,Int)]] = {
    var w = n
    var h = 0
    while(w>1) {
      h += 1; w /= 2
    }
    scan(h-1,List(IndexedSeq((0,n/2),(n/2,n)))).to[Iterator]
  }

  def main(args:Array[String]) = {
    val n = readInt
    val a = readLine.split(" ").map(_.toInt)
    val l = getSortIndex(n)
    for(e2<-l) {
      for(e<-e2) {
        val mid = (e._2+e._1)/2
        if(e._2-e._1>1)
          merge(a,e._1,mid,e._2)
      }
    }
    merge(a,0,n/2,n)
    println(cnt)
  }
}