Sum of all Maximum and minimum in all segments - c++

Question

I am doing a problem in which there is a need to find sum of maximum elements in a segment - sum of minimum elements in a segment.I tried using Sparse Table ,but it is two slow for the time limit.So i did something like this:
If n=4 segments are [1,2],[1,3],[1,4],[2,3],[2,4],[3,4].
The problem is similar to an RMQ problem but i have to do it for all segments and find the
sum=max(a[1],a[2])+
max(a[1],a[2],a[3])+max(a[1],a[2],a[3],a[4])+max(a[2],a[3])+m‌​ax(a[2],a[3],a[4])+max(a[3],a[4])-min(a[1],a[2])+min(a[1],a[2],a[3])+min(a[1],a[2‌​],a[3],a[4])+min(a[2],a[3])+min(a[2],a[3],a[4])+min(a[3],a[4])
for(i=1;i<n;i++)
{
maxtilli[i-1]=INT_MIN;
mintilli[i-1]=INT_MAX;
for(k=1,j=i;j<=n;k++,j++)
{
if(a[j]>maxtilli[k-1])
{
maxtilli[k]=a[j];
}
else
{
maxtilli[k]=maxtilli[k-1];
}
if(a[j]<mintilli[k-1])
{
mintilli[k]=a[j];
}
else
{
mintilli[k]=mintilli[k-1];
}
if(i!=j)
{
ans+=(maxtilli[k]-mintilli[k]);
}
}
}
Here n is of the order of 100,000. So is there any way to optimize it.
Suppose n=4 then segments are [1,2],[1,3],[1,4],[2,3],[2,4],[3,4].
The thing required is
sum=max(a[1],a[2])+max(a[1],a[2],a[3])+max(a[1],a[2],a[3],a[4])+max(a[2],a[3])+m‌​ax(a[2],a[3],a[4])+max(a[3],a[4])-min(a[1],a[2])+min(a[1],a[2],a[3])+min(a[1],a[2‌​],a[3],a[4])+min(a[2],a[3])+min(a[2],a[3],a[4])+min(a[3],a[4])

Answer 1

We can try to finish the first problem, sum of the max value in all segments.
Algorithm
First, you can find the max value a[i] in the whole sequence. All segments which contain a[i] would be considered. The answer plus A[i] * (i * (n - i)). And the problem is split into two small sequences [1, i - 1] and [i + 1， n], you can do it in the same way.
Code
void cal(int L, int R){
max_index = find_max(L, R); // O(logN), using Sparse Table or Segment Tree
int all_segments = (max_index - L + 1) * (R - max_index)
ans += a[max_index] * all_segments;
cal(L, max_index - 1);
cal(max_index + 1, R);
}
// call max_index N times, so the total complexity is O(N * logN)