Frequency of each element of an array considering all contiguos subarrays

Question

Consider an array A = [5,1,7,2,3]

All contiguos subarrays = { [5], [1], [7], [2], [3], [5,1], [1,7], [7,2], [2,3], [5,1,7], [1,7,2], [7,2,3], [5,1,7,2], [1,7,2,3],

Answer 1

Traverse your value-to-index map from bottom to top - maintain an augmented tree of intervals. Each time an index is added, adjust the appropriate interval and calculate the total from the relevant segment:

A = [5,1,7,2,3] => {1:1, 2:3, 3:4, 5:0, 7:2}

indexes     interval     total sub-arrays with maximum exactly
1           (1,1)        1 =>           1
1,3         (3,3)        2 =>           1 
1,3,4       (3,4)        3 =>           2
1,3,4,0     (0,1)        5 =>           2
1,3,4,0,2   (0,4)        7 => 3 + 2*3 = 9

Insertion and deletion in augmented trees are of O(log n) time-complexity. Worst-case total time-complexity is O(n log n).