One possible data structure to store the set P is a balanced binary search tree. We can assume that the binary search tree is sorted in ascending order.
To compute R(a, b), we can start at the root of the tree and traverse down the tree as follows:
- If the current node's value is less than a, then traverse to the right subtree.
- If the current node's value is greater than b, then traverse to the left subtree.
- If the current node's value is between a and b (inclusive), then increment a counter and traverse to both the left and right subtrees.
At the end of the traversal, the counter will contain the value of R(a, b).
The time complexity of this algorithm is O(log n), since at each step we are discarding half of the remaining tree to search.
To construct the binary search tree from the set P, we can use an algorithm like binary search tree insertion. The time complexity of this algorithm is O(n log n) in the worst case, since each insertion operation takes O(log n) time and we need to perform n insertions. However, the actual time complexity may be less than O(n log n) if the input set P is already sorted or has some other favorable property.
