Segment Tree Modify

Problem

For a Maximum Segment Tree, which each node has an extra value max to store the maximum value in this node's interval.

Implement a modify function with three parameter root, index and value to change the node's value with [start, end] = [index, index] to the new given value. Make sure after this change, every node in segment tree still has the max attribute with the correct value.

Example

For segment tree:

                          [1, 4, max=3]
                        /                \
            [1, 2, max=2]                [3, 4, max=3]
           /              \             /             \
    [1, 1, max=2], [2, 2, max=1], [3, 3, max=0], [4, 4, max=3]

if call modify(root, 2, 4), we can get:

                          [1, 4, max=4]
                        /                \
            [1, 2, max=4]                [3, 4, max=3]
           /              \             /             \
    [1, 1, max=2], [2, 2, max=4], [3, 3, max=0], [4, 4, max=3]

or call modify(root, 4, 0), we can get:

                          [1, 4, max=2]
                        /                \
            [1, 2, max=2]                [3, 4, max=0]
           /              \             /             \
    [1, 1, max=2], [2, 2, max=1], [3, 3, max=0], [4, 4, max=0]

Note

We suggest you finish problem Segment Tree Build and Segment Tree Query first.

Challenge

Do it in O(h) time, h is the height of the segment tree.