Is it possible to perform a range addition update, adding a linear function to a max-segment tree?

Question

I was thinking about this and was encountering a lot of bugs while trying to do this. Is it possible?

Answer 1

I believe you are asking whether you can do the following update: Given the update A B C, you want to add C to all elements from A to B.

The problem is to do the update on the segment tree would normally take O(N * logN) time given that N is the maximum number of elements. However, the key idea about implementing a segment tree is that you would like to suppose range queries and that normally you are not interested in all O(N^2) ranges but rather a much smaller subset of them.

You can enhance the range updates using lazy propagation which would generally mean that you do an update, but you do not update all of the nodes in the segment tree -> you update to some point, but you do not continue futher down the tree since it not needed.
Say that you have updated everything up to a node K which is responsible for the range [10; 30] for example. Later, you do a "get info" query on [20;40]. Obviously, you will have to visit node K, but you are not interested on the whole range, but rather the range [20;30], which is actually its right child.
What you have to do is "push" the update of node K to its left child and then its right child and continue as needed.

In general this would mean that when doing an update, you will do an update only until you find a suitable node(s) for the interval you update, but not go any further. This yields O(logN) time.
Then, when quering you continue propagating the updates down the tree when you reach a node for which you know you have saved some update for later. This also yields O(logN) time.

Some good material: Lazy propagation on segment trees