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Let $a_{1} , ... , a_{m} $ be real numbers $\geq 1$, where $m$ is at least 1. I am supposed to store them in an augmented AVL structure with the following operations:

-PartialSum (i): Return the $i_{th}$ partial sum.

-Change (i, y): Change the $a_{i}$ element for y.

I need this structure to retain the AVL properties and both of these functions to be in $O(logn)$. The problem is that my approach doesn't let me have that asymptotic bound on CHANGE.

My approach goes like this: Let i be the key, and also store the relative $i_{th}$ partial sum in each node, the value for a as well.

Thus each of my nodes has:

  1. Key = The $i_{th}$ element of the series input.
  2. Sum = The $i_{th}$ partial sum so far.
  3. Val = The value of $a_{i}$ for this node.

By doing that, the operation PartialSum will be identical to Search, thus being in $O(logn)$.

Unfortunately my approach doesn't work so well with change. By storing the partial sums in every node, if a lower node changes, all of the tree's sums are to be recomputed, being in $O(n)$.

I am trying to take a different approach but I'm confused. Can someone HINT me here?

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Instead of storing the whole $i_{th}$ sum in each node, just store the partial sum of the left sub tree.

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  • $\begingroup$ Alternatively, in each node, store the sum of the elements in that subtree. $\endgroup$ – Pseudonym Feb 3 '15 at 7:15

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