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Suppose I have a 2D array M[n][n] of integers (in fact, binary is fine, but I doubt it matters). I am interested in repeated queries of the form: given a coordinate pair $k,l$, what is $$ \sum_{i = 0}^{k-1} \sum_{j = 0}^{l-1} M[i][j]? $$ Of course, all these values can be computed in $\mathcal O(n^2)$ time total, and after that queries take $\mathcal O(1)$. However, my array is mutable, and each time I change a value, the obvious solution requires a $\mathcal O(n^2)$ update.

We can create a quad tree over M; the preprocessing takes $\mathcal O(n^2\log(n))$, and this allows us to do queries in $\mathcal O(n\log(n))$, and updates in $\mathcal O(\log(n))$.

My question is:

Can we improve significantly on the queries without sacrificing too much on the updates?

I am especially interested in getting both the update and query operations sub-linear, and in particular getting them both to $\mathcal O(n^\epsilon)$.

Edit: for some more information, although I think the problem is interesting even without this further restriction, I expect to do roughly $\mathcal O(n^3)$ queries, and about $\mathcal O(n^2)$ updates. The ideal goal is to get the full runtime down to about $\mathcal O(n^{3+\epsilon})$. Thus, a situation where an update takes $\mathcal O(n \log(n))$ while a query takes $\mathcal O(\log(n))$ would also be interesting to me.

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    $\begingroup$ With a 2D Fenwick tree (a.k.a. BIT tree), you can get both updates and queries in $O(\log n)$ time. There's a page describing it here: geeksforgeeks.org/…. Disclaimer: I haven't fully understood how this works myself. $\endgroup$ – j_random_hacker Mar 1 '18 at 13:16
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There is a relatively straightforward solution where each query and each update can be done in $O(\log^2 n)$ time. The data structure uses $O(n^2)$ space.

We will have $\lg n$ "granularities" of data structure, one for each power of two $2^m$ such that $1 \le 2^m \le n$. The data structure for granularity $2^m$ stores the sums

$$\sum_{i=k_0 \cdot 2^m}^{(k_0+1) \cdot 2^m -1} \sum_{j=0}^{l-1} M[i,j]$$

for each $k_0,l$. This data structure for granularity $2^m$ can in turn be represented using $n/2^m$ balanced trees (one for each possible value of $k_0$) to store prefix sums.

Now to look up the prefix sum for $k,l$, we break up the interval $[0,k-1]$ into a union of intervals of power-of-two length; at most $\lg n$ intervals are needed. For each such interval of length $2^m$, we do a lookup into the data structure of granularity $2^m$. Thus queries can be answered by doing $O(\log n)$ lookups into a balanced tree, each of which takes $O(\log n)$ time, for a total time of $O(\log^2 n)$ per query.

Updates can also be done in $O(\log^2 n)$ time. To update $M[i,j]$, for each granularity $2^m$, you update the appropriate balanced tree in the data structure of granularity $2^m$. This is $O(\log n)$ updates oin $O(\log n)$ balanced trees; each such takes $O(\log n)$ time, so the total time is $O(\log^2 n)$ time.

Finally, the data structure of granularity $2^m$ contains $n/2^m$ trees, each taking up $O(n)$ space, so the total space usage is $O(n^2 \cdot (1 + 1/2 + 1/4 + \cdots)) = O(n^2)$.

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