Treaps, Segment Tree, 2d trees
In 1D, this can be solved by a fairly straightforward use of a segment tree: each node stores the GCD of its two children. Since R can be quite big, this needs to be a sparse segment tree; another alternative would be a balanced binary tree.
In 2D, it is tempting to use a quadtree, but in fact that doesn’t guarantee poly-logarithmic time. A 1xC query will force refinement down to the individual non-zero cell entries. Instead, we can use a range tree, which is a tree of trees: an outer tree is built over the columns, and for each column span corresponding to a node in this tree, we have an inner tree over the rows. Each node in this inner tree corresponds to a rectangle of the grid, and stores the GCD of elements from this rectangle. A query now uses the columns to select O(log C) nodes from the outer tree i.e., O(log C) inner trees, and applies a query to each of them. Queries thus take O(log R·log C) time when the implementation uses segment trees for the inner and outer trees. With balanced binary trees, it would only be O((log Nu)2).
Unfortunately, using segment trees also requires O(log R·log C) memory per non-zero element, which just exceeds the available memory. Using balanced binary trees instead should fit within memory, but is a lot more painful to implement. I think it might also be possible to make it work by sticking with segment trees, but compressing the representation by compacting chains of nodes with one child.
source : http://blog.brucemerry.org.za/2013/07/ioi-2013-day-2-analysis.html
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