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I read somewhere that you can compute range queries in O(1) with O(N log N) preprocessing for any associative operation (but not necessarily invertible or idempotent). How do you do this?

I know this is possible for range min and range sum. But both of these are special because min is idempotent and sum is invertible.

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Sparse Tables

For each index i of array, we store minimum of blocks of 2 powers till block index's doesn't exceed array length.

Any non-negative number can be uniquely represented as a sum of decreasing powers of two. This is just a variant of the binary representation of a number.

E.g. $13 = (1101)_2 = 8 + 4 + 1$.

E.g. $[2, 14] = [2, 9] \cup [10, 13] \cup [14, 14]$,

https://discuss.codechef.com/t/tutorial-disjoint-sparse-table/17404

https://cp-algorithms.com/data_structures/sparse-table.html

https://github.com/e-maxx-eng/e-maxx-eng/blob/master/src/data_structures/sparse-table.md

Edit: Updating with codechef's link for handling general associativity, thanks D.W.

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