Concepts5

Sqrt Tree

A sqrt tree is a layered block-decomposition data structure that answers range queries in O(1) time after O(n \log \log n) preprocessing.

#sqrt tree#range query#associative operation+11

∑MathIntermediate

Harmonic Lemma

The Harmonic Lemma says that the values of \lfloor n/i \rfloor only change about 2\sqrt{n} times, so you can iterate those value blocks in O(\sqrt{n}) instead of O(n).

#harmonic lemma#integer division trick#block decomposition+12

∑MathAdvanced

Divisor Function Sums

Summing the divisor function d(i) up to n equals counting lattice points under the hyperbola xy ≤ n, which can be done in O(√n) using floor-division blocks.

#divisor function#euler totient#mobius function+11

⚙️AlgorithmIntermediate

Mo's Algorithm

Mo's algorithm answers many range queries offline by reordering them to minimize pointer movement along the array.

#mo's algorithm#offline queries#range queries+12

🗂️Data StructureIntermediate

Sqrt Decomposition

Sqrt decomposition partitions an array into about \(\sqrt{n}\) blocks, each of size about \(\sqrt{n}\), to speed up range queries and updates.

#sqrt decomposition#block decomposition#bucket decomposition+11