Little tricks

Revision en2, by adamant, 2022-07-03 05:02:53

Hi everyone!

Here's another collection of little tricks and general ideas that might make your life better. Some of them might be well-known to many community members, but I hope they will be useful for someone as well.

Evaluating polynomial modulo $$$p$$$ in all points in $$$O(p \log p)$$$. You can Evaluate $$$P(x)$$$ in every possible $$$x$$$ modulo $$$p$$$ as $$$P(0), P(g^0), P(g^1), \dots, P(g^{p-2})$$$ with chirp Z-transform, where $$$g$$$ is a primitive root modulo $$$p$$$.

Generalized Euler theorem. Let $$$a$$$ be a number, not necessarily co-prime with $$$m$$$, and $$$k > \log_2 m$$$. Then

$$$ a^k \equiv a^{\phi(m) + k \mod \phi(m)} \pmod m, $$$

where $$$\phi(m)$$$ is Euler's totient. This follows from the Chinese remainder theorem, as it trivially holds for $$$m=p^d$$$.

Range queries on a plane with Fenwick tree. Fenwick tree, generally, allows for range sum/point add queries.

Let $$$s_{xy}$$$ be a sum on $$$[1,x] \times [1,y]$$$. If we add $$$c$$$ on $$$[a, n] \times [b, m]$$$, the sum $$$s_{xy}$$$ would change as

$$$ s_{xy} \mapsto s_{xy} + (x-a+1)(y-b+1)c, $$$

for $$$x \geq a$$$ and $$$y \geq b$$$. To track these changes, we may represent $$$s_{xy}$$$ as

$$$ s_{xy} = s_{xy}^{(0)}+ x \cdot s_{xy}^{(x)} + y \cdot s_{xy}^{(y)} + xy \cdot s_{xy}^{(xy)}, $$$

which allows us to split the addition of $$$c$$$ on $$$[a,n] \times [b,m]$$$ into additions in $$$(a;b)$$$:

$$$\begin{align} s_{xy}^{(0)} &\mapsto s_{xy}^{(0)} + (a-1)(b-1)c, \\ s_{xy}^{(x)} &\mapsto s_{xy}^{(x)} - (b-1)c, \\ s_{xy}^{(y)} &\mapsto s_{xy}^{(y)} - (a-1)c, \\ s_{xy}^{(xy)} &\mapsto s_{xy}^{(xy)} + c. \end{align}$$$
code

The solution generalizes 1-dimensional Fenwick range updates idea from Petr blog from 2013.

Tags tutorial, i love tags

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