Hello everyone!

While I am solving this problem 1967C - Fenwick Tree, I read Elegia matrix solution and finally came up with my own solution that is based on his approach, using the matrix of linear transformation. Throughout the solution is tons of "heavy-math" knowledges and assumptions that is considered to be hard to see. Motivated enough, I have my heart set on writing an entire blog here with an ambition to prove every details in my solution.

Let's first quote the problem here

Problem Statement

Let $$$\operatorname{lowbit}(x)$$$ denote the value of the lowest binary bit of $$$x$$$, e.g. $$$\operatorname{lowbit}(12)=4, \operatorname{lowbit}(8)=8.$$$

For an array $$$a$$$ of length $$$n$$$, if an array $$$s$$$ of length $$$n$$$ satisfies $$$s_k=\left(\sum\limits_{i=k-\operatorname{lowbit}(k)+1}^{k}a_i\right)\bmod 998\,244\,353$$$ for all $$$k$$$, then $$$s$$$ is called the Fenwick Tree of $$$a$$$. Let's denote it as $$$s=f(a).$$$

For a positive integer $$$k$$$ and an array $$$a$$$, $$$f_k(a)$$$ is defined as follows:

You are given an array $$$b$$$ of length $$$n$$$ and a positive integer $$$k$$$. Find an array $$$a$$$ that satisfies $$$0\leq a_i <998244353$$$ and $$$f_k(a)=b$$$. It can be proved that an answer always exists. If there are multiple possible answers, you may print any of them.

Solution

The definition of $$$s_k$$$ makes us think of the linear map $$$T: \mathbb{R_n}\rightarrow \mathbb{R_n}$$$ which map a vetor array $$$a$$$ to the Fenwick Tree of $$$a$$$, in other words $$$T(a)=f(a)$$$. Note that $$$R_n$$$ here is the set of all matrices with size $$$n\times 1$$$ and if $$$a \in \mathbb{R_n}$$$ then

From the definition of $$$s_k$$$ we can see that the matrix $$$T$$$ of the linear map is

and satisfy

For example, with $$$n=10$$$, we have

Because $$$T$$$ is an lower triangular matrix, we have $$$\det T = 1$$$ for all $$$n\geq 1$$$ and the characteristicpolynomial of $$$T-I$$$ is

So all the eigenvalues of $$$T-I$$$ are $$$0$$$, $$$T-I$$$ is nilpotent and according to the Cayley-Hamilton theorem, we have

However,all of these above observations are just not enough to solve this problem, as will be seen later on. Indeed, we need to find the smallest integer $$$m$$$ such that

We claim that $$$m = \lfloor\log_2(k) \rfloor +1$$$