# A quick barycentric evaluation tutorial
One powerful technique when working with polynomials is taking a set of evaluations of that polynomial and using that _directly_ to compute an evaluation at a different point. For example, if $P(x)$ is a degree-100 polynomial, you can use the evaluations $P(0), P(1) ... P(100)$ to directly compute $P(101)$, or $P(1247130)$, in $O(N)$ time, without ever reconstructing the polynomial. This post describes how this is done.
_See also, this earlier and more detailed paper by Oxford researchers on the topic: https://people.maths.ox.ac.uk/trefethen/barycentric.pdf_
## General technique
Let $P(x)$ be the polynomial, and $x_1 ... x_N$ as the set of points for which we have evaluations of $P(x)$. Call these evaluations $y_1 ... y_N$.
Think of $P(x)$ as a linear combination $\sum_i y_i L_i(x)$, where $L_i(x)$ is the polynomial that equals 1 at $x_i$ and 0 at all the other x-coordinates in the set.
Now, let us explore how these $L_i$ can be computed. Each $L_i(x)$ can be expressed as:
$$\frac{(x - x_1)(x - x_2)...(x - x_{i-1})(x - x_{i+1}) ... (x - x_N)}{(x_i - x_1)(x_i - x_2)...(x_i - x_{i-1})(x_i - x_{i+1}) ... (x_i - x_N)}$$
To see why this is true, notice that the numerator of the fraction is a polynomial that equals 0 at every evaluation point except $x_i$, and the denominator is just whatever the numerator equals at $x_i$. Hence, the result equals 0 at every evaluation point except $x_i$, and 1 at $x_i$.
Let us denote this denominator $d_i$. Let us also denote the product $(x - x_1)(x - x_2) ... (x - x_N)$ (_without_ skipping any indices) $M(x)$. We can now express $L_i(x)$ as:
$$\frac{M(x)}{d_i(x - x_i)}$$
The $d_i$ values take $O(N^2)$ time to compute (because each one takes $O(N)$ time and there are $N$ of them), but their values depend only on the evaluation points, not the specific polynomial. Hence, they can be precomputed and stored and don't need to be re-computed each time. $M(x)$ takes $O(N)$ time to compute at any specific point. Given these values, each $L_i(x)$ can be computed in $O(N)$ time. Hence, the expression:
$$P(x) = M(x) * \sum_i \frac{y_i}{d_i(x - x_i)}$$
can be computed in $O(N)$ time.
## Special case: roots of unity
In many applications of barycentric evaluation, we can choose the evaluation points. One very convenient choice is the set $\{1, \omega, \omega^2 ... \omega^{N-1}\}$, for an $\omega$ where $\omega^N = 1$ and $N$ is a power of two. The main advantage of this choice is that it removes the need to pre-compute the $d_i$ values, because there is a simple closed-form value for them.
To see why, let us try to simplify the expression for $d_i$:
$$\prod_{0 \le j \ne i < N} (\omega^i - \omega^j)$$
Notice that $\omega^{\frac{N}{2}} = -1$. We can re-express the above as:
$$2 \omega^i * \prod_{0 \le j \ne i < \frac{N}{2}} (\omega^i - \omega^j)(\omega^i + \omega^j)$$
Each term inside the product groups together $(\omega^i - \omega^j)$ and $(\omega^i - \omega^{j + \frac{N}{2}})$, as $\omega^{j + \frac{N}{2}} = -\omega^j$. The term on the left is for $(\omega^i - \omega^{i + \frac{N}{2}})$, which simplifies to $2\omega^i$ for the same reason.
Now, notice that $(\omega^i - \omega^j)(\omega^i + \omega^j)$ combines into $(\omega^{2i} - \omega^{2j})$. So we have:
$$2 \omega^i * \prod_{0 \le j \ne i < \frac{N}{2}} (\omega^{2i} - \omega^{2j})$$
Repeat this, and we have:
$$2 \omega^i * 2 \omega^{2i} * \prod_{0 \le j \ne i < \frac{N}{4}} (\omega^{4i} - \omega^{4j})$$
Keep repeating, and we get:
$$2 \omega^i * 2 \omega^{2i} * 2\omega^{4i} * ... * 2\omega^{\frac{N}{2}i}$$
The product term on the right falls away because the product eventually becomes empty. The remaining term simply becomes $N * \omega^{(N-1)i}$, or:
$$d_i = \frac{N}{\omega^i}$$
Additionally, note that $M(x) = \prod_i (x - \omega^i)$ simply becomes $x^N - 1$, so the entire calculation simplifies down to:
$$P(x) = \frac{x^N - 1}{N} * \sum_i \frac{y_i * \omega^i}{x - \omega^i}$$

WIP The purpose of this document is to describe in more detail a proposal for how phase 1 can be structured based on a "data-availability-focused" approach. The main addition to the beacon chain will be a Vector of ShardDataHeader objects, one for each shard. A ShardDataHeader is a small object which represents a large amount of underlying data (roughly 0-512 kB in size). A block is only valid if the underlying data that the ShardDataHeader points to is available - that is, it has been published to the network and anyone can download it. However, to preserve scalability, clients will not try to download the full underlying data of every ShardDataHeader to verify the block. Instead, they will verify that the data is available using an indirect technique called data availability sampling. Parameters Parameter Value Description

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10/26/2022
Published on ** HackMD**