# Hsin-Po's Website

# Distributed Storage Comments

The following are my works on distributed storage systems.

Abbreviation | Title |
---|---|

MoulinAlge20 | Multilinear Algebra for Distributed Storage |

Atrahasis20 | Multilinear Algebra for Minimum Storage Regenerating Codes |

Both MoulinAlge20 and Atrahasis20 concern regenerating codes that have applications in distributed storage systems.

A **regenerating code** consists of

- a file of size $M$ symbols and
- a system of $n$ storage devices, called
**nodes**.

The configuration of the nodes satisfies the following conditions:

- Each node stores $\alpha$ symbols of the file.
- Any $k$ nodes contains sufficient information to recover the file.
- When a node fails, any $d$ other nodes will each sends it $\beta$ symbols to repair.

The code is named regenerating mainly due to the last bullet point—the nodes regenerate themselves.

The theory of regenerating codes concerns
the relation among $n, k, d, \alpha, \beta, M$.
For example, since any $k$ nodes contain $k\alpha$ symbols
and can recover the file, the file size $M$ is at most $k\alpha$.
Similarly, since $d\beta$ symbols repair a failing node,
the node size $\alpha$ is at most $d\beta$.
(Exercise)
One can also show that $k - 1$ nodes ($\alpha$)
plus $d - k + 1$ help messages ($\beta$) is at least $M$.
There is a family of bounds of this type.
They are called *cut-set bounds* and restrict where those parameters can live.

The opposite approach is to construct regenerating codes
that aim to achieve low $\alpha$, low $\beta$, and high $M$.
MoulinAlge20 utilizes multilinear algebra to do this.
We construct a series of regenerating codes which we call **moulin codes**.
They achieve the best known $\alpha/M$-versus-$\beta/M$ trade-off to date.
And it is conjectured that this trade-off is optimal.

See Figure 1 on page 3 in MoulinAlge20 for the $\alpha/M$-versus-$\beta/M$ trade-off for the $(n, 3, 3)$ case. Here is another $\alpha/M$-versus-$\beta/M$ trade-off for the $(n, 3, 4)$ case. (In a newer version of MoulinAlge20 that I am still working on.)

See also Table 2 on page 29 for the relations among some competitive constructions.

Atrahasis20 exploits multilinear algebra to construct MSR codes,
which we called **Atrahasis codes**.
Formally, an **MSR code** is a regenerating code
with $M = k\alpha$ and $\beta = \alpha/(d - k + 1)$.
From the constraint on $M$ one sees that there is no wastes of storage
(hence the name **minimum storage regeneration** = MSR).
Some researchers see MSR codes as the intersection of regenerating codes
and MDS codes.

MSR alone attracts significant attentions because people want to minimize node size ($\alpha \geq M/k$), and only then they minimize help messages ($\beta \geq \alpha/(d - k + 1)$ given that $\alpha \geq M/k$). See Table 1 on page 5 in Atrahasis20 for a comparison of some existing contraptions.

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