# Hsin-Po Wang Math PhD @ UIUC

From the oldest to the newest.

Abbreviation Title
MoulinAlge20 Multilinear Algebra for Distributed Storage
Atrahasis20 Multilinear Algebra for Minimum Storage Regenerating Codes
PlutoCharon20 Parity-Checked Strassen Algorithm

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.

They satisfy 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 restrict where those parameters can live.

The opposite approach is to construct regenerating codes that aim to achieve low $\alpha$ and $\beta$ 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. It is conjectured that this trade-off is optimal.

See Figure 1 on page 3 in MoulinAlge20 for an example of $\alpha/M$-versus-$\beta/M$ trade-off. 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. PlutoCharon20 concerns distributed computation. To be precise, it deals with distributed matrix-matrix multiplication (MMM) where the workers might crash or straggle. By MMM we mean that we want to compute $C=AB$, where $A, B$ are huge matrices. By crashing and straggling we mean that an entry multiplication, for instance $A_{12}\times A_{23}$, might be available very late, if at all.

To compensate, one needs to hire more-than-necessary workers and asks them to do redundant computations. A possibility to generate redundancy is to draw random vectors $g, h$ and then ask extra workers to compute $(gA)\times(Bh)$ on top of $A\times B$.

The contribution of PlutoCharon20 is three-fold. One: We obverse that the computation of $A\times B$ can be carried-out by fast matrix multiplication (FMM). This construction is named Pluto codes. Two: Applying Pluto codes recursively, we obtain a code that behaves like tensor product codes. Three: The computation of $(gA)\times(Bh)$, if $g, h$ are matrices, can be carried-out by FMM. This is named Charon construction.