From 2d0dae44612bdc4ed07ae7f03932fc6cf74ace21 Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Thu, 16 Oct 2025 12:46:47 -0500 Subject: [PATCH] updates? --- content/CSE5313/CSE5313_L14.md | 297 ++++++++++++++++++ content/CSE5313/_meta.js | 1 + public/CSE5313/Storage_bandwidth_tradeoff.png | Bin 0 -> 133490 bytes 3 files changed, 298 insertions(+) create mode 100644 content/CSE5313/CSE5313_L14.md create mode 100644 public/CSE5313/Storage_bandwidth_tradeoff.png diff --git a/content/CSE5313/CSE5313_L14.md b/content/CSE5313/CSE5313_L14.md new file mode 100644 index 0000000..da6498a --- /dev/null +++ b/content/CSE5313/CSE5313_L14.md @@ -0,0 +1,297 @@ +# CSE5313 Coding and information theory for data science (Lecture 14) + +## The repair problem + +Main challenge: + +- Locality (number of contacted servers) +- Bandwidth (number of bits transferred) + +From last lecture we build optimal Local Recoverable Codes (LRCs) for storage systems. + +Let $\mathcal{C} = [n, k]_q$ which is $r$-LRC, with minimum distance $d$. + +- Bound 1: $\frac{k}{n}\leq \frac{r}{r+1}$. +- Bound 2: $d\leq n-k-\frac{k}{r}+2$. + +- Optimal LRC: +- Let $\mathcal{A} = \{\alpha_1, \ldots, \alpha_n\}$, partition $\mathcal{A}$ to $\mathcal{A}_i$ for $i=1$ to $\frac{n}{r+1}$. +- $g\in \mathbb{F}_q[x]$ is good if $\deg(g) = r+1$ and $g$ is constant on all $\mathcal{A}_i$'s. +- $\mathcal{C} = \{f_a(\alpha_i)\}_{i=1}^{n}|a\in \mathbb{F}_q^k\}$, where + - $f_a(x) = \sum_{i=0}^{r-1} f_{a,i}(x)\cdot x^i$, + - $f_{a,i}(x) = \sum_{j=0}^{k/r-1} a_{i,j}\cdot g(x)^j$, where $g$ is a good polynomial. + - $g$ is a "good" polynomial. +- $\dim \mathcal{C} = k$ and $d = n-k-\frac{k}{r}+2$. + +## Minimizing the repair bandwidth + +Goal: understand repair bandwidth. + +- What is the minimum repair bandwidth? +- Is repair bandwidth in trade-off with other parameters? +- Tool: The information flow graph. + +Spoiler alert: + +- Tradeoff: Storage Repair and bandwidth. +- Codes which achieve an optimal tradeoff. + +### Information flow graph + +We can model the repair problem as a directed graph. + +- Source: System admin. +- Sink: Data collector. +- Nodes: Storage servers. + - Nodes leave/crash + - Newcomer replaces them $\to$ new nodes. +- Edges: Represents transmission of information. (Number of $\mathbb{F}_q$ elements is weight.) + +Main observation: + +- $k$ elements from $\mathbb{F}_q$ must "flow" from the source (system admin) to the sink (data collector). +- Any cut $(U,\overline{U})$ which separates source from sink must have capacity at least $k$. + +Roadmap: + +Information flow graph $\to$ Minimum cut analysis $\to$ Bound on file size $\to$ Storage/bandwidth tradeoff. + +#### Basic definitions for information flow graph + +> [!WARNING] +> +> This is not the same as definitions in linear codes. $k$ is not the dimension of the code and $d$ is not the minimum distance of the code for general cases. + +Parameters: + +- $n$ is the number of nodes in the initial system (before any node leaves/crashes). +- $k$ is the number of nodes required to reconstruct the file $k$. +- $d$ is the number of nodes required to repair a failed node. +- $\alpha$ is the storage at each node. +- $\beta$ is the edge capacity for repair. +- $B$ is the file size. + +Goal: Find the trade off between $n,k,d,\alpha,\beta,B$ using min-cut analysis of the information flow graph. + +Initial system: + +We denote the system admin as $S$ + +Sever as $1,2,\ldots,n$, each with edge capacity $\alpha$. + +For each new server: + +We have two nodes $in$ and $out$, the edge weight is $\alpha$. + +Connect to $d$ previous nodes $out$'s with edge capacity $\beta$. + +Data collector:connects to $k$ arbitrary nodes with each edge capacity $\alpha$. + +Observe that: + +- File size $B$. +- Any cut separating $S$ form $DC$ must have capacity at least $B$. +- Otherwise, two different files are indistinguishable to $DC$. + +#### Bound on bandwidth + +Claim: $mincut\geq \sum_{i=1}^{k-1}\min\{(d-i)\beta, \alpha\}$. + +Intuition: Let $(U, \overline{U})$ be a cut separating $S$ form $DC$. + +- The DC contacts newest $k$ nodes, say $n_1,n_2,\ldots,n_k$. +- The cut must decide if to cross $\alpha$ edges or $\beta$ edges. +- At east $d-i$ edges to go to $U$. + +
+ +Proof + +Let $(U, \overline{U})$ be a cut separating $S$ form $DC$, assuming $S\in U$ and $DC\in \overline{U}$. + +Every directed acyclic graph has **topological sort**. + +Let $x_{out}^1$ be the first $out$ node in $\overline{U}$. There are two cases: + +- $x_{in}^1\in U$. Then $\alpha$ edges must be crossed. +- $x_{in}^1\in \overline{U}$. Then all $d$ incoming edges to $x_{in}^1$ must be crossed. (Otherwise, there exists an earlier $out$ node $x_{out}^j$ with $x_{in}^j\in U$, contradicting the topological sort.) + +So $x_{out}^1$ contributes at least $\min\{d\beta, \alpha\}$ to the cut capacity. + +Let $x_{out}^2$ be the second $out$ node in $\overline{U}$. There are two cases: + +- $x_{in}^2\in U$. Then $\alpha$ edges must be crossed. +- $x_{in}^2\in \overline{U}$. Then at least $d-1$ incoming edges ($1$ edge may come from $x_{out}^1$) to $x_{in}^2$ must be crossed. (Otherwise, there exists an earlier $out$ node $x_{out}^j$ with $x_{in}^j\in U$, contradicting the topological sort.) + +So $x_{out}^2$ contributes at least $\min\{(d-1)\beta, \alpha\}$ to the cut capacity. + +By repeating this process, we can show that the minimum cut capacity is at least $\sum_{i=1}^{k-1}\min\{(d-i)\beta, \alpha\}$. + +
+ +#### Storage/bandwidth tradeoff + +Claim: There exists an information graph with $mincut = \sum_{i=1}^{k-1}\min\{(d-i)\beta, \alpha\}$. + +Homework: Build this graph as follows: + +- Construct the initial graph with $n$ nodes and the system admin. +- Add $n+k$ nodes, each node connects to the most recent $d$ nodes. +- Find the minimum cut capacity. + +Corollary: $B\leq \sum_{i=1}^{k-1}\min\{(d-i)\beta, \alpha\}$. + +#### Definition of regenerate code + +A code which attains $B=\sum_{i=1}^{k-1}\min\{(d-i)\beta, \alpha\}$ is called a regenerate code. + +Goal: Find tradeoff between storage $\alpha$ to repair bandwidth $d\beta$. + +Let $\gamma = d\beta$, then $B \leq \sum_{i=0}^{k-1}\min\{(1-i/d)\gamma, \alpha\}$. + +Tool: Fix $\gamma$ and $d$, and minimize for $\alpha$ (not shown). + +Result: The storage/bandwidth tradeoff. + +- Each point on/above the line is feasible. +- Points on the line = regenerating codes. +- One endpoint: Minimum Bandwidth Regenerating (MBR) codes. +- another endpoint: Minimum Storage Regenerating (MSR) codes + +![Storage/bandwidth tradeoff](https://notenextra.trance-0.com/CSE5313/Storage_bandwidth_tradeoff.png) + +For Minimum Storage Regenerating (MSR) codes, we have $\alpha = \frac{B}{k}$, $\beta = \frac{B}{k(d-k+1)}$ + +For Minimum Bandwidth Regenerating (MBR) codes, we have $\alpha = \frac{dB}{kd-\frac{k(k-1)}{2}}$, $\beta = \frac{B}{kd-\frac{k(k-1)}{2}}$ + +Notes: + +- In MSR $\alpha=B/k$, Data collector contacts $k$ nodes and downloads $B/k$ from each to reconstruct the file of size $B$, that is optimal. +- In MBR $\beta=B/(kd-\frac{k(k-1)}{2})$, new comer download exactly what it stores, which is the same as replication. This has much smaller storage overhead in the replication. + +Regenerating codes, Magic #1: + +- MBR: Same repair-bandwidth as replication ($\alpha$), at lower storage costs. +- MSR: Same reconstruction-bandwidth ($B/k$) as replication, at lower storage costs. + +Regenerating codes, Magic #2: + +- In MSR: $\gamma = d\beta = \frac{dB}{k(d-k+1)}$, $\alpha = \frac{B}{k}$ +- In MBR: $\gamma = d\beta = \frac{dB}{kd-\frac{k(k-1)}{2}}$, $\alpha = \frac{dB}{kd-\frac{k(k-1)}{2}}$ +- Both decreasing functions of $d$. +- $\Rightarrow$ Less repair-bandwidth by contacting more nodes, minimized at $d = n - 1$. + +### Constructing Minimum bandwidth regenerating (MBR) codes from Minimum distance codes + +Observation: For MBR code with parameters $n, k, d$ and $\beta = 1$, one can construct MBR with parameters $n, k, d$ and any $\beta$. + +Next: Construct MBR for $[n, k, d = n - 1]$ and $\beta = 1$. + +In any MBR: $\alpha, \beta = \frac{dB}{kd-\frac{k(k-1)}{2}}, \frac{B}{kd-\frac{k(k-1)}{2}}$ + +Specifically: + +- Storage $\alpha = d\beta = d = n - 1$. +- File size $B = kd - \binom{k}{2}\beta = kd - \binom{k}{2}$ + +Take an $[\binom{n}{2}, B]$ MDS code (e.g., Reed-Solomon). + +Need $q\geq \frac{n}{2}$. + +Consider a complete graph $K_n$ on $n$ nodes. + +- $\binom{n}{2}$ edges. +- Place each codeword symbol on a distinct edge. +- Storage server $i$ stores all codeword symbols adjacent with node $i$. + - $\alpha = n - 1$. + +#### Repairing on MBR codes + +New comer contacts each node $j\neq i$; + +And downloads the symbol on edge $(i,j)$. + +We get $\alpha=n-1=d\beta$ which is optimal. + +#### Reconstruction on MBR codes + +We use $[\binom{n}{2}, B]_q$ MDS code. So any $B$ symbols suffice to reconstruct the file. + +Any $t$ nodes have $\binom{t}{2}$ edges between them, and $(n-1)t-2\binom{t}{2}$ edges to other nodes. + +Overall $(n-1)t-\binom{t}{2}$. For $t=k$, we get $kd-\binom{k}{2}=B$. + +### Constructing Minimum bandwidth regenerating (MBR) codes from Product-Matrix MBR codes + +Recall: File size in MBR $B=kd-\binom{k}{2}=\binom{k+1}{2}+k(d-k)$. + +Step 1: Arrange the $B=\binom{k+1}{2}+k(d-k)$ symbols in a matrix $M$ follows: + +$$ +M=\begin{pmatrix} +S & T\\ +T^T & 0 +\end{pmatrix}\in \mathbb{F}_q^{d\times d} +$$ + +- $S$ is a $k\times k$ symmetric matrix. contains $\binom{k+1}{2}$ symbols. +- $T$ is a $k\times (d-k)$ matrix. contains $k(d-k)$ symbols. + +So there are $B$ elements overall. + +Step 2: Construct the encoding matrix $C=(\Psi,\Delta)\in \mathbb{F}_q^{n\times d}$ + +$\Psi\in \mathbb{F}_q^{n\times k}$ such that + +- Any $k$ rows are linearly independent. +- Example: Vandermonde matrix. + +$\Delta\in \mathbb{F}_q^{n\times (d-k)}$ such that + +- Any $d$ rows of $C$ are linearly independent. +- Example: Complete $\Psi$ to a full $n\times d$ Vandermonde matrix. + +Step 3: Encoding of the data $M\in \mathbb{F}_q^{d\times d}$ using the encoding matrix $C\in \mathbb{F}_q^{n\times d}$. + +- Multiply $M$ by $C$. +- Store the $i$ the row of $CM$ in the node $i$. +- Note $CM=(\Psi,\Delta)M=(\Psi S+\Delta T, \Psi T)$ + +#### Repairing on Product-Matrix MBR codes + +Assume node $i$ storing $c_iM$ is lost. + +Repair from (any) nodes $H = \{h_1, \ldots, h_d\}$. + +- Node $h_j$ stores $c_{h_j}M$. + +Newcomer contacts each $h_j$: “My name is $i$, and I’m lost.” + +Node $h_j$ sends $c_{h_j}M c_i^T$ (inner product). + +Newcomer assembles $C_H Mc_i^T$. + +$CH$ invertible by construction! + +- Recover $Mc_i^T$. + +- Recover $c_i^TM$ ($M$ is symmetric) + +#### Reconstruction on Product-Matrix MBR codes + +Data Collector (DC) contacts (any) nodes $D = \{d_1, \ldots, d_k\}$. + +- Node $d_j$ stores $c_{d_j}M$. + +Downloads $c_{d_j}M$ from node $d_j$. + +DC assembles $C_D M$. + +- Recall $CM=(\Psi S,\Delta)M=(\Psi S+\Delta T, \Psi T)$ +- $C_D M=(\Psi_D S,\Delta_D)M=(\Psi_D S+\Delta_D T, \Psi_D T)$ + +$\Psi_D$ invertible by construction. + +- DC computes $\Psi_D^{-1}C_DM = (S+\Psi_D^{-1}\Delta_D^T, T)$ +- DC obtains $T$. +- Subtracts $\Psi_D^{-1}\Delta_D T^T$ from $S+\Psi_D^{-1}\Delta_D T^T$ to obtain $S$. diff --git a/content/CSE5313/_meta.js b/content/CSE5313/_meta.js index c7a5b5b..c037cf0 100644 --- a/content/CSE5313/_meta.js +++ b/content/CSE5313/_meta.js @@ -16,4 +16,5 @@ export default { CSE5313_L11: "CSE5313 Coding and information theory for data science (Recitation 11)", CSE5313_L12: "CSE5313 Coding and information theory for data science (Lecture 12)", CSE5313_L13: "CSE5313 Coding and information theory for data science (Lecture 13)", + CSE5313_L14: "CSE5313 Coding and information theory for data science (Lecture 14)", } \ No newline at end of file diff --git a/public/CSE5313/Storage_bandwidth_tradeoff.png b/public/CSE5313/Storage_bandwidth_tradeoff.png new file mode 100644 index 0000000000000000000000000000000000000000..4e5efe1c38351107e9fcc686ff51149ada4892f2 GIT binary patch literal 133490 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