CSE5313 Computer Vision (Lecture 16: Exam Review)

Exam Review

Information flow graph

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.

Graph construction

Source: System admin.

Sink: Data collector.

Nodes: Storage servers.

Edges: Represents transmission of information. (Number of \mathbb{F}_q elements is weight.)

Main observation:

k elements (number of servers required to reconstruct the file) The message size is B. 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.

Bounds for local recoverable codes

Turan's Lemma

Let G be a graph with n vertices. Then there exists an induced directed acyclic subgraph (DAG) of G on at least \frac{n}{1+\operatorname{avg}_i(d^{out}_i)} nodes, where d^{out}_i is the out-degree of vertex i.

Bound 2

Consider the induced acyclic graph G_U on U nodes.

By the definition of $r$-locally recoverable code, each leaf node in G_U must be determined by other nodes in G\setminus G_U, so we can safely remove all leaf nodes in G_U and the remaining graph is still a DAG.

Let N\subseteq [n]\setminus U be the set of neighbors of U in G.

|N|\leq r|U|\leq k-1.

Complete n to be of the size k-1 by adding elements not in U.

|C_N|\leq q^{k-1}

Also |N\cup U'|=k-1+\lfloor\frac{k-1}{r}\rfloor

All nodes in G_U can be recovered from nodes in N.

So |C_{N\cup U'}|=|C_N|\leq q^{k-1}.

Therefore, \max\{|I|:C_I<q^k,I\subseteq [n]\}\geq |N\cup U'|=k-1+\lfloor\frac{k-1}{r}\rfloor.

Using reduction lemma, we have d= n-\max\{|I|:C_I<q^k,I\subseteq [n]\}\leq n-k-1-\lfloor\frac{k-1}{r}\rfloor+2=n-k-\lceil\frac{k}{r}\rceil +2.

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