ecc3 - Why Expander Based Codes? 15-853:Algorithms in the...

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1 15-853 Page1 15-853:Algorithms in the Real World Error Correcting Codes III (expander based codes) –Expander ±graphs – Low density parity check (LDPC) codes – Tornado codes Thanks to Shuchi Chawla for many of the slides 15-853 Page2 Why Expander Based Codes? Linear codes like RS & random linear codes The other two give nearly optimal rates But they are slow : Assuming an (n, (1-p)n, (1- ε )pn+1) 2 tornado code *does not necessarily fix (d-1)/2 errors O(n) O(n 2 ) or better LDPC O(n log 1/ ε ) O(n log 1/ ε ) Tornado O(n 2 ) O(n log n) RS O(n 3 ) O(n 2 ) Random Linear Decoding* Encoding Code 15-853 Page3 Error Correcting Codes Outline Introduction Linear codes Read Solomon Codes Expander Based Codes – Expander Graphs – Low Density Parity Check (LDPC) codes – Tornado Codes 15-853 Page4 Expander Graphs (non-bipartite) Properties Expansion: every small subset ( k α n ) has many ( β k ) neighbors Low degree – not technically part of the definition, but typically assumed k α n β k G
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2 15-853 Page5 Expander Graphs (bipartite) Properties Expansion: every small subset ( k α n ) on left has many ( β k ) neighbors on right Low degree – not technically part of the definition, but typically assumed k bits (k α n) β kb its 15-853 Page6 Expander Graphs Useful properties: – Every set of vertices has many neighbors – Every balanced cut has many edges crossing it – A random walk will quickly converge to the stationary distribution (rapid mixing) – The graph has “high dimension” – Expansion is related to the eigenvalues of the adjacency matrix 15-853 Page7 Expander Graphs: Applications Pseudo-randomness : implement randomized algorithms with few random bits Cryptography : strong one-way functions from weak ones. Hashing: efficient n-wise independent hash functions Random walks: quickly spreading probability as you walk through a graph Error Correcting Codes: several constructions Communication networks: fault tolerance, gossip- based protocols, peer-to-peer networks 15-853 Page8 d-regular graphs An undirected graph is d-regular if every vertex has d neighbors. A bipartite graph is d-regular if every vertex on the left has d neighbors on the right. The constructions we will be looking at are all d- regular.
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3 15-853 Page9 Expander Graphs: Eigenvalues Consider the normalized adjacency matrix A ij for an undirected graph G (all rows sum to 1) The (x i , λ i ) satisfying A x i = λ i x i are the eigenvectors (x i ) and eigenvalues ( λ i ) of A. Consider the eigenvalues λ 0 λ 1 λ 2 For a d-regular graph, λ 0 = 1. Why?
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ecc3 - Why Expander Based Codes? 15-853:Algorithms in the...

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