Error Detection and Correction in Communication Networks Chong - PowerPoint PPT Presentation

Error Detection and Correction in Communication Networks Chong Shangguan Joint work with Itzhak (Zachi) Tamo Department of Electrical Engineering-Systems Tel Aviv University ISIT 2020 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 1 / 18

Outline 1. Motivation 2. Error detection and correction for repetition codes 3. Error detection for arbitrary codes 4. Open questions Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 2 / 18

Motivation Regenerating codes 1 Data stored across n nodes (servers) with redundancy Any two servers can communicate Task: Efficiently repair a failed (erased) server 1 Dimakis, Godfrey, Wu, Wainwright, Ramchandran TIT10 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 3 / 18

Motivation Regenerating codes 1 Data stored across n nodes (servers) with redundancy Any two servers can communicate Task: Efficiently repair a failed (erased) server Graph Code Task K n MDS code 1 erasure correction 1 Dimakis, Godfrey, Wu, Wainwright, Ramchandran TIT10 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 3 / 18

Motivation Testing Equality in Communication Graphs 2 2 Alon, Efremenko, Sudakov TIT17 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 4 / 18

Motivation Testing Equality in Communication Graphs 2 Connected graph G on n vertices 2 Alon, Efremenko, Sudakov TIT17 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 4 / 18

Motivation Testing Equality in Communication Graphs 2 Connected graph G on n vertices vertex v i holds an m -bit string x i 2 Alon, Efremenko, Sudakov TIT17 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 4 / 18

Motivation Testing Equality in Communication Graphs 2 Connected graph G on n vertices vertex v i holds an m -bit string x i Communication along edges 2 Alon, Efremenko, Sudakov TIT17 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 4 / 18

Motivation Testing Equality in Communication Graphs 2 Connected graph G on n vertices vertex v i holds an m -bit string x i Communication along edges Task: Efficiently determine if x 1 = · · · = x n ? 2 Alon, Efremenko, Sudakov TIT17 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 4 / 18

� � � Motivation Testing Equality in Communication Graphs 2 Connected graph G on n vertices vertex v i holds an m -bit string x i Communication along edges Task: Efficiently determine if x 1 = · · · = x n ? x 2 f 1 ( x ) f 2 ( x ) x 1 x 3 f 3 ( x ) 2 Alon, Efremenko, Sudakov TIT17 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 4 / 18

� � � Motivation Testing Equality in Communication Graphs 2 Connected graph G on n vertices vertex v i holds an m -bit string x i Communication along edges Task: Efficiently determine if x 1 = · · · = x n ? x 2 f 1 ( x ) f 2 ( x ) x 1 x 3 f 3 ( x ) Graph Code Task Connected graph Repetition code Error detection 2 Alon, Efremenko, Sudakov TIT17 Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 4 / 18

Motivation The model A problem in the model is defined as follows: Graph Code Task graph C an Error detection or G , | V | = n ( n , k , d ≥ 2) code Error correction Vertex v i holds x i , ( x 1 , ..., x n ) ∈ C Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 5 / 18

Motivation The model A problem in the model is defined as follows: Graph Code Task graph C an Error detection or G , | V | = n ( n , k , d ≥ 2) code Error correction Vertex v i holds x i , ( x 1 , ..., x n ) ∈ C Communication along the edges Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 5 / 18

Motivation The model A problem in the model is defined as follows: Graph Code Task graph C an Error detection or G , | V | = n ( n , k , d ≥ 2) code Error correction Vertex v i holds x i , ( x 1 , ..., x n ) ∈ C Communication along the edges Given G , C and a task, what is the minimum communication cost? Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 5 / 18

Motivation The model A problem in the model is defined as follows: Graph Code Task graph C an Error detection or G , | V | = n ( n , k , d ≥ 2) code Error correction Vertex v i holds x i , ( x 1 , ..., x n ) ∈ C Communication along the edges Given G , C and a task, what is the minimum communication cost? This talk: Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 5 / 18

Motivation The model A problem in the model is defined as follows: Graph Code Task graph C an Error detection or G , | V | = n ( n , k , d ≥ 2) code Error correction Vertex v i holds x i , ( x 1 , ..., x n ) ∈ C Communication along the edges Given G , C and a task, what is the minimum communication cost? This talk: Error detection Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 5 / 18

Motivation The model A problem in the model is defined as follows: Graph Code Task graph C an Error detection or G , | V | = n ( n , k , d ≥ 2) code Error correction Vertex v i holds x i , ( x 1 , ..., x n ) ∈ C Communication along the edges Given G , C and a task, what is the minimum communication cost? This talk: Error detection Single error correction for repetition codes Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 5 / 18

Motivation Outline 1. Motivation 2. Error detection and correction for repetition codes 3. Error detection for arbitrary codes 4. Open questions Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 6 / 18

Error detection and correction for repetition codes The trivial protocol Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 7 / 18

Error detection and correction for repetition codes The trivial protocol G = K n , x = ( x 1 , . . . , x n ), x i ∈ { 0 , 1 } m Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 7 / 18

Error detection and correction for repetition codes The trivial protocol G = K n , x = ( x 1 , . . . , x n ), x i ∈ { 0 , 1 } m ( n − 1) m bits to detect an error Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 7 / 18

Error detection and correction for repetition codes The trivial protocol G = K n , x = ( x 1 , . . . , x n ), x i ∈ { 0 , 1 } m ( n − 1) m bits to detect an error ( n − 1) m + im bits to correct i errors, i ≤ ⌊ d − 1 2 ⌋ Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 7 / 18

Error detection and correction for repetition codes The trivial protocol G = K n , x = ( x 1 , . . . , x n ), x i ∈ { 0 , 1 } m ( n − 1) m bits to detect an error ( n − 1) m + im bits to correct i errors, i ≤ ⌊ d − 1 2 ⌋ Is it optimal? Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 7 / 18

Error detection and correction for repetition codes Example: G = K 3 , C = Rep Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

Error detection and correction for repetition codes Example: G = K 3 , C = Rep Trivial protocol: Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

Error detection and correction for repetition codes Example: G = K 3 , C = Rep Trivial protocol: 2 m bits for error detection Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

Error detection and correction for repetition codes Example: G = K 3 , C = Rep Trivial protocol: 2 m bits for error detection 3 m bits for correcting a single error Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

Error detection and correction for repetition codes Example: G = K 3 , C = Rep Trivial protocol: 2 m bits for error detection 3 m bits for correcting a single error “Nontrivial” protocol: Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

Error detection and correction for repetition codes Example: G = K 3 , C = Rep Trivial protocol: 2 m bits for error detection 3 m bits for correcting a single error “Nontrivial” protocol: (1 . 5 + o (1)) m for error detection (Alon et al.) Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18

Error detection and correction for repetition codes Example: G = K 3 , C = Rep Trivial protocol: 2 m bits for error detection 3 m bits for correcting a single error “Nontrivial” protocol: (1 . 5 + o (1)) m for error detection (Alon et al.) (2 . 5 + o (1)) m for correcting a single error (this work) Chong Shangguan (Tel Aviv University) Error Detection and Correction in Communication Networks 8 / 18