Optimal Index Codes with Near-Extreme Rates Vitaly Skachek (joint - PowerPoint PPT Presentation

Optimal Index Codes with Near-Extreme Rates Vitaly Skachek (joint work with Son Hoang Dau and Yeow Meng Chee ) Estonian Theory Days October 25th, 2013 This work is supported by Research Grant NRF-CRP2-2007-03 (Singapore) S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

b b b Introduction Index Coding with Side Information (ICSI) has n messages Sender S R 1 R n R 2 each receiver has some messages requests one message S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

b b b Introduction Index Coding with Side Information (ICSI) has n messages Sender Receiver Demand Side Info. S R 1 x 1 { x 2 } R 2 x 2 { x 3 } { x 1 , x 4 } R 3 x 3 R 1 R n R 4 x 4 { x 5 } R 2 { x 2 , x 4 } R 5 x 5 each receiver has some messages requests one message S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

b b b Introduction Index Coding with Side Information (ICSI) has n messages Sender Receiver Demand Side Info. S R 1 x 1 { x 2 } R 2 x 2 { x 3 } { x 1 , x 4 } R 3 x 3 R 1 R n R 4 x 4 { x 5 } R 2 { x 2 , x 4 } R 5 x 5 each receiver has some messages requests one message Questions: How can S satisfy all the demands in a minimum number of transmissions? S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Motivation Data Distributor C 1 C 2 C 3 P2, P3 P1, P3 P1, P2 Receive P1 P2 P3 Lose Data Delivery S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Motivation Data Distributor P1 + P2 + P3 C 1 C 2 C 3 P2, P3 P1, P3 P1, P2 Received Data Delivery S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Motivation Data Distributor P1 + P2 + P3 C 1 C 2 C 3 P2, P3 P1, P3 P1, P2 Received P1 P2 Retrieve P3 Data Delivery S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

b b b b b b Motivation Network Coding Network Coding (NC): Ahlswede et al. , 2000 x 1 x 2 x n s 1 s 2 s n y 2 y 1 y 3 y 1 + y 2 y 2 + y 3 t 1 t 2 t m S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

b b b b b b Motivation Network Coding Index Coding and Network Coding Network Coding (NC): Ahlswede et Index coding proposed by Birk 1 al. , 2000 and Kol (1998) ICSI is a special case of 2 x 1 x 2 x n non-multicast network coding s 1 s 2 s n ICSI and NC are equivalent 3 (El Rouayheb, Sprintson, Georghiades, 2008; Effros, y 2 El Rouayheb, Langberg, 2012) y 1 y 3 y 1 + y 2 y 2 + y 3 t 1 t 2 t m S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Literature Overview Bar-Yossef et al. (2006) Associate each ICSI instance with a digraph Optimal scalar linear transmission rate = minrank of the digraph S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Literature Overview Bar-Yossef et al. (2006) Associate each ICSI instance with a digraph Optimal scalar linear transmission rate = minrank of the digraph Peeters (1996): Finding minrank of a graph is NP-hard (deciding whether minrank of a graph is three is NP-hard) S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Literature Overview Bar-Yossef et al. (2006) Associate each ICSI instance with a digraph Optimal scalar linear transmission rate = minrank of the digraph Peeters (1996): Finding minrank of a graph is NP-hard (deciding whether minrank of a graph is three is NP-hard) Chaudhry and Sprintson (2008): exact and approximate algorithms for minranks S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Literature Overview Bar-Yossef et al. (2006) Associate each ICSI instance with a digraph Optimal scalar linear transmission rate = minrank of the digraph Peeters (1996): Finding minrank of a graph is NP-hard (deciding whether minrank of a graph is three is NP-hard) Chaudhry and Sprintson (2008): exact and approximate algorithms for minranks Bar-Yossef et al. (2006); Berliner and Langberg (2011): polynomial time computation of minranks for some families of graphs S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Other works Alon et al. (2008) Lubetzky, Stav (2009) Dau, Skachek, Chee (2011) Haviv, Langberg (2012) Ong, Lim, Ho (2012) Brahma, Fragouli (2012) Tehrani, Dimakis (2012) Neely, Tehrani, Zhang (2012) Shum, Dai, Sung (2012) Maleki, Cadambe, Jafar (2012) Arbabjolfaei, Bandemer, Kim, Sasoglu (2013) Shanmugam, Dimakis, Caire (2013) S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Our contributions In this work we S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Our contributions In this work we characterize families of digraphs with some extremely high or low minranks S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Our contributions In this work we characterize families of digraphs with some extremely high or low minranks show that deciding whether minrank of a digraph is two is NP-hard S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Our contributions In this work we characterize families of digraphs with some extremely high or low minranks show that deciding whether minrank of a digraph is two is NP-hard (trivial for graphs) S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Definitions and Notation An Example Consider the following ICSI instance: S x 1 + x 2 x 2 + x 3 x 4 + x 5 R 1 R 5 requests x 1 requests x 5 R 4 R 2 owns x 2 R 3 owns x 2 , x 4 requests x 4 requests x 2 requests x 3 owns x 5 owns x 3 owns x 1 , x 4 S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Definitions and Notation An Example Consider the following ICSI instance: S x 1 + x 2 x 2 + x 3 x 4 + x 5 R 1 R 5 requests x 1 requests x 5 R 4 R 2 owns x 2 R 3 owns x 2 , x 4 requests x 4 requests x 2 requests x 3 owns x 5 owns x 3 owns x 1 , x 4 S transmits x 1 + x 2 , x 2 + x 3 , and x 4 + x 5 (an IC of length 3). S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Definitions and Notation An Example Consider the following ICSI instance: S x 1 + x 2 x 2 + x 3 x 4 + x 5 R 1 R 5 requests x 1 requests x 5 R 4 R 2 owns x 2 R 3 owns x 2 , x 4 requests x 4 requests x 2 requests x 3 owns x 5 owns x 3 owns x 1 , x 4 S transmits x 1 + x 2 , x 2 + x 3 , and x 4 + x 5 (an IC of length 3). R 1 decodes: x 1 = x 2 + ( x 1 + x 2 ) 1 R 2 decodes: x 2 = x 3 + ( x 2 + x 3 ) 2 R 3 decodes: x 3 = x 1 + ( x 1 + x 2 ) + ( x 2 + x 3 ) 3 R 4 decodes: x 4 = x 5 + ( x 4 + x 5 ) 4 R 5 decodes: x 5 = x 4 + ( x 4 + x 5 ) 5 S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Definitions and Notation Describe an ICSI Instance via Side Information Digraphs S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Definitions and Notation Describe an ICSI Instance via Side Information Digraphs Vertex set: V ( D ) = [ n ] = { 1 , 2 , . . . , n } ( n messages, n receivers) S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Definitions and Notation Describe an ICSI Instance via Side Information Digraphs Vertex set: V ( D ) = [ n ] = { 1 , 2 , . . . , n } ( n messages, n receivers) � � Arc set: E ( D ) = ( i , j ) : i ∈ [ n ] , R i has x j as side information S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Definitions and Notation Describe an ICSI Instance via Side Information Digraphs Vertex set: V ( D ) = [ n ] = { 1 , 2 , . . . , n } ( n messages, n receivers) � � Arc set: E ( D ) = ( i , j ) : i ∈ [ n ] , R i has x j as side information Receiver Demand Side Info. R 1 1 R 1 x 1 { x 2 } R 5 R 2 x 2 { x 3 } 5 2 R 2 R 3 x 3 { x 1 , x 4 } R 4 x 4 { x 5 } R 5 x 5 { x 2 , x 4 } R 4 4 3 R 3 S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Definitions and Notation Definition (Haemer, 1978) Let D be a digraph where V ( D ) = [ n ]. S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Definitions and Notation Definition (Haemer, 1978) Let D be a digraph where V ( D ) = [ n ]. A matrix M = ( m i , j ) ∈ F n × n is said 1 q to fit D if S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Definitions and Notation Definition (Haemer, 1978) Let D be a digraph where V ( D ) = [ n ]. A matrix M = ( m i , j ) ∈ F n × n is said 1 q to fit D if � m i , j � = 0 , i = j , m i , j = 0 , i � = j , ( i , j ) / ∈ E ( D ) . S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates

Definitions and Notation Definition (Haemer, 1978) 1 Let D be a digraph where V ( D ) = [ n ]. A matrix M = ( m i , j ) ∈ F n × n is said 1 q 5 2 to fit D if � m i , j � = 0 , i = j , 4 3 m i , j = 0 , i � = j , ( i , j ) / ∈ E ( D ) . A digraph of minrank 3 The minrank of D over F q is 2 defined to be △ � rank( M ) : M ∈ F n × n � minrk q ( D ) = min and M fits D . q     1 1 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0         A + I = 1 0 1 1 0 = ⇒ M = 1 0 1 0 0         0 0 0 1 1 0 0 0 1 1     0 1 0 1 1 0 0 0 1 1 S. H. Dau, V. Skachek, Y. M. Chee Optimal Index Codes of Near-Extreme Rates