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B IPARTITE I NDEX C ODING Arash Saber Tehrani Alexandros G. Dimakis Michael J. Neely Department of Electrical Engineering University of Southern California (USC)

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Outline Index Coding Problem – Introduction – Bipartite model Our Scheme: Partition Multicast – Formulation Partition Multicast is NP-hard – Connection to clique cover

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Index Coding Problem Introduced in [Birk and Kol 98], and further developed in [Bar-Yossef, Birk, Jayram, and Kol 06 and 11]. Broadcast station Set of m packets P ={x 1, x 2, …, x m } from a finite alphabet X Set of n users U ={u 1, u 2, …, u n } Each user demands exactly one packet Each user i knows a subset of packets denoted by N out (u i ) as side info Objective: Minimize the amount of broadcast data so that all users decode their designated packets.

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Bipartite model for IC The system can be represented by a bipartite graph A directed edge from packet x j to user u i indicates that user u i demands packet x j. A directed edge from user u i to packet x j indicates that user u i knows packet x j as side info.

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Index Coding Problem A solution of the problem – A finite alphabet W X – an encoding function E: X m W X – each user u i is able to decode its designated packet from the broadcast message w and its side information. Optimal solution is HARD to compute.

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Our Scheme: Partition Multicast When each user knows at least d packets as side information – We call d “minimum out-degree” or “minimum knowledge” Then there are at most m – d unknowns for each user. With transmission of m - d independent equations in the form a 1 x 1 + a 2 x 2 + … + a m x m where a i 's are taken from some finite field F, each user can decode the packet it demands as shown in Ho et al. (Given that |F| is large enough)

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Our Scheme: Partition Multicast Induced subgraph by a subset of packets S X1X1 X2X2 X3X3 X4X4 U1U1 U2U2 U3U3 U4U4 U5U5 X1X1 X2X2 U1U1 U2U2 U3U3

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Our Scheme: Partition Multicast We are looking for a partition (valid packet decomposition) X1X1 X2X2 X3X3 X4X4 U1U1 U2U2 U3U3 U4U4 U5U5 X1X1 X2X2 X3X3 X4X4 |{X 1,X 2 }| = 2, d 1 = 1|{X 3,X 4 }| = 2, d 1 = 1 X 1 +X 2 X 3 +X 4

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Our Scheme: Partition Multicast Partition Multicast:

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Our Scheme: Partition Multicast The scheme is optimal for known cases such as – Cliques – trees – Directed cycles It has cycle cover schemes proposed by Chaudhry et al. and Neely et al. as a special case and outperforms them.

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Partition Multicast is NP-hard Undirected case: – We want to find a partition for which the sum of minimum knowledge is maximized – We call this problem “sum-degree cover” U 1, X 1 U 2, X 2 U 3, X 3 U 4, X 4 U 5, X 5 X1X1 U1U1 U2U2 X2X2 X3X3 X4X4 X5X5 U3U3 U4U4 U5U5

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Partition Multicast is NP-hard Sum-degree cover and clique cover are equivalent – Partitioning a clique is strictly suboptimal For any graph T(G S ) ≥1. If G S is a clique, then T(G S ) = 1, i.e., the minimum knowledge d = |S| - 1. – We need to show that Solution of sum-degree cover gives the solution of clique cover Solution of the clique cover gives the solution of sum-degree cover

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SD cover Clique cover Let the solution of SD cover be G S1, …, G SK induced by subsets S 1, S 2, …, S k. Clique cover is also a graph partition where each subgraph requires exactly one transmission, so Consider subgraph G S1 with minimum knowledge d 1. The complement of G S1 has maximum degree |S 1 | - d 1 - 1. As is well known, any graph of maximum degree d has a vertex coloring of size d + 1.

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SD cover Clique cover The complement of G S1 has a vertex coloring with |S 1 | - d 1 color. Thus, G S1 has a clique cover of size |S 1 | - d 1. That is Repeating the same procedure over all k subgraphs, gives Jointly with the previous inequality we get

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Partition Multicast is NP-hard Maps an undirected graph G to a bipartite graph. Solve the partition multicast. Find the clique cover of all partitions through coloring of complements of the subgraphs. Find the clique cover.

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Conclusion We introduced the bipartite graph model for the index coding problem We presented a new scheme “partition multicast” for index coding problem. We introduced the sum-degree cover problem. We showed that finding the optimal partition is NP-hard. Future work: finding a ‘good’ partition

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Thanks, Questions?

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Partition Multicast Partition or Cover: – Let x ∈ S 1, x ∈ S 2 – Delete x from S 1 to get set S 1 ’ – New minimum knowledge for G S1, namely, d 1 ’. – |S 1 ’| =|S 1 |-1 and d 1 -1 ≤ d 1 ’ ≤ d 1. G S1 G S2 G Sk T(G S1 )=|S 1 |-d 1 T(G S2 )=|S 2 |-d 2 T(G Sk )=|S k |-d k

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Our Scheme: Partition Multicast Bipartite case (Painful stuff) – For set S ⊆ P, define G S = (U S,S,E S ) to be the subgraph induced by S: – A valid packet decomposition is set of k disjoint subgraphs such that – It can be checked that for a valid packet decomposition

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Index Coding Problem A solution of the problem – A finite alphabet W X – an encoding function E: X m W X – each user u i is able to decode its designated packet from the broadcast message w and its side information. The minimum coding length of the solution per input symbol: where the minimum is over all encoding functions E. Optimal broadcast rate

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