Bipartite Index Coding
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Transcript of Bipartite Index Coding
BIPARTITE INDEX CODING
Arash Saber TehraniAlexandros G. DimakisMichael J. Neely
Department of Electrical Engineering University of Southern California (USC)
Outline
• Index Coding Problem– Introduction– Bipartite model
• Our Scheme: Partition Multicast– Formulation
• Partition Multicast is NP-hard– Connection to clique cover
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 ={x1, x2, … , xm} from a finite alphabet X • Set of n users U ={u1, u2, … , un} • Each user demands exactly one packet• Each user i knows a subset of packets denoted by Nout(ui) as
side info• Objective: Minimize the amount of broadcast data so that all
users decode their designated packets.
Bipartite model for IC
• The system can be represented by a bipartite graph
• A directed edge from packet xj to user ui indicates that user ui demands packet xj.
• A directed edge from user ui to packet xj indicates that user ui knows packet xj as side info.
Index Coding Problem
• A solution of the problem– A finite alphabet WX – an encoding function E: Xm WX
– each user ui is able to decode its designated packet from the broadcast message w and its side information.
• Optimal solution is HARD to compute.
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 a1x1 + a2x2 + … + amxm where ai'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)
Our Scheme: Partition Multicast
• Induced subgraph by a subset of packets S
X1
X2
X3
X4
U1
U2
U3
U4
U5
X1
X2
U1
U2
U3
Our Scheme: Partition Multicast
• We are looking for a partition (valid packet decomposition)
X1
X2
X3
X4
U1
U2
U3
U4
U5
X1
X2
X3
X4
|{X1,X2}| = 2, d1 = 1 |{X3,X4}| = 2, d1 = 1X1+X2 X3+X4
Our Scheme: Partition Multicast
• Partition Multicast:
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.
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”
U1, X1
U2, X2
U3, X3 U4, X4
U5, X5
X1U1
U2 X2
X3
X4
X5
U3
U4
U5
Partition Multicast is NP-hard
• Sum-degree cover and clique cover are equivalent– Partitioning a clique is strictly suboptimal
• For any graph T(GS) ≥1.• If GS is a clique, then T(GS) = 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
SD cover Clique cover• Let the solution of SD cover be GS1, … , GSK induced by
subsets S1, S2, …, Sk.• Clique cover is also a graph partition where each subgraph
requires exactly one transmission, so
• Consider subgraph GS1 with minimum knowledge d1. The complement of GS1 has maximum degree |S1| - d1 - 1.
• As is well known, any graph of maximum degree d has a vertex coloring of size d + 1.
SD cover Clique cover
• The complement of GS1 has a vertex coloring with |S1| - d1 color.
• Thus, GS1 has a clique cover of size |S1| - d1.• That is• Repeating the same procedure over all k
subgraphs, gives
• Jointly with the previous inequality we get
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.
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
Thanks, Questions?
Partition Multicast
• Partition or Cover:
– Let x S∈ 1, x S∈ 2
– Delete x from S1 to get set S1’– New minimum knowledge for GS1, namely, d1’.– |S1’| =|S1|-1 and d1-1 ≤ d1’ ≤ d1.
GS1 GS2 GSk
T(GS1)=|S1|-d1 T(GS2)=|S2|-d2 T(GSk)=|Sk|-dk
Our Scheme: Partition Multicast• Bipartite case (Painful stuff)
– For set S P,⊆ define GS = (US,S,ES) 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
Index Coding Problem
• A solution of the problem– A finite alphabet WX – an encoding function E: Xm WX
– each user ui 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