BIPARTITE INDEX CODING
Arash Saber Tehrani
Alexandros G. Dimakis
Michael 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
U1
U2
U3
U4
U5
X1
U1
X2
U2
X3
U3
X4
X1
X2
Our Scheme: Partition Multicast
• We are looking for a partition (valid packet decomposition)
U1
X1
U2
X2
U3
X3
U4
X4
U5
|{X1,X2}| = 2, d1 = 1
X1+X2
X1
X3
X2
X4
|{X3,X4}| = 2, d1 = 1
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:
U2, X2
U1 , X1
U5, X5
U3, X3 U4, X 4
U1
X1
U2
X2
U3
X3
U4
X4
U5
X5
– We want to find a partition for which the sum of
minimum knowledge is maximized
– We call this problem “sum-degree cover”
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:
GS1
GS2
T(GS1)=|S1|-d1 T(GS2)=|S2|-d2
–
–
–
–
GSk
T(GSk)=|Sk|-dk
Let x∈S1, x∈S2
Delete x from S1 to get set S1’
New minimum knowledge for GS1, namely, d1’.
|S1’| =|S1|-1 and d1-1 ≤ d1’ ≤ d1.
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