Bipartite Index Coding - PowerPoint Presentation

Slide1

Bipartite Index Coding

Arash Saber Tehrani

Alexandros

G.

Dimakis

Michael J. Neely

Department of Electrical Engineering University of Southern

California (

USC) Slide2

Outline

Index Coding Problem

Introduction

Bipartite model

Our Scheme: Partition Multicast

Formulation

Partition Multicast is NP-hard

Connection to clique coverSlide3

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.Slide4

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 ui demands packet xj.A directed edge from user ui to packet xj indicates that user ui knows packet xj as side info.Slide5

Index Coding Problem

A solution of the problem

A finite alphabet W

X

an encoding function E:

X

m WXeach user ui is able to decode its designated packet from the broadcast message w and its side information.Optimal solution is HARD to compute.Slide6

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)Slide7

Our Scheme: Partition Multicast

Induced

subgraph

by a subset of packets S

X

1

X

2

X

3

X

4

U

1

U

2

U

3

U

4

U

5

X

1

X

2

U

1

U

2

U

3Slide8

Our Scheme:

Partition

Multicast

We are looking for a partition (valid packet decomposition)

X

1

X

2

X

3

X

4

U

1

U

2

U

3

U

4

U

5

X

1

X

2

X

3

X

4

|{X

1

,X

2

}| = 2, d

1

= 1

|{X

3

,X

4

}| = 2, d

1

= 1

X

1

+X

2

X

3

+X

4Slide9

Our Scheme:

Partition

Multicast

Partition Multicast:Slide10

Our Scheme: Partition Multicast

The scheme is optimal for known cases such as

Cliques

t

rees

Directed cycles

It has cycle cover schemes proposed by

Chaudhry et al. and Neely et al. as a special case and outperforms them.Slide11

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

X

1

U

1

U

2

X

2

X

3

X

4

X

5

U

3

U

4

U

5Slide12

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 coverSlide13

SD cover Clique cover

Let the solution of SD cover be G

S1

, … , G

SK

induced by subsets S

1

, S2, …, Sk.Clique cover is also a graph partition where each subgraph requires exactly one transmission, soConsider 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.Slide14

SD cover Clique cover

T

he complement of G

S1

has a vertex coloring with

|S

1

| - d1 color.Thus, GS1 has a clique cover of size |S1| - d1.That isRepeating the same procedure over all k subgraphs, givesJointly with the previous inequality we getSlide15

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.Slide16

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’ partitionSlide17

Thanks,

Questions?Slide18

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

’.

|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

)=|Sk|-dkSlide19

Our Scheme: Partition Multicast

Bipartite

case (Painful stuff)

For set

S⊆

P,

define G

S = (US,S,ES) to be the subgraph induced by S:A valid packet decomposition is set of k disjoint subgraphs such thatIt can be checked that for a valid packet decompositionSlide20

Index Coding Problem

A solution of the problem

A finite alphabet W

X

an encoding function E:

X

m WXeach 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