On the solvability of 4s/3t sum-networks

Project Report Phase-1

By

Amit Kumar V. Jha

Roll No - 134102037

Supervisor - Dr. Brijesh Kumar Rai

DEPARTMENT OF ELECTRONICS AND ELECTRICAL ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI

Abstract

We consider directed acyclic networks where each terminal requires sum of all

the symbols generated by each source independently and one symbol at a time

from given field F. Such a class of networks has been termed as sum-networks in

the literature. A sum-network having m sources and n terminals has been termed

as ms / nt sum-network. A necessary and sufficient condition for a 3-sources

and 3-terminals also termed as 3s /3t sum-network to allow all the terminals to

recover the sum of source symbols over any field has already been given by Sagar

Shenvi and Bikash Kumar Dey[3]. It will be interesting to find a necessary and

sufficient condition for a sum-network with more number of sources and more

number of terminals. In this paper, we give a necessary and sufficient condition

for a 4s /3t sum-network.

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Contents

1 Introduction 3

2 Terms and definitions 4

3 5

4 System modelling and Definitions 5

List of Figures

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1 Introduction

In a network, mixing or coding of information at the intermediate nodes is referred

as a network coding. It has been shown by Ahlswede et al. [1] that network coding

has two main advantages over multi-cast or broadcast networks like saving in overall

bandwidth and increase in throughput when network coding is allowed. Network

coding can be implemented to fulfil requirement of all terminals or some set of termi-

nals. The requirement may be any function of source symbols but we consider only

sum function in this paper because sum function is the simplest form and which

can be extended to reveal the intricacies of other functions. In [5], A. Ramamoor-

thy demonstrated that as long as there exist at most two sources and n terminals

or m sources and at most two terminals, it is possible to recover sum of all source

symbols over any F by all terminals if and only if all source-terminal pairs are con-

nected. However, the source-terminal pair connectivity is proved to be insufficient

by B.K. Rai, B.K. Dey and A. Karandikar [4]. A necessary and sufficient condition

for a sum-network with exactly three sources and three terminals (also referred as

3s/3t sum network) is given by Sagar Shenvi and Bikash Kumar Dey[3] whereas, the

problem is still open for more general sum-networks with more number of sources

and terminals. In this paper, we give a necessary and sufficient condition for 4s/3t

sum-network.

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2 Terms and definitions

In this section, we will present some well known terms and definitions which will

be used through out the work of this paper. We will also present some theorems and

lemmas form the previous papers which will be useful for our work.

Sum-Network: A network in which every terminal wants to recover sum of all

source symbols over a finite field F is called sum-network.

Connected: A network is said to be connected if and only if there exist at least one

path for every source-terminal pair.

Linear Network Coding: The network coding in which all nodes are allowed to per-

form only linear operations is called linear network coding. Otherwise, it is termed

as non linear network coding.

Fractional Network Code: A coding technique in which knumber of source sym-

bols encoded in a block of length n such that k 6= n is called fractional network coding.For k=n, the coding scheme is referred as vector network coding. In vector network

network coding, a block or vector of n source symbols is treated as a single symbol

at all nodes and edges and also receivers recover a vector of n symbols.

Code Rate: For a network having (k,n) fractional network code solution, it is possi-

ble to communicate k source symbols in n use of the network. For such a class of

network, code rate is defined as the ratio of k and n i.e k/n.

Coding Capacity: It is defined as the supremum of all achievable rates.

Solvable: A network is said to be solvable if and only if its coding capacity is at least

equal to 1.

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34 System modelling and Definitions

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References

[1] R.Ahlswede,N.Cai,S.-Y.R.Li,andR.W.Yeung,Networkinformationflow,IEEE Trans.

Inf. Theory, vol. 46, no. 4, pp. 12041216, 2000.

[2] S. Shenvi and B. K. Dey, On the solvability of 3-source 3-terminal sum-

networks,available at http://arxiv.org/abs/1001.4137.

[3] A Necessary and Sufficient Condition for Solvability of a3s/3tSum-network.

[4] B. K. Rai, B. K. Dey, and A. Karandikar, Some results on communicating the

sum of sources over a network, inProc. NetCod, 2009.

[5] A. Ramamoorthy, Communicating the sum of sources over a network, in Proc.

ISIT, (Toronto, Canada), pp. 16461650, 2008.

[6] B. K. Rai, B. K. Dey, and S. Shenvi, Some bounds on the capacity of commu-

nicating the sum of sources, inProc. ITW, 2010.

[7] R. Appuswamy, M. Franceschetti, N. Karamchandani, and K. Zeger, Network

coding for computing, in Proceedings of Annual Allerton Conference, (UIUC, IIlinois,

USA), 2008.

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