Retargeting LT Codes using XORs at the Relay

Shirish Karande

Algorithms and Optimization Group, Systems Research Lab,

Tata Research Development and Design Center

Hadapsar, Pune – 411013, India

shirish.karande@tcs.com

Abstract— We consider a network where multiple sources

communicate via a single relay to sinks with non-uniform and unequal demands. The LT distributions employed at the sources

may be ill-matched to the demands of the sinks connected to the relay. We consider a probabilistic "morphing" of two or more

fountain-encoded streams into a single stream better suited for the demand patterns downstream of the relay. The relay observes

symbols generated from two distinct fountain codes, and can decide to forward a symbol from one source, or the other source,

or the X-OR of the two symbols. We propose a linear

programming based design of Generalized LT codes, which with appropriate substitutions is utilized to design the probabilities for

the relay. Simulation results show that the designs obtained on the basis of the proposed optimization problem, when compared with multiplexing or simple mixing, reduce the number of

symbols that have to be downloaded to guarantee a desired probability of successful decoding.

I. INTRODUCTION

Consider a network, such as the one shown in Figure 1, where the sinks asynchronously download multiple files from distinct sources via a common relay. It has been shown (e.g.[1]) that in a rateless download of multiple files, network coding [2] can provide delay and throughput gains. It can be verified that in the Y-network shown in Figure 1, random network coding (RNC) at the relay, reduces the delay in complete download of the source files. However, in general, the decoding complexity associated with RNC is significantly high. Hence, one seeks low-complexity alternatives that can facilitate download of large files.

LT Codes [3] are low-complexity capacity approaching rateless codes ([4]-[5]), which have found utility for asynchronous broadcast (e.g. [6]) over channels with heterogeneous and unknown behavior. LT encoding lends itself to distributed implementation, and hence can be used to improve the communication efficiency, e.g. by enabling parallel downloads from multiple mirror sites. Nevertheless, in networks with more than one-hop, LT coding cannot be easily combined with other modern information theoretic techniques such as network coding [2]. The decoding efficiency and time complexity of an LT code are sensitive to the degree distribution, and hence an intermediate relay cannot naively employ an operation which significantly perturbs the effective distribution.

Motivated by the above observations, in [7] Puducheri et. al. investigated the problem of constructing a Distributed LT code (DLT). Specifically, they consider a network where

multiple sources communicate to the sinks via a single relay. Traditionally, the operation at the relay would be limited to forwarding the LT encoded symbols received from the sources. However, selective XOR-ing at the relay can create a statistical effect of constructing an LT code on the concatenation of all source files. Such a DLT code has been shown to reduce the total download time [7].

The design methods proposed in [7] have been tailored for the Robust Soliton Distribution (RSD) [3] and are limited to the case where all sources have equal priority and all the sinks have an equal demand. In this work we consider a significantly generic scenario. We highlight that, while the degree distributions suggested by Luby [3] were optimized for complete recovery of the message word, in recent years, multimedia and sensor network applications have motivated several researchers to design LT distributions that enable partial and prioritized data recovery (e.g. [8]-[16]).

In the network analyzed in this work, we assume that the LT degree distributions at the sources are optimized for the non-uniform

1 demands of the “local” sinks. Nevertheless,

these sources are further subscribed to by sinks connected to a

1 Nonuniform demand implies that different sets of sinks are interested in

downloading distinct fractions of the source file. The downloaded fraction

may determine the quality of media. Consequently, the non-uniformity can be

seen to be emerging from a rate-distortion tradeoff.

Figure 1 Network consisting of two sources [1,2] and a common relay. Each source employs an LT encoded fountain to meet the local non-uniform demands (e.g. half of the local sinks connected to source 1 desire to recover only 60% of the file, while the other half desires to recover 80% ). An additional set of sinks with unequal and non-uniform demands subscribe to the source via the common relay.

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common relay. The demands of the sinks connected to the relay can be non-uniform as well as unequal

2. Furthermore, the

non-uniform demands of the local sinks and those connected to the relay may be differing, thus creating a mismatch. We seek to design a selective mixing strategy at the relay which has the effect of retargeting the effective degree distribution.

The output of the relay can be modeled by a Generalized LT (GLT) distribution, parameterized by the mixing probabilities at the relay. GLT codes use vector degrees to explicitly indicate the sampling of input symbols from distinct data segments. The use of ripple based analysis of GLT codes [14] is the key to our design strategy. We propose a linear program (LP) based design of GLT codes for non-uniform and unequal demands. With suitable substitutions, the LP can be extended to design the mixing probabilities for the relay.

In [16] , Talari et. al. have also considered the design of DLT codes capable of providing prioritized recovery. They employ AND-OR tree analysis for the design. Nevertheless, they require a limiting relaxation that the probability of combining two encoded symbols at the relay is independent of the degree of the received encoding symbol. We show that such a relaxation affects the efficiency of combining LT codes.

The remainder of the paper is organized as: Section II provides a primer on GLT codes and establishes the network model. Section III, describes a LP based design of GLT codes. Section IV describes the design of mixing probabilities. Section IV reports the numerical results.

II. PRELIMINARIES

A. Generalized LT Codes

The encoding/decoding of GLT codes is similar to the

traditional LT codes. Consquently, we only provide a brief

description here, and refer the reader to [3], [14] for details.

Suppose we want to transmit a message comprising of k

input symbols, partitioned into m segments, such that,

[ ]1,i m∀ ∈ segment-i contains i i

k kα= symbols, than the

GLT code is defined by a generalized degree polynomial

( ) ( )( )

1

1

1

1 1, ,, ,

, , m

m

m

dd

m md d

d d

x x x xβ β= ∑ ⋯

⋯

⋯ ⋯ (1)

where ( )1 , , md dβ

⋯

represents the probability of choosing a vector

degree [ ]1, , md d d= ⋯ . The encoding symbol is generated by

XOR-ing i

d randomly chosen input symbols from segment-i.

Prior to decoding, a receiver downloads sufficient number

of encoding symbols, denoted here by kγ , to permit the

recovery of i iz k input symbols from each segment-i. The

collection of all encoding symbols with a single undecoded neighbor are said to form a ripple. Since each symbol in the ripple is connected to a single input symbol, the ripple can be

2 The term unequal demand describes a scenario where a sink desires to

download distinct proportions of different source files.

partitioned into mulitple colored sub-ripples by associating a color i with each data segment-i. Similar to traditional LT decoding, in each step, the decoder utilizes an encoding symbol from any of the colored ripples to decode an input symbol, thus making other input symbols decodable. The decoding stops either when the desired fractions are recovered or when all the colored ripples become empty.

B. Network Model

We employ a network model similar to the one used in [7].

As shown in Figure 2, we consider a network where two

disjoint sources communicate to the sinks via a common relay.

We assume that the communication takes place in terms of

epochs. Each epoch consists of an LT encoded transmission

from each source to the relay A, followed by a single broadcast

from the relay. The coding at the sources is determined by the

LT distributions

( ) ( )

1

1 1 11,1

k

d

d

d

x xβ β=

=∑ , ( ) ( )

2

2 2 22,1

k

d

d

d

x xβ β=

=∑ .

Meanwhile, the relay is assumed to have storage capacity of

one symbol per source and processing capability that is limited

to simple XORs. At each epoch the relay receives a symbol

from the individual sources. The relay probabilistically chooses

to either forward one of the received symbols or to transmit an

XOR-ed combination of the received symbols. Upon

transmission, the relay discards the received symbols. The

probabilistic decisions taken by the relay can be described by

“mixing rules” ( )( )

( )( )

( )( ){ }

1 2 1 2 1 2

1 2 1,2

, , ,, ,

d d d d d dΛ = Λ Λ Λ , where each rule

represents a conditional probability distribution:

Mixing Rule Λ : Given degrees 1

d , 2

d of encoding symbols

transmitted by source 1 and 2 resp.,

( )( )

( )( )

( )( )

1 2

1 2

1 2

1

,

2

,

1,2

,

1

2

-

d d

d d

d d

probability of forwarding symbol from source

probability of forwarding symbol from source

probability of transmitting an X OR

Λ =

Λ =

Λ =

where ( )( )

( )( )

( )( )

1 2 1 2 1 2

1 1,2 2

, , ,1 , , 0

d d d d d d≥ Λ Λ Λ ≥

( )( )

( )( )

( )( )

1 2 1 2 1 2

1 1,2 2

, , ,1

d d d d d dand Λ + Λ + Λ = (2)

A

1

2

T

1xβ

2xβ

1 2d dΛ

1

1

k

⋮

1

1 2

1k

k k k

+

= +

⋮

Figure 2 Network consisting of two sources [1,2] and a relay A. Network

operation is governed by1x

β , 2x

β and 1 2,d d

Λ .

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We say that the relay employs a multiplexing policy if

( )( )

( )( )

1 2 1 2

1 2

, ,0.5

d d d dΛ = Λ = and

( )( )

1 2

1

,0

d dΛ = . The analysis in [16]

requires that mixing probabilities be independent of the degrees

of encoded symbols received at the relay,

i.e.( )( )

( )( )

1 2 1 2

1 2

1 2, ,,

d d d dp pΛ = Λ = and

( )( )

1 2

1,2

3 1 2,1

d dp p pΛ = = − − .

We refer to such a constrained strategy as “simple mixing”.

Furthermore, we assume that the data broadcasted by the

relay is subscribed to by m sets of sinks 1 m

T T⋯ . Each set of

sinks may contain an arbitrarily large number of sinks. The set

jT has a demand described by the pair ( )1, 2,,j jz z , which

implies that a sink in set j

T desires to recover ij i

z k message

symbols from source i. Let j

γ be a parameter associated with

each sink set, such that, when a sink receives jkγ symbols

from the relay it is successfully able to recover the desired

fraction of message symbols. The design problem explored in

the paper can thus be stated as:

Problem 1 (P1):

*

1 1 2 2

arg min maxj

jj j

k

z k z k

γ

Λ

Λ = +

given ( ) ( ){ }1 1 2 2,x xβ β

P1 states that for the given source degree distributions, we

seek to find the mixing probabilities that minimize the worst

ratio, of the number of encoding symbols received by a sink to the number of message symbols recovered.

III. DESIGN OF GENERALIZED LT CODES

We design GLT codes using the following fluid approximations [14] for the size of the colored ripples.

( ) ( ) ( ) ( ) ( )1,0

1 1 2 1 1 2 1

1

, 1 , log 1r t t t t t tγ

βα

= − + −

(3)

( ) ( ) ( ) ( ) ( )0,1

2 1 2 2 1 2 2

2

, 1 , log 1r t t t t t tγ

βα

= − + −

(4)

In the above equations, ( )1 2,i ik r t t gives size of the ith ripple

when i i

k t input symbols have been decoded from the ith

source.

It is worth noting that the symbols from a particular segment can be decoded iff the corresponding ripple is non-empty. For

example, a decoder can transit from state ( )1 2,t t to

( )( )1 1 21 ,t k t+ iff ripple-1 is not empty. Therefore, it is

possible to recover a demand ( )1, 2,,i iz z if and only if there

exists a sequence of feasible transitions (i.e. a feasible path

[14]) from ( )0, 0 to ( )1, 2,,i iz z . Analyzing all the possible paths

increases the complexity of the design step. Hence, as shown in Figure 3 we employ a relaxation and seek degree distributions

which would make the straight segment from

( )0, 0 to ( )1, 2,,

i iz z a feasible path.

The problem of designing a GLT distribution for a single

demand ( )1 2,z z can be stated as:

Problem 2 (P2): ( ) ( )* *

,

, arg maxβ

β∆

∆ = ∆

subject to 1

[0, )t z∀ ∈

( )

( )( )

( )1,0 2 1

1 1 1 2 2 1

, log 1 01

z tt t t

z z z k t

αβ

α α

∆+ − − ≥ + −

( )

( ) ( )0,1 2 2 2

1 1 1 2 2 1 2

, log 1 01

z z tt t t

z z z z k t

αβ

α α

∆+ − − ≥ + −

with ( ) ( ) ( ) [ ]1 2 1 21 2, ,

1, , 0,1d d d d

d dβ β= ∀ ∈∑ and 0∆ ≥ .

The constraints in the above LP are obtained by rearranging the terms in Eq. (3) and (4). Note that, we have employed a

substitution ( )( )1 1 2 2z zγ α α= + ∆ and heuristic that

requires ( ) ( )( )1 2, 1j j j

r t t t t k≥ − . This heuristic, motivated

by the analysis in [8], employs a lower bound on the ripple-size in order to account for the variance.

In order to extend the above LP to the case of multiple non-uniform demands, we highlight that for a given degree distribution, a non-uniform demand can be feasibly supported

if the paths ( )0, 0 to ( )1, 2,,i iz z are all feasible with their

respective parameter i

γ . The LP for the single demand case

can thus be extended to multiple non-uniform demands with

the substitution ( )( )1 1, 2 2,i i iz zγ α α= + ∆ .

Figure 3 Relaxation checks feasibility only along the straight segment.

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Problem 3 (P3): ( ) ( )* *

,

, arg maxβ

β∆

∆ = ∆

subject to 1,

, [0, )i

i t z∀ ∀ ∈

( )

( )( )1,0 2, 1

1

1, 1 1, 2 2,

, , 0i

i i i

zt t k t

z z z

αβ µ

α α

∆+ ≥ +

( )

( )0,1 2, 2,2

2

1, 1,1 1, 2 2,

, , 0i i

i ii i

z zt t k t

z zz z

αβ µ

α α

∆+ ≥ +

with ( ) ( ) ( ) [ ]1 2 1 2

1 2, ,1, , 0,1

d d d dd dβ β= ∀ ∈∑ and 0∆ ≥ .

Where we define ( ) ( ) ( )( )( ), log 1 1x t t t x tµ = − − − .

A. The generalized-degree terms

The generalized-degree terms in the GLT distributions, for

example the terms such as 1 2

0.2x x and 2 3

1 20.2x x in the

distribution: ( ) 2 3

1 2 1 2 1 2 1 2, 0.4 0.2 0.2 0.2x x x x x x x xβ = + + + , lead

to the creation of encoding symbol formed by XORing input

symbols from different segments. In the absence of these terms, the generalized distributions can be viewed as a simple

multiplexing of two conventional LT distributions operating

on distinct message segments. Hence it is appropriate to make

the inquiry that when is mixing the most advantageous, i.e. if

one were to restrict the design of generalized degrees to

distributions without any generaliuzed-degree term, when is

the performance loss greatest? In order to answer this question

we solved P2 for various demands at 1 2 0.5α α= =

and

1 2,k k → ∞ . It was observed that, in the asymptotic regime,

when the demand is uniform, mixing provides no advantage.

However, in case of un-equal demand mixing can provide

gains even in the asymptotic regime. The results of our

enumeration are shown in Figure 4. It can be observed that the

absence of mixing can increase overhead by upto 12%.

IV. DESIGN OF MIXING PROBABILITIES

In order to design the mixing rules observe that if the

sources employ degree distributions ( )1 1xβ , ( )2 2

xβ and the

relay employs the rules Λ then the output of the relay is described by the following GLT distribution:

( )( )( )

( ) ( ) ( )( )

( ) ( )

( )( )

( ) ( )( )

1 2

1 2 1 2 1 2 1 2

1 21 2

1 2 1 2

1 2

1, 2 1 1, 2, 2, , ,

1 2 1,2,

1, 2 1 2, ,

,

d d

d d d d d d d d

d dd d

d d d d

x xx x

x x

β β β βλ

β β

Λ + Λ = Λ

∑

(5)

The LP for designing the mixing rules is obtained as follows

by substituting Eq. (5) in problem P3.

Problem 4 (P4): ( ) ( )* *

,

, arg maxΛ ∆

Λ ∆ = ∆

subject to 1,

, [0, )i

i t z∀ ∀ ∈

( )

( )( )1,0 2, 1

1

1, 1 1, 2 2,

, , 0i

i i i

zt t k t

z z z

αλ µ

α α

∆+ ≥ +

( )

( )0,1 2, 2,2

2

1, 1,1 1, 2 2,

, , 0i i

i ii i

z zt t k t

z zz z

αλ µ

α α

∆+ ≥ +

with ( ) ( ) ( ) [ ]1 2 1 2

(.) (.)

1 2 , ,, : 1, 0,1

d d d dd d∀ Λ = Λ ∈∑ and 0∆ ≥ .

V. NUMERICAL RESULTS

We considered a network with 1 750k =

and

2 1250k = ,

where the source 1 employs an LT code for an local non-

uniform demand of [0.6,0.8], while source 2 employs the LT

code for an local demand of [0.3, 0.7]. Thus, the distributions

employed at the sources are given by:

( ) 1 2 3

1 1 1 1 10.1 0.89 0.01x x x xβ = + +

(5)

( ) 1 2

2 2 2 20.57 0.43x x xβ = +

(6)

We consider that the relay is subscribed to by two set of sinks

with demands ( )1 0.95,0.95z = and ( )2 0.8,0.6z = .

The optimal mixing rules for the network were found to be:

( )( )

( )( )1 1

1,1 2,11, 0.0215 Λ = Λ =

( )( )

( )( )

( )( )

( )( )1,2 1,2 1,2 1,2

1,2 2,1 2,2 3,11, 0.9785, 0.0265, 0.8532Λ = Λ = Λ = Λ =

( )( )

( )( )

( )( )2 2 2

3,1 2,2 3,20.1468, 0.9735, 1Λ = Λ = Λ =

If the operation of the relay is restricted to simple mixing

than the optimal parameters were 1 2

0.12, 0.47p p= = and

30.41p = . In order to compare the various designs, we define

“failure to recover rate” as the probability with which a sink

fails to recover its desired demand. Figure 5 provides a

comparison of proposed morphing designs with simple mixing

0.290.61

0.930.29

0.61

0.93

2

7

12% ove

rhea

d with

out m

ixing

Figure 4 The percentage overhead without the generalized-terms as a

function of the demand (z1,z2).

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and multiplexing. It can be clearly observed that mixing at the

relay provides significant gains. The simple mixing as well as

the generalized morphing proposed in this paper significantly

outperform the forwarding based multiplexing strategy.

Furthermore, we can observe that generalized morphing when

compared with simple mixing, reduces the overhead by ~10%

for sink set 1 and by ~7% for sink set 2. Hence it can be

concluded that the degree sensitive XORing conducted by

morphing leads to improved designs compared to the

probabilistic mixing proposed in [16].

REFERENCES

[1] A. Eryilmaz, A. Ozdaglar and M. Medard, “On the Delay and Throughput Gains of Coding in Unreliable Networks”, IEEE Trans. on

Information Theory, vol. 54, no. 12, pp. 5511-5524, 2008.

[2] C. Fragouli and E. Soljanin, “Network Coding Fundamentals”, Foundations and Trends in Networking. Hanover, MA: now Publishers Inc., vol. 2, no. 1, pp. 1-133, 2007

[3] M. Luby, “LT codes”, 43rd

Annual IEEE Symposium on Foundations of Computer Science, 2002.

[4] A. Shokrollahi, “Raptor Codes”, IEEE Trans. on Information Theory, vol. 52, no. 6, pp. 2551-2567, June 2006.

[5] P. Maymounkov, “Online Code”, NYU Technical Report TR2003-883,

2002.

[6] J. W. Byers, M. Luby and M. Mitzenmacher, “A Digital Fountain Approach to Asynchronous Reliable Multicast”, IEEE J-SAC, 20 (8),

pp. 1528-1540, October 2002.

[7] S. Puducheri, J. Kliewer and T. Fuja, "The design and performance of distributed LT codes", IEEE Trans. on Information Theory, vol. 53, no. 10, pp. 3740-3754, October, 2007.

[8] B. Hajek, “Connections between network coding and stochastic network theory", Stochastic Networks Conference, June 19-24, 2006, Urbana.

[9] S. Sanghavi, “Intermediate Performance of Rateless Codes”, Information Theory Workshop 2007, Tahoe.

[10] A. Kamra, J. Feldman, V. Misra and D. Rubenstein, “Growth Codes:

Maximizing Sensor Network Data Persistence”, ACM Sigcomm, 2006.

[11] N. Rahnavard, B. N. Vellambi, and F. Fekri, “Rateless codes with unequal error protection property”, IEEE Trans. on Information

Theory, vol. 53, no. 4., pp. 1521-1532, April 2007.

[12] A. G. Dimakis, J. Wang and K. Ramchandran, "Unequal Growth Codes: Intermediate Performance and Unequal Error Protection for Video Streaming", MMSP, October 2007.

[13] S. S. Karande, “Network Channel Coding: Importance of in-network processing and side-information”, Ph.D. dissertation, Dept. of Elec.

Eng., Michigan State Univ., East Lansing, 2007.

[14] S S. Karande, K. Misra, S. Soltani and H. Radha, “Design and analysis of Generalized LT codes using colored ripples”, IEEE Symposium on Information Theory, 2008

[15] Y. Li and E. Soljanin, "Rateless Codes for Single-Server Streaming to Diverse Users", Allerton Conference, 2009.

[16] A. Talari and N. Rahnavard, "Distributed Rateless Codes with UEP Property", IEEE Symposium on Information Theory, 2010..

(a)

(b)

Figure 5 The failure recovery rate of (a) Sink set with demand [0.95,0.95] and (b) Sink set with demand [0.8,0.6].

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