An Optimal Adaptive Network Coding Scheme for Minimizing Decoding Delay in Broadcast Erasure Channels

Abstract

We are concerned with designing feedback-based adaptive network coding schemes with the aim of minimizing decoding delay in each transmission in packet-based erasure networks. We study systems where each packet brings new information to the destination regardless of its order and require the packets to be instantaneously decodable. We first formulate the decoding delay minimization problem as an integer linear program and then propose efficient algorithms for finding its optimal solution(s). We show that our problem formulation is applicable to memoryless erasures as well as Gilbert-Elliott erasures with memory. We then propose a number of heuristic algorithms with worst case linear execution complexity that can be used when an optimal solution cannot be found in a reasonable time. We verify the delay and speed performance of our techniques through numerical analysis. This analysis reveals that by taking channel memory into account in network coding decisions, one can considerably reduce decoding delays.

1. Introduction

In this paper, we are concerned with designing feedback-based adaptive network coding schemes that can deliver high throughputs and low decoding delays in packet erasure networks. We first present some background on existing work and emphasize that the notion of delay and the choice of a suitable network coding strategy are highly entangled with the underlying application.

1.1. Motivation and Background

Consider a broadcast packet-based transmission from one source to many destinations where erasures can occur in the links between the source and destinations. Two main throughput optimal schemes to deal with such erasures are fountain codes [1] and random linear network codes (RLNC) [2]. In the latter scheme, for example, the source transmits random linear mixtures of all the packets to be delivered. It is well-known that if the random coefficients are chosen from a finite field with a sufficiently large size, each coded packet will almost surely become linearly independent of all previously received coded packets and hence, innovative for every destination [2]. The scheme is therefore almost surely throughput optimal. Another benefit of fountain codes and RLNC is that they do not require feedback about erasures in individual links in order to operate.

However in these schemes, throughput optimality comes at the cost of large decoding delays, as the receiver needs, in general, to collect all coded packets in a block before being able to decode. Despite this drawback, there are applications which are insensitive to such delays. Consider, for example, a simple software update (file download). The update only starts to work when the whole file is downloaded. In this case, the main desired properties are throughput optimality and the mean completion time and there is often little or no incentive to aim for partial "premature" decoding. The completion time performance of RLNC for rateless file download applications has been considered in [3]. In [3], the mean completion time of RLNC is shown to be much shorter than scheduling. Reference [4] considers time division duplex systems with large round-trip link latencies and proposes solutions for the number of coded packet transmissions before waiting for acknowledgement on the received number of degrees of freedom.

There are applications where partial decoding can crucially influence the end user's experience. Consider, for example, broadcasting a continuous stream of video or audio in live or playback modes. Even though fountain codes and RLNC are throughput optimal, having to wait for the entire coded block to arrive can result in unacceptable delays in the application layer. But, we also note that partial decoding of packets out of their natural temporal order does not necessarily translate into low delivery delays desired by the application layer. The authors in [5, 6] have proposed feedback-based throughput-optimal schemes to deal with the transmitter queue size, as well as decoding and delivery delays at the destinations. When the traffic load approaches system capacity, their methods are shown to behave "gracefully" and meet the delay performance benchmark of single-receiver automatic repeat request (ARQ) schemes.

There is yet another set of applications for which partial decoding is beneficial and can result in lower delays irrespective of the order in which packets are being decoded. Consider, for example, a wireless sensor network in which there is a fusion/command center together with numerous sensors/agents scattered in a region. Each sensor/agent has to execute or process one or more complex commands. Each command and its associated data is dispatched from the center in a packet. For coordination purposes, each agent needs to know its own and other agents' commands. Therefore, commands are broadcast to everyone in the network. In this application, in-order processing/execution of commands may not be a real issue. However, fast command execution may be crucial and therefore, it is imperative that innovative packets arrive and get decoded at the destinations as quickly as possible regardless of their order. As another example, consider emergency operations in a large geographical region where emergency-related updates of the map of the area need to be dispatched to all emergency crew members. In such situations too, updates of different parts of the map can be decoded in any order and still be useful for handling the emergency.

Finally, some applications may be designed in such a way that they are insensitive to in-order delivery. This can be particularly useful where the transport medium is unreliable. In such a case, it may be natural to use multiple-description source coding techniques [7], in which every decoded packet brings new information to the destination, irrespective of its order. In light of the emergency applications described above, one can perform multiple-description coding for map updates, so that updates of different subregions can be divided into multiple packets and each packet can provide an improved view of one region in a truly order-insensitive fashion.

1.2. Contributions

In this paper, we are inspired by the last set of order-insensitive packet delivery applications and hence, focus on designing network coding schemes that, with the help of feedback, can deliver innovative packets in any order to the destination and also guarantee fast decoding of such packets. As a first step towards such goal, we limit ourselves to broadcast erasure channels, but emphasize that the ideas can be extended to other more complicated scenarios. We also consider the class of instantaneously decodable network coding schemes, in which each coded transmission contains at most one new source packet that a receiver has not decoded yet. The rationale is that in an order-insensitive application, any innovative packet that cannot be decoded immediately incurs a unit of delay. Obviously, one other source of delay is when a coded packet does not contain any new information for a receiver and hence, is not innovative. A similar definition of the decoding delay was first considered in [8], where the authors presented a number of heuristic algorithms to reduce order-insensitive decoding delay. In this context, our main contributions are the following.

• (i) In Section 1.1, we have motivated the problem in light of possible applications in sensor and ad hoc networks. To the best of our knowledge, such application-dependent classification of network coding delays did not previously exist in the literature.

• (ii) In Section 3.1, we present a systematic framework for the minimization of decoding delay in each transmission subject to the instantaneous decodability constraint. We show that this problem can be cast into a special integer linear programming (ILP) framework, where instantaneously decodable packet transmission corresponds to a set packing problem [9] on an appropriately defined set structure.

• (iii) In Section 3.2, we provide a customized and efficient method for finding the optimal solution to the set packing problem (which is in general NP-hard). Our numerical results in Section 6 show that for reasonably sized number of receivers, the optimum solution(s) can be found in a time that is linearly proportional to the total number of packets.

• (iv) In Section 4, we discuss decoding delay minimization for an important class of erasure channels with memory, which can occur in wireless communication systems due to deep fades and shadowing [10]. We show that the general set packing framework in Section 3 can be easily modified to account for the erasure memory. Our results in Section 6 reveal that by adapting network coding decisions based on channel erasure conditions, significant improvements in delay are possible compared to when decisions are taken irrespective of channel states.

• (v) In Section 5, we provide a number of heuristic variations of the optimal search for finding (possibly suboptimal) solutions faster, if needed. Our results in Section 6 show that such heuristics work very well and often provide solutions that are very close to the search algorithm. Moreover, they improve on the proposed random opportunistic method in [8].

2. Network Model

Consider a single source that wants to broadcast some data to receivers, denoted by for . The data to be broadcast is divided into packets, denoted by for . Time is slotted and the source can transmit one (possibly coded) packet per slot.

A packet erasure link connects the source to each individual receiver . Erasures in different links can be independent or correlated with each other. Different erasures in a single link can be independent (memoryless) or correlated with each other (with memory) over time.

For memoryless erasures, an erasure in link can occur with a probability of in each packet transmission round independent of previous erasures.

For correlated erasures, we consider the well-known Gilbert-Elliott channel (GEC) [11], which is a Markov model with a good and a bad state. If the channel is in the good state, packets can be successfully received, while in the bad state packets are lost (e.g., due to deep fades or shadowing in the channel). The probability of moving from the good state to the bad state in link is and the probability of moving from the bad state to the good state is , where is the time slot index. Steady-state probabilities are given by and . Following [12], we define the memory content of the GEC in link as , which signifies the persistence of the channel in remaining in the same state. A small means a channel with little memory and a large means a channel with large memory.

Before transmission of the next packet, the source collects error-free and delay-free 1-bit feedback from each destination indicating if the packet was successfully received or not. A successful reception generates an acknowledgement (ACK) and an erasure generates a negative acknowledgement (NAK). This feedback is used for optimizing network coding decisions at the source for the next packet transmission round, as described in future sections.

In this work, we consider linear network coding [2] in which coded packets are formed by taking linear combinations of the original source packets. Packets are vectors of fixed size over a finite field . The coefficient vector used for linear network coding is sent in the packet header so that each destination can at some point recover the original packets. Since in this paper we are only dealing with instantaneously decodable packet transmission, it suffices to consider linear network coding over . That is, coded packets are formed using binary XOR of the original source packets. Thus, network coding is performed in a similar manner as in [13].

Definition 1.

A transmitted packet is instantaneously decodable for receiver if it is a linear combination of source packets containing at most one source packet that has not decoded yet. A scheme is called instantaneously decodable if all transmissions have this property for all receivers.

Definition 2.

At the end of transmission round in an instantaneously decodable scheme, the knowledge of receiver is the set consisting of all packets that the receiver has decoded so far. The receiver can therefore, compute any linear combination of the packets that it has decoded for decoding future packets.

Definition 3.

In an instantaneously decodable scheme, a coded packet is called non-innovative for receiver if it only contains source packets that the receiver has decoded so far. Otherwise, the packet is innovative.

Definition 4.

A scheme is called rate or throughput optimal if all transmissions are innovative for the entire set of receivers.

Definition 5.

In time slot , receiver experiences one unit of delay if it successfully receives a packet that is either non-innovative or not instantaneously decodable. If we impose instantaneous decodability on the scheme, a delay can only occur if the received packet is not innovative.

Note that in the last definition, we do not count channel inflicted delays due to erasures. The delay only counts "algorithmic" overhead delays when we are not able to provide innovative and instantaneously decodable packets to a receiver.

As an example, if the knowledge of is , receiving will cause to experience one unit of delay, whereas is innovative and instantaneously decodable, hence does not incur any delay.

We note that a packet that is not transmitted yet or transmitted but not received by any receiver can be transmitted in an uncoded manner at any transmission slot without incurring any algorithmic delay. In fact, this is how the transmission starts: by sending uncoded, for example.

A zero-delay scheme would require all packets to be both innovative and instantaneously decodable to all receivers. Thus zero-delay implies rate optimality, but not vice versa. As the authors show in [8, Theorem  1] for the case of and receivers, there exists an offline algorithm that is both rate optimal and delay-free. For the authors prove that a zero-delay algorithm does not exist. By offline we mean that the algorithm needs to know future realizations of erasures in broadcast links. In contrast, an online algorithm decides on what to send in the next time slot based on the information received in the past and in the current slot. In this paper, we focus on designing online algorithms.

3. Optimization Framework

3.1. Problem Formulation Based on Integer Linear Programming

Instantaneous decodability can be naturally cast into the framework of integer optimization. To this end, let us fix the packet transmission round to and consider the knowledge of all receivers, which is also available at the source because of the feedback. The state of the entire system at time index (in terms of packets that are still needed by the receivers) can be described by an binary receiver-packet incidence matrix with elements

(1)

Columns of matrix are denoted by to . We assume that packets received by all receivers are removed from the receiver-packet incidence matrix. Hence, does not contain any all-zero columns.

Example 1.

Consider receivers and packets. Before the transmission begins, the receiver-packet incidence matrix is an all-one matrix. If we send packet in the first transmission round and assuming that only receiver successfully receives it, will become

(2)

If we send packet in the next transmission round and assuming that only receiver successfully receives it, will then be

(3)

The condition of instantaneous decodability means that at any transmission round we cannot choose more than one packet which is still unknown to a receiver . In the example above, at , we cannot send because it contains more than one packet unknown to .

Let represent a binary decision vector of length that determines which packets are being coded together. The transmitted packet consists of the binary XOR of the source packets for which . More formally, we can define the instantaneous decodability constraint for all receivers as , where represents an all-one vector of length and the inequality is examined on an element-by-element basis (Note that although is a binary or Boolean vector, is calculated in real domain. Hence, is in fact a pseudo-Boolean constraint.). This condition ensures that a transmitted coded packet contains at most one unknown source packet for each receiver. A vector is called infeasible if it does not satisfy the instantaneous decodability condition. In other words, is called infeasible if and only if there exists at least one for which in . A vector is called a solution if and only if it satisfies . In the rest of this paper, "" and " is a solution" are used interchangeably.

Now consider sets , where is the nonempty set of receivers that still need source packet . Note that these sets can be easily determined by looking at the columns of matrix . The "importance" of packet can be, for example, taken to be the size of set , which is the number of receivers that still need .

We now formally describe the optimization procedure that should be performed at the transmitter. Maximizing the number of receivers for which a transmission is innovative, subject to the constraint of instantaneous decodability, can be posed as the following (binary-valued) integer linear program (ILP):

(4)

where . This is a standard problem in combinatorial optimization, usually called set packing [9]. Here the universe is the set of all receivers and we need to find disjoint (due to instantaneous decodability condition) subsets with the largest total size. In the (most desirable) case when equality holds in for every receiver, we also speak of a set partition. This is equivalent to a zero-delay transmission.

In Section 4, we will consider other measures of packet importance and discuss the role of in tailoring the optimization problem according to the application requirements or channel conditions, such as memory in erasure links.

We assume that elements of , which signify packet importance, are all positive. If one has already found a solution such as with , then changing this solution into by changing into can only result in a strictly smaller than . We say that given solution , is clearly suboptimal and hence, can be discarded in an algorithm that searches for the optimal solution(s).

3.2. Efficient Search Methods for Finding the Optimal Solution of (4)

It is well known that the set packing problem is NP-hard [9]. Here, we present an efficient ILP solver designed to take advantage of the specific problem structure. Later, we will see that for many practical situations of interest, our method performs well empirically. Based on this framework, we will also present some heuristics in Section 5 to deal with more complicated and time-consuming problem instances.

We begin presenting our method by first defining constrained and unconstrained variables.

Definition 6.

Two binary-valued variables are said to be constrained if they cannot be simultaneously in a solution. Or formally, and are constrained if for any satisfying , (Again, note that the addition of variables takes place in real domain.). We also say that is constrained to and vice versa. It can be proven that and are constrained if and only if there exits at least one row index in for which .

Definition 7.

The set of all variables constrained to is called the constrained set of and is denoted by . That is,

(5)

If and are not constrained to each other ( and ), then columns and in cannot have nonzero elements in the same row position. That is, for each row index , and .

Definition 8.

A variable is said to be unconstrained if . The set of all unconstrained variables is denoted by and is referred to as the unconstrained set.

If is an unconstrained variable, then for each row index , for all (otherwise, and would become constrained).

Example 2.

Consider the following receiver-packet incidence matrix

(6)

One can easily verify the relations defined above. For example, variables and are constrained because for , . Variables and are not constrained to each other because columns and do not have a nonzero element in the same row position. Variable is unconstrained because no other column has a nonzero element in rows 6 or 7. In summary, , , , and .

To design an efficient search algorithm, one needs to efficiently prune the parameter space and reduce the problem size. We make the following observations for pruning of the parameter space.

• (1) Unconstrained variables must be set to 1. In other words, setting those variables to 0 does not contribute to the optimal solution (note that the elements in are positive). In the above example, and must be set to 1 because no other variable is constrained to them (we will make this statement formal in the optimality proof of the algorithm in the appendix).

• (2) If a constrained variable is set to 1 all members of its constrained set must be set to 0. In the above example, setting forces and to zero.

• (3) At a given step, the parameter space can be pruned most by resolving the variable with the largest constrained set.

Application of the third observation, in a search algorithm results in greedy pruning of the parameter space. We note that greedy pruning is only optimal for a given step of the algorithm and is not guaranteed to result in the optimal reduction of the overall complexity of the search.

We now make a final remark before presenting the search algorithm. In particular, we have observed that finding constrained sets for each variable in each step of the algorithm can be somewhat time consuming. A very effective alternative is to first sort matrix , column-wise, in descending order of the number of 's in each column. Setting the "most important" head variable (with the highest ) to is likely to result in the largest constrained set (because it potentially overlaps with many other variables) and hence, many variables will be resolved in the next recursion. We will refer to the approach based on finding the largest constrained set as the greedy pruning strategy and to the alterative approach as the sorted pruning search strategy.

The greedy pruning search strategy is shown in Figure 1, which with appropriate modifications can also represent the sorted pruning variation.Let denote the problem of size whose input is an receiver-packet incidence matrix and whose output is a set of solutions of the form of length which satisfy the instantaneous decodability condition . The algorithms can be described as shown in Algorithm 1.

Algorithm 1: Recursive search for the optimal solution(s) of (4).

() Start with the original problem of size.

() if sorted pruning strategy is desired then

()  Rearrange the variables inin descending order of packet importance (number of's in each column).

() end if

• (5) Solve :

• (6) if   then

• (7)  Return (since the variable is not constrained).

• (8) else

• (9)    if greedy pruning strategy is desired then

• (10)   Determine the constrained set for all variables to .

• (11)   Denote the index of the variable with the largest constrained set by and the cardinality of its constrained

    set by .

• (12)  else

(13)  Determine the constrained set for the head variablewith cardinalityand also the set of unconstrained

     variables (Note that we have overused index 1 to refer to the head variable in the reordered matrix at each

     recursion.). Set.

• (14)  end if

(15)  Denote the cardinality of the unconstrained setby.

(16)  Set all the unconstrained variables to 1.

(17)  Setand the variables in its corresponding constrained setto 0.

(18)  Reduce the problem by removing resolved variables. Reduceaccordingly.

(19) Solve(Note thatunconstrained variables are set to one,andvariables constrained by

    are set to zero, hence a total ofvariables are resolved.).

(20) Combine the solution with previously resolved variables. Save solution.

(21)  Set.

(22) Reduce the problem by removing resolved variables. Reduceaccordingly.

(23) Solve(Note thatunconstrained variables are set to one and, hence a total ofvariables

   are resolved.).

(24) Combine the solution with previously resolved variables. Return solution(s).

(25)  end if  

Figure 1

A schematic of Algorithm 1 with greedy pruning for finding the optimal network coding solution of (4). Note that the algorithm is recursive as it calls and within itself.

In the appendix, we prove by structural induction that Algorithm 1 is guaranteed to return all optimal solutions of (4). However, we note that not every solution returned by Algorithm 1 is optimal. The nonoptimal solutions can be easily discarded by testing against the objective function (4) at the end of the algorithm. We also note that in Algorithm 1, we can simply remove those packets received by every receiver from the problem. If there are such variables, we can start step above from instead of . The Matlab code for both the greedy and sorted pruning algorithms can be found at http://users.rsise.anu.edu.au/~parastoo/netcod/.

We conclude this section by a brief note on the computational complexity of Algorithm 1. Let us denote the number of recursions required to solve the problem of size by . According to Algorithm 1, this problem is always broken into two smaller problems of size and . Therefore, one can find the number of recursions required to solve by recursively computing . The recursion stops when one reaches a problem of size 1 (only one packet to transmit) where .

4. Adaptive Network Coding in the Presence of Erasure Memory

Here, we present a generalization of the set packing approach for coded transmission in erasure channels with memory. The idea is that the importance of a packet is no longer determined by how many receivers need , but by the probability that will be successfully decoded by the receivers that need it. In computing this probability, one can use the fact that successive channel erasures in a link are usually correlated with each other and hence, their history can be used to make predictions about whether a receiver is going to experience erasure or not in the next time slot. To present the idea, we focus on the GEC model for representing channel erasures. More general memory models for erasure can also be incorporated into our framework.

We define the reward of sending a packet to receiver as the probability of successful reception by in the next time slot: , where is the state of in the previous transmission round (Statements like "state of " should be interpreted as the state of the physical link connecting the source to .). The total reward or importance of sending packet is then

(7)

The above weight vector gives higher priority to a packet for which there is a higher chance of successful reception, because the receivers that need are more likely to be in good state in the next time slot. With this newly defined weight vector, one can try to solve the optimization problem given in (4) under the same instantaneous decodability condition.

Remark 1.

We conclude this section by emphasizing that the optimization framework in (4) is very flexible in accommodating other possibilities for the weight vector , which can be appropriately determined based on the application. For example, instead of allocating the same weight to a packet needed by a subset of receivers, one can allocate different weights to the same packet (looking column-wise at ) depending on the priorities or demands of each user. In the map update example described in the Introduction, different emergency units can adaptively flag to the base station different parts of the map as more or less important depending on their distance from a certain disaster zone. The task of the base station is then to send a packet combination that satisfies the largest total priority. One can also combine user-dependent packet weights with the channel state prediction outcomes in a GEC. One possibility is to multiply the probabilities by the receiver priority. It could then turn out that although a receiver is more likely to be in erasure in the next transmission round, it may be served because of a high priority request.

5. Heuristic Search Algorithms

In Section 3.2, we proposed efficient search algorithms for finding the optimal solution(s) of (4). However, there may be situations where one would like to obtain a (possibly suboptimal) solution much more quickly. This may be the case, for example, when the total number of packets to be transmitted is very large. Therefore, designing efficient heuristic algorithms to complement the optimal search is important. In this section, we propose a number of such heuristics.

5.1. Heuristic 1—Weight Sorted Heuristic Algorithm

The idea behind this recursive algorithm is very simple. As in Algorithm 1, we start with the original problem of size . We then rearrange the columns of the matrix in descending order of (starting from the packet with the highest weight). Note that this is different from the sorted pruning version of the Algorithm 1, in which the columns of were sorted in descending order of to potentially result in large constrained sets. We then set the head variable and find its corresponding constrained set to resolve variables that are to be set to zero. We then solve the smaller problem of size and continue until the problem cannot be further reduced. One main difference between Heuristic 1 and Algorithm 1 is that at each recursion, the head variable is only set to one; the other possibility of is not pursued at all. In a sense, this heuristic algorithm finds greedy solutions to the problem at each recursion by serving the highest priority packet. In this heuristic algorithm, all unconstrained variables are naturally set to 1 in the course of the algorithm. The computational complexity of this method is at worst proportional to , which can happen when there is no constraint between packets.

5.2. Heuristic 2—Search Algorithm 1 with Maximum Recursions/Elapsed Time

It is possible to terminate the recursive search Algorithm 1 prematurely once it reaches a maximum number of allowed recursions/elapsed time. If the algorithm reaches this value and the search is not complete, it performs a termination procedure whereby it heuristically resolves the remaining unresolved packets in the current incomplete solution. That is, it performs Heuristic 1 on a smaller problem, which is yet to be solved. It then returns the best solution that has been found so far. We note that due the extra termination procedure, the actual number of recursions/elapsed time can be (slightly) higher than the preset value.

Two comments are in order here. Firstly, Algorithm 1 is designed to sort the matrix based on the number of receivers that need a packet. It only reverts to sorting the unresolved variables based on the vector in the termination process. Secondly, if the maximum number of recursions is set to one, Algorithm 1 just performs the termination process and becomes identical to Heuristic 1.

5.3. Heuristic 3—Dynamic Number of Recursions

This heuristic is based on Heuristic 2, where we dynamically increase the number of allowed recursions as needed. At each transmission round, we start with only one allowed recursion (effectively run Heuristic 1). If the throughput (Let denote the index of receivers that still need at least one packet and denote such receivers. The achieved throughput at time slot is defined as , where is the found solution and is an appropriate function of receivers' needs. For memoryless erasures and for GEC's (refer to Section 4 and (7)).) is higher than a desired value, there is no need to proceed any further. Otherwise, we can gradually increase the number of recursions by an appropriate step size. This heuristic stops when it either reaches the maximum allowed recursions or when increasing the number of recursions does not result in a noticeable improvement in the throughput.

6. Numerical Results and Secondary Coding Considerations

We start this section by presenting end-to-end decoding delay results for memoryless erasure channels. We then specialize to erasure channels with memory. The end-to-end problem is the complete transmission of packets. End-to-end decoding delay of a receiver is the sum of decoding delays for the receiver in each transmission step. In the following, when we say "the delay performance of method X", we are referring to the delay performance of the end-to-end transmission, where method X is applied at each step.

In the course of presenting the results and based on the observed trends, we will discuss some secondary coding techniques and post processing considerations that can improve the decoding delay. Throughout the analysis of this section, we assume independent erasures in different links with identical probabilities. Hence, we can drop subscript when referring to link erasure probabilities.

Figure 2 shows the median of decoding delay for the transmission of packets to to receivers. Channel erasures are memoryless and occur with a high probability of independently in every link. The median of delay is computed across all receivers and is, in fact, also the median across many stochastic runs of the algorithms. The first curve from below shows the delay obtained from Algorithm 1 (Throughout the numerical evaluations, we used the sorted pruning version of Algorithm 1.). The middle curve is the delay obtained by performing Heuristic 1. The top curve shows a reproduction of delay results reported in [8] which are based on a random opportunistic instantaneous network coding strategy. In this case, the transmitter first selects a packet needed by at least one receiver at random. Then, it goes over other packets in some order and adds a packet to the current choice only if their addition still results in instantaneous decodability. In comparison, Heuristic 1 performs noticeably better than that in [8] and more importantly, is not much far away from the results of Algorithm 1. This is specially important since for some number of receivers, Heuristic 1 can run considerably faster than Algorithm 1, which will be shown in the coming figures shortly.

Figure 2

Median of decoding delay for the transmission of packets to to receivers. Channel erasures are memoryless and occur with a high probability of independently in every link. Algorithm 1, Heuristic 1 and random Heuristic [8] are compared with each other.

Figure 3 compares the mean delay performance of different heuristics presented in Section 5 with that of Algorithm 1. Similar to the previous figure, mean delay is computed across all receivers. The delay performance of Heuristic 2, Heuristic 3, and Algorithm 1 are close, whereas Heuristic 1 results in the largest delay. A careful reader may notice that the end-to-end performance of Heuristic 2 is at times better than Algorithm 1. While the difference is practically insignificant, this deserves some explanation. The end-to-end transmission problem involves making packet transmission decisions at each step. While all algorithms start with the same packet incidence matrix (all-ones), due to packet erasures and as they make decisions about transmission of packets at each step, they take diverging paths in the solution space. As a result, they end up with different packet incidence matrices to solve over time. Hence, it is conceivable for an algorithm to make suboptimal decisions at one or more steps and yet end up with a better end-to-end delay than Algorithm 1 that strictly makes optimal decisions at every step. Intuition suggests that an algorithm such as Heuristic 1 that consistently makes suboptimal decisions is unlikely to outperform Algorithm 1 end-to-end, which is confirmed by the numerical results. However, an algorithm such as Heuristic 2 which almost always makes optimal decisions with only infrequent exceptions, may outperform Algorithm 1. According to Figure 3, these perturbations in end-to-end performance are practically insignificant and the intuitive choice of the optimal or a largely optimal algorithm at each step will result in the best end-to-end performance.

Figure 3

Mean decoding delay for the transmission of packets to to receivers. Algorithm 1 is compared with Heuristics 1–3. Both Heuristics 2 and 3 perform very closely to Algorithm 1. The maximum number of recursions for both Heuristic 2 and 3 is set to 100.

We note that the delays presented here (and also in the following figures) are, in fact, excess median or mean delays beyond the minimum required number of transmissions, which is . For example, a mean delay of 10 slots for packets signifies on average overhead, which is the price for guaranteeing instantaneous decodability. In other words, one measure of throughput is , where is the mean delay across all receivers. An example is shown in Figure 3. For up to around receivers in the system, Algorithm 1, Heuristics 2, and 3 ensure an average throughput loss of .

It is quite possible that Algorithm 1 returns multiple network coding solutions all of which have the same objective value . A natural question that arises is whether systematic selection of a solution with a particular property is better than others in the presence of erasures in the channel. Our experiments verify that indeed some secondary post processing on the solutions can improve the end-to-end delay. In particular, we compare two post processing techniques: selecting a solution which involves minimum amount of coding (lowest number of 1's in the solution vector ) and selecting a solution with maximum amount of coding (highest number of 1's in the solution vector ). Figure 4 shows the effects of such processing on the overall decoding delays. It is clear that maximum coding is not a reasonable choice and results in worse delays compared with minimum coding. We attempt to explain this behavior by means of an example and intuitive reasoning. Let us assume that there are packets to be transmitted to receivers and at the beginning of the third transmission round, matrix is given as follows

(8)

It is clear that there are two optimal solutions: we can either send packets or packet by itself, where the former involves coding and latter is uncoded. Now let us assume that we select the maximum coding strategy and send . If in the third transmission round only successfully receives, will become

(9)

and clearly the optimal solution is sending packet . If in the fourth transmission round only successfully receives, will become

(10)

where it is evident that in the fifth transmission round, we cannot find a packet which is innovative and instantaneously decodable for all the three receivers. On the other hand, one can verify that if we adopt a minimum coding strategy and send packet in the third transmission round, we can always find innovative and instantaneously decodable packets for all three receivers in the future regardless of erasures in the channel. In summary, solutions with less coding tend to cause less constrains on the problem in the future.

Figure 4

The effect of post processing on mean delay. Whenever Algorithm 1 returns multiple solutions, minimum amount of coding should be chosen. Heuristic 1 is shown for reference.

It is noted in Figure 4 that the first solution returned by Algorithm 1 performs almost the same as the minimum coding solution. The reason for this is that Algorithm 1 first ranks the packets based on the number of receivers that need them. Therefore, the first solution picked by the algorithm is likely to contain packets with largest constrained sets and hence, many resolved packets are set to zero, which often translates into small amount of coding. Throughout this section, unless otherwise stated, we have shown the delay results based on the first returned solution of Algorithm 1.

It is interesting to analyze the actual number of recursions that the search in Algorithm 1 takes to find the optimum solution. This is shown in Figure 5 for packets along with the number of recursions required in Heuristics 1, 2, and 3. Algorithm 1 shows three modes of behavior: low, medium, and high number of recursions. When the number of receivers is larger than , Algorithm 1 finds the optimal solution very quickly and the number of recursions is very close to the number of packets . However, when the number of receivers is lower, the constraints that each receiver imposes on the network coding decisions cannot limit the search space enough and hence, a large number of combinations have to be tested. Obviously, Heuristic 1 has the lowest number of recursions. Compared to Heuristic 2 with 100 fixed recursions, dynamic Heuristic 3 can almost halve the number of recursions with negligible effect on delay performance (see Figure 3). By referring to Figure 3, we conclude that for the system under consideration, the excessive number of recursions in Algorithm 1 is not warranted as it does not result in any noticeable delay improvement compared to Heuristics 2 or 3.

Figure 5

Average number of recursions in Algorithm 1 and Heuristics 1–3. The maximum number of recursions for both Heuristic 2 and 3 is set to 100. By referring to Figure 3, we observe that for small number of receivers, Heuristics 2-3 can provide same decoding delays at a fraction of computational complexity.

Figure 6 shows the effect of increasing the number of packets on the computational complexity of Algorithm 1 in terms of number of recursions to complete the search.Three different numbers of receivers , , and are considered. The complexity remains linear with the number of packets for well-sized receiver populations (30 and 40 receivers). This is in agreement with observations in Figure 5. When the number of receivers is not so large (see the blue curve in Figure 6 for ), we see a sudden growth in complexity, in terms of number of recursions, when packets. In such situations, truncating the number of recursion to be linear with the number of packets (Heuristic 2) is a good alternative.

Figure 6

The effect of increasing the number of packets on the computational complexity of Algorithm 1 in terms of number of recursions. The complexity remains linear with the number of packets for well-sized receiver populations (30 and 40 receivers).

Figure 7 shows the impact of the number of packets and also erasure probability on the decoding delay. The normalized mean delay versus number of packets is plotted for three different erasure probabilities , , and , which are still high erasure probabilities. The number of receivers is fixed to . The delay performance of Heuristics 1 and 3 are shown. A few observations are made. Firstly, as expected, the delay (both absolute and normalized measures) decreases as the erasure probability decreases. Secondly, the difference in the delay performance between Heuristics 1 and 3 decreases as the erasure probability decreases. This trend has also been observed for other number of receivers. Moreover, the difference between heuristics and Algorithm 1 decreases with erasure probability, which is not shown here for clarity of figure. Finally, the normalized delay decreases as the number of packets increases. We noted, however, that the absolute delay may increase or decrease depending on the number of receivers in the system. We attribute possible decrease in the normalized delay to the fact that when there are more packets to transmit, the transmitter has more options to choose from and hence, encounters delays less often in a normalized sense.

Figure 7

The effect of number of packets and erasure probabilities on the normalized delay. The maximum number of recursions for Heuristic 3 is set to 100. As the erasure probability decreases, the delay decreases as expected. The normalized delay decreases with for this particular (this is not always the case).

An important question that may arise in practical situations is how to choose the "block size" or the number of packets that are taken into account for making network coding decisions. If one has a total of packets to transmit, does it make sense to divide them into subblocks of smaller sizes or does it make sense to treat them as one single block of packets? The short answer is to include all "order-insensitive" packets in making transmission decisions and only break the packets into subblocks when the assumption of order insensitivity between subblocks breaks down. In the extreme case, an infinite number of order-insensitive packets provides an infinite pool of packets to choose from that can satisfy the demands of all receivers and are instantaneously decodable. Figure 8 shows the end-to-end delay when the number of packets in a block is finite and packets is chosen as the reference for comparison. We can see that although the delay of transmitting packets, , can be larger than that of transmitting packets , the delay does not increase by a factor of . That is and one does not benefit from breaking packets into subblocks of size packets each. By treating subblocks of size as one block of size , we add more degrees of freedom in making decisions.

Figure 8

The effect of block size on the mean delay. If the delay of transmitting packets in Heuristic 1, , is taken as the reference, we can see that the delay of including packets in transmission is less than . The same observation applies to the delay of Algorithm 1. In general, it is recommended to include all "order-insensitive" packets in making transmission decisions and only break the packets into subblocks when the assumption of order insensitivity between subblocks breaks down.

Now we turn our attention to the delay performance of our algorithms in channels with memory. Figure 9 shows the mean delay of different algorithms for packets and receivers. The GEC parameters for all links are identical with . The horizontal axis shows the memory content .The first curve from above shows the performance of Algorithm 1 when the transmitter does not take channel conditions into account in making coding decisions. In other words, is used in Algorithm 1 as if the channel states were memoryless. For relatively large memory contents, this method results in the largest mean delay. The next curve shows the delay performance of Heuristic 1. The next two curves, which are almost indistinguishable, show the performance of Algorithm 1 which takes channel states into account (using (7)) and Heuristic 2 with 100 recursions. The last curve shows the best delay that can be achieved by occasionally violating the instantaneous decodability rule for one receiver in favor of the other two receivers that are predicted to be in good state in the next transmission round. More details can be found in [14].

Figure 9

Delay performance of different algorithms in Gilbert-Elliott channels. The maximum number of recursions for Heuristic 2 is set to 100. By predicting next channel states and defining packet weights accordingly (see (7)), one can achieve considerably lower delays.

Figure 10 shows the delay performance of Algorithm 1 using packet weights according to (7) for to receivers. Both the mean delay and mean delay plus one standard deviation of delay (across 1000 stochastic runs of the transmission) are shown. As expected, the delay increases as the number of receivers increases. Comparing the delay's standard deviation with its mean, we observe that when the number of receivers is 3–5, the delay is relatively more variant than when the number of receivers is 10–15. For example, for and , the ratio of standard deviation to mean delay is around , whereas for and this ratio reduce to only . One should keep these variations in mind when designing the transmission system.

Figure 10

Delay performance of Algorithm 1 with weights defined using (7) for different number of receivers. As expected, the delay increases with the number of receivers. Both the mean delay (solid curves) and mean delay plus one standard deviation of delay (dashed-dotted curves) across 1000 stochastic runs of the transmission are shown.

We conclude this section with a brief look at the effect of post processing on the delay performance in channels with memory. Figure 11 shows different delays for receivers and packets. The figure confirms our earlier finding that selecting the maximum amount of coding among the optimal solutions provided by Algorithm 1 can result in larger end-to-end delays. We also note that serving the maximum number of receivers can have an adverse effect on the delay in GEC's. To explain this, consider an example where there are left packets to be transmitted to receivers. Packet 1 is needed by to and packet 2 is needed by and . Since both packets are needed by , we can either send packet 1 or 2, but not both. Now assume that to are all predicted to be in good state with probability and is predicted to be in good state with probability , so that according to (7). Therefore, transmission of either packet seems to be equally optimal. However, one can easily verify that the probability of at least one receiver among to receiving packet 1 is only , whereas the probability of either or receiving packet 2 is . Therefore, it makes sense to satisfy only two receivers, one of which has a high priority due its good channel conditions.

Figure 11

The effect of post processing on mean delay. As explained in the main text, whenever Algorithm 1 returns multiple solutions, choosing the maximum amount of coding and serving maximum number of receivers can often have adverse effects on the delay.

7. Conclusions

In this paper, we provided an online optimal network coding scheme with feedback to minimize decoding delay in each transmission round in erasure broadcast channels. Efficient search algorithms for the optimal network coding solution, as well as heuristic methods were presented and their delay and computational performance were tested in several system scenarios. We found that adopting an optimized approach using as much information about the channel as possible, such as memory, leads to a significantly better decoding delay. An interesting problem for future research is to relax the instantaneous decodability condition to -step decodability and investigate the delay-throughput tradeoff.

Appendix

Here we prove by structural induction that (a) every result returned by Algorithm 1 is a solution of (4) and (b) the set of solutions returned by the algorithm contains all the optimal solutions. We note that the algorithm is designed to discard infeasible vectors and those solutions that are clearly suboptimal at each recursion to improve performance. The latter is based on positiveness of the elements of as explained below.The algorithm generates a binary tree. Each node represents a problem of size and , and branches into two subproblems of size and . The former subproblem is a result of setting and the latter a result of setting . A leaf is reached when we need to solve . Without loss of generality let us assume that the variable to be examined is the first variable () which is followed by variables ( to ) that are constrained to , variables ( to ) that are constrained but not to , and finally unconstrained variables to . This can be easily accomplished by rearranging the columns of .For , it is clear that the only optimal solution to is which is returned by the algorithm. Hence, the minimal structure of the algorithm returns the optimal solution and our claim is true for . The induction hypothesis is that the two subproblems and have only discarded infeasible vectors and some suboptimal solutions. We need to prove that the same statement applies to the parent problem .We first look at the left branch where . According to the construction of the algorithm, any solution such as of length provided by the left branch is appended by the parent problem to form

(A.1)

where the head variable is set to one, variables constrained to are set to zero and all unconstrained variables are set to one. We first prove that is indeed a solution and then show that changing any element of results in either an infeasible or a clearly suboptimal . We use Definitions 6–8.(i)For

(A.2)

we write the condition as a weighted sum of columns of . That is, , where is a submatrix of of size , which is input to , and according to the induction hypothesis . But since no variable in is constrained to , no column in and can have ones in the same row position. Therefore, .(ii)Since to are unconstrained, no column to can have ones in the same row position. Hence, .(iii)Using similar arguments, we can assert that no column in or can have ones in the same row positions as to do. Therefore, and is a solution.(iv)We now argue that variables to cannot be anything other than zero. This directly follows from the fact that is constrained with for and hence, in any given solution they cannot be simultaneously one.(v)Since we have already found a solution where the first and last variables are one, we know that any other solution such as with one or more zeros in these positions becomes suboptimal and can be discarded. That is, due to positiveness of elements of .(vi)Finally, according to induction hypothesis, we know that cannot be changed into anything other than what provides without making it either infeasible or suboptimal. In summary, for each solution provided by the left branch , the constructed vector

(A.3)

is the only solution that is not trivially suboptimal.Now we look at the right branch where . According to the construction of the algorithm, a given solution such as of length provided by the right branch is appended by the parent problem to form

(A.4)

where the head variable is set to zero and all unconstrained variables are set to one. We need to show that for a given this is indeed a solution. We then show that changing any element of can only result in an infeasible vector, a clearly suboptimal solution, or a duplicate solution already provided by the left branch and hence, can be discarded. We use Definitions 6–8.(i)We write as , where is a submatrix of of size , which is input to , and according to the induction hypothesis . Similar to the arguments for the left branch, we can assert that no column to corresponding to unconstrained variables can have ones in the same row position. Hence, . Furthermore, that no column in can have ones in the same row positions as to . Therefore, and is a solution.(ii)Since we have already found a solution where the last variables are one, we know that any other solution such as with one or more zeros in these positions becomes suboptimal and can be discarded.(iii)Finally, we show that any vector of the form

(A.5)

with a one in the first variable is either infeasible or is already constructed based on solutions from the left branch and hence, need not be considered twice. We consider two possibilities for . If for any , then we have already shown in the analysis of the left branch that

(A.6)

is infeasible because and are constrained to each other. If none of to are one, then will be of the form

(A.7)

for some . But, has to be a solution of . Hence, considering vectors of the form

(A.8)

does not lead to any new solution.In summary, for each solution provided by the right branch , the constructed vector

(A.9)

is the only novel solution that is not trivially suboptimal. By combining the arguments of left and right branch, the induction claim is proven.