Alex Sprintson

On the Index Coding Problem and Its Relation to Network Coding and Matroid Theory

The index coding problem has recently attracted a significant attention from the research community due to its theoretical significance and applications in wireless ad hoc networks. An instance of the index coding problem includes a sender that holds a set of information messages X={x1,...,xk} and a set of receivers R. Each receiver (x,H) in R needs to obtain a message x X and has prior side information consisting of a subset H of X . The sender uses a noiseless communication channel to broadcast encoding of messages in X to all clients. The objective is to find an encoding scheme that minimizes the number of transmissions required to satisfy the demands of all the receivers. In this paper, we analyze the relation between the index coding problem, the more general network coding problem, and the problem of finding a linear representation of a matroid. In particular, we show that any instance of the network coding and matroid representation problems can be efficiently reduced to an instance of the index coding problem. Our reduction implies that many important properties of the network coding and matroid representation problems carry over to the index coding problem. Specifically, we show that vector linear codes outperform scalar linear index codes and that vector linear codes are insufficient for achieving the optimum number of transmissions.

The encoding complexity of network coding

In the multicast network coding problem, a source s needs to deliver h packets to a set of k terminals over an underlying communication network G. The nodes of the multicast network can be broadly categorized into two groups. The first group includes encoding nodes, i.e., nodes that generate new packets by combining data received from two or more incoming links. The second group includes forwarding nodes that can only duplicate and forward the incoming packets. Encoding nodes are, in general, more expensive due to the need to equip them with encoding capabilities. In addition, encoding nodes incur delay and increase the overall complexity of the network. Accordingly, in this paper, we study the design of multicast coding networks with a limited number of encoding nodes. We prove that in a directed acyclic coding network, the number of encoding nodes required to achieve the capacity of the network is bounded by h/sup 3/k/sup 2/. Namely, we present (efficiently constructible) network codes that achieve capacity in which the total number of encoding nodes is independent of the size of the network and is bounded by h/sup 3/k/sup 2/. We show that the number of encoding nodes may depend both on h and k by presenting acyclic coding networks that require /spl Omega/(h/sup 2/k) encoding nodes. In the general case of coding networks with cycles, we show that the number of encoding nodes is limited by the size of the minimum feedback link set, i.e., the minimum number of links that must be removed from the network in order to eliminate cycles. We prove that the number of encoding nodes is bounded by (2B+1)h/sup 3/k/sup 2/, where B is the minimum size of a feedback link set. Finally, we observe that determining or even crudely approximating the minimum number of required encoding nodes is an /spl Nscr/P-hard problem.

On the Minimum Number of Transmissions in Single-Hop Wireless Coding Networks

The advent of network coding presents promising opportunities in many areas of communication and networking. It has been recently shown that network coding technique can significantly increase the overall throughput of wireless networks by taking advantage of their broadcast nature. In wireless networks, each transmitted packet is broadcasted within a certain area and can be overheard by the neighboring nodes. When a node needs to transmit packets, it employs the opportunistic coding approach that uses the knowledge of what the nodes neighbors have heard in order to reduce the number of transmissions. With this approach, each transmitted packet is a linear combination of the original packets over a certain finite field. In this paper, we focus on the fundamental problem of finding the optimal encoding for the broadcasted packets that minimizes the overall number of transmissions. We show that this problem is NP-complete over GF(2) and establish several fundamental properties of the optimal solution. We also propose a simple heuristic solution for the problem based on graph coloring and present some empirical results for random settings.

On coding for cooperative data exchange

We consider the problem of data exchange by a group of closely-located wireless nodes. In this problem each node holds a set of packets and needs to obtain all the packets held by other nodes. Each of the nodes can broadcast the packets in its possession (or a combination thereof) via a noiseless broadcast channel of capacity one packet per channel use. The goal is to minimize the total number of transmissions needed to satisfy the demands of all the nodes, assuming that they can cooperate with each other and are fully aware of the packet sets available to other nodes. This problem arises in several practical settings, such as peer-to-peer systems and wireless data broadcast. In this paper, we establish upper and lower bounds on the optimal number of transmissions and present an efficient algorithm with provable performance guarantees. The effectiveness of our algorithms is established through numerical simulations.

Efficient algorithms for Index Coding

The index coding problem has attracted a considerable amount of attention in recent years. The problem is motivated by several applications in wireless networking and distributed computing, including wireless architectures that utilize network coding and opportunistic listening. In this paper, we propose efficient exact and heuristic solutions for this problem. Our numerical study shows that exact solutions can be efficiently obtained for small instances of the problem, while heuristic solutions with low computation time can achieve near-optimal performance for large instances.

Algorithms for computing QoS paths with restoration

There is a growing interest among service providers to offer new services with Quality of Service (QoS) guarantees that are also resilient to failures. Supporting QoS connections requires the existence of a routing mechanism, that computes the QoS paths, i.e., paths that satisfy QoS constraints (e.g., delay or bandwidth). Resilience to failures, on the other hand, is achieved by providing, for each primary QoS path, a set of alternative QoS paths used upon a failure of either a link or a node. The above objectives, coupled with the need to minimize the global use of network resources, imply that the cost of both the primary path and the restoration topology should be a major consideration of the routing process. We undertake a comprehensive study of problems related to finding suitable restoration topologies for QoS paths. We consider both bottleneck QoS constraints, such as bandwidth, and additive QoS constraints, such as delay and jitter. This is the first study to provide a rigorous solution, with proven guarantees, to the combined problem of computing QoS paths with restoration. It turns out that the widely used approach of disjoint primary and restoration paths is not an optimal strategy. Hence, the proposed algorithms construct a restoration topology , i.e., a set of bridges, each bridge protecting a portion of the primary QoS path. This approach guarantees to find a restoration topology with low cost when one exists.

Secure Network Coding for Wiretap Networks of Type II

We consider the problem of securing a multicast network against a wiretapper that can eavesdrop on the packets on a limited number of network edges of its choice. We assume that the network employs network coding to simultaneously deliver the packets available at the source to all the destinations. We show that this problem can be looked at as a network generalization of the wiretap channel of type II introduced in a seminal paper by Ozarow and Wyner. In particular, we show that the transmitted information can be secured by using the Ozarow-Wyner approach of coset coding at the source on top of the existing network code. This way, we quickly and transparently recover some of the results available in the literature on secure network coding for wiretap networks. Moreover, we use this framework to derive new bounds on the code alphabet size that are independent of the network size, and provide algorithms for explicit construction of secure network codes. We also analyze the amount of information that can be leaked to the wiretapper as a function of the number of wiretapped edges.

On the hardness of approximating the network coding capacity

This work addresses the computational complexity of achieving the capacity of a general network coding instance. It has been shown [Lehman and Lehman, SODA 2005] that determining the “scalar linear” capacity of a general network coding instance is NP-hard. In this paper we address the notion of approximation in the context of both linear and nonlinear network coding. Loosely speaking, we show that given an instance of the general network coding problem of capacity C , constructing a code of rate αC for any universal (i.e., independent of the size of the instance) constant α ≤ 1 is “hard”. Specifically, finding such network codes would solve a long standing open problem in the field of graph coloring. Our results refer to scalar linear, vector linear, and nonlinear encoding functions and are the first results that address the computational complexity of achieving the network coding capacity in both the vector linear and general network coding scenarios. In addition, we consider the problem of determining the (scalar) linear capacity of a planar network coding instance (i.e., an instance in which the underlying graph is planar). We show that even for planar networks this problem remains NP-hard.

Efficient algorithms for computing disjoint QoS paths

Networks are expected to meet a growing volume of requirements imposed by new applications such as multimedia streaming and video conferencing. Two essential requirements are support of quality of service (QoS) and resilience to failures. In order to satisfy these requirements, a common approach is to use two disjoint paths between the source and the destination nodes, the first serving as a primary path and the second as a restoration path. Such approach, referred to as path restoration, has several advantages, the major one being the ability to switch promptly from one path to another in the event of a failure. A major issue in this context is how to identify two paths that satisfy the QoS constraints imposed by network applications. Since network resources, e.g., bandwidth, are allocated along both primary and restoration paths, we need to consider also the overall network performance. Accordingly, in this paper we study the fundamental problem of finding two disjoint paths that satisfy the QoS constraints at minimum cost. We present approximation algorithms with provable performance guarantees for this fundamental network problem.

A randomized algorithm and performance bounds for coded cooperative data exchange

We consider scenarios where wireless clients are missing some packets, but they collectively know every packet. The clients collaborate to exchange missing packets over an error-free broadcast channel with capacity of one packet per channel use. First, we present an algorithm that allows each client to obtain missing packets, with minimum number of transmissions. The algorithm employs random linear coding over a sufficiently large field. Next, we show that the field size can be reduced while maintaining the same number of transmissions. Finally, we establish lower and upper bounds on the minimum number of transmissions that are easily computable and often tight as demonstrated by numerical simulations.