Minimum-cost Subgraph Algorithms for Static and Dynamic Multicasts with Network Coding

Abstract

Network coding, introduced by Ahlswede et al. in their pioneering work [1], has generated considerable research interest in recent years, and numerous subsequent papers, e.g., [2–6], have built upon this concept. One of the main advantages of network coding over traditional routed networks is in the area of multicast, where common information is transmitted from a source node to a set of terminal nodes. Ahlswede et al. showed in [1] that network coding can achieve the maximum multicast rate, which is not achievable by routing alone. When coding is used to perform multicast, the problem of establishing minimum cost multicast connection is equivalent to two effectively decoupled problems: one of determining the subgraph to code over and the other of determining the code to use over that subgraph. The latter problem has been studied extensively in [5, 7–9], and a variety of methods have been proposed, which include employing simple random linear coding at every node. Such random linear coding schemes are completely decentralized, requiring no coordination between nodes, and can operate under dynamic conditions [10]. These papers, however, all assume the availability of dedicated network resources. In this chapter, we focus on the former problem, which is to find the min-cost subgraph that allows the given multicast connection to be established (with appropriate coding) over coded packet networks. This problem has been studied in [11,12]. The analogous problem for routed network is the Steiner tree problem, which is known