In this paper, we consider a strongly reachable multiple unicast network with$k$pairs of sender and receivers, and we show that the Langberg-Médard multiple unicast conjecture holds for such a network under the assumption that it is supported on a collapsed backbone that takes the form of rooted binary tree.

The Langberg-Médard multiple unicast conjecture claims that for a strongly reachable$k$-pair network, there exists a feasible multi-flow with rate$(1,1, {\dots },1)$. In this paper, we confirm the conjecture for$k=3$.

The Langberg-Médard multiple unicast conjecture claims that for a strongly reachable k-pair network, there exists a multi-flow with rate (1,1,...,1). In this paper, we show that the conjecture holds true for stable 3-pair networks.

The Langberg-Médard k-Unicast Conjecture states that for any strongly reachable k-pair network, there exists a multi-flow with rate (1, 1, ..., 1). In this paper, for k=3,4, we construct multi-flows with rate ([11/12], [11/12],..., [11/12]), which improves the previous result ([8/9], [8/9], ..., [8/9]), and we further prove that our constructions are optimal within the proposed framework.

We consider the locally repairable codes (LRCs), aiming at sequentially recovering multiple erasures; in particular, we propose and study the so-called (n, k, r, t)-sequential LRCs (SLRC) as an [n, k] linear code, where any t' (≤ t) erasures can be sequentially recovered, each by r (2 ≤ r <; k) other code symbols. Here, sequential recovering means that the erased symbols are recovered one by one, and an already recovered symbol can be used to recover the remaining erased symbols. This important recovering method, in contrast with the extensively studied parallel recovering, is currently far from being thoroughly understood; more specifically, there are to date no codes constructed for arbitrary t ≥ 3 erasures and bounds to evaluate the performance of such codes. We first derive a tight upper bound on the code rate of the (n, k, r, t)-SLRC for t = 3 and r ≥ 2. We then propose two constructions of binary (n, k, r, t)-SLRCs for general r, t ≥ 2 (existing constructions only deal with t ≤7 erasures). The first construction generalizes the method of direct product construction. The second construction is based on the resolvable configurations and yields SLRCs for any r ≥ 2 odd t ≥ 3. For both constructions, the rates are optimal for t ∈ {2, 3} and are higher than most of the existing LRC families for arbitrary t ≥ 4.

We consider the problem of network coding advantage in a communication scenario where information exchange is bi-directional and peers communicate via multiple unicast sessions. In such a setting, we study the overall performance of all multiple unicast sessions and propose a version of the multiple unicast conjecture. One of our main results is a weaker version of the proposed conjecture: Consider all the multiple unicast sessions associated with a number of terminals in an undirected network. Then, the common transmission rate of all these multiple unicast sessions achieved by network coding in the sense of Langberg and Médard [13] can also be achieved by fractional routing.

In this paper, by studying the feasible fractional routing solution under the so-called full reachability condition, we give bounds on the network coding advantage for undirected multiple unicast networks. More precisely, we prove that, for certain class of fully reachable networks, the network coding advantage is upper bounded by 9/8, improving the previous bound 3 by M. Langberg and M. Médard.

We consider the solvability problem under network coding and derive a sufficient and necessary condition for 3-pair networks with common “bottleneck links” being solvable. We show that, for such networks: (1) the solvability can be determined in polynomial time; (2) being solvable is equivalent to being linear solvable; (3) finite fields of size 2 or 3 are sufficient to construct linear solutions.

The encoding complexity of network coding for single multicast networks has been intensively studied from several aspects: e.g., the time complexity, the required number of encoding links, and the required field size for a linear code solution. However, these issues as well as the solvability are less understood for networks with multiple multicast sessions. Recently, Wang and Shroff showed that the solvability of networks with two unit-rate multicast sessions (2-URMS) can be decided in polynomial time . In this paper, we prove that for the 2-URMS networks: 1) the solvability can be determined with time O(|E|); 2) a solution can be constructed with time O(|E|); 3) an optimal solution can be obtained in polynomial time; 4) the number of encoding links required to achieve a solution is upper-bounded by max{3,2N - 2}; and 5) the field size required to achieve a linear solution is upper-bounded by max{2, ⌊√{2N-7/4}+1/2⌋}, where |E| is the number of links and N is the number of sinks of the underlying network. Both bounds are shown to be tight.

We consider a directed acyclic network with two source-sink pairs {s1, t1} and {s2, t2}. The source s1wishes to communicate a message X1to the sink t1and the source s2wishes to communicate two messages X2and X3to the sink t2, where Xi, i = 1,2,3, are independent random variables of unit rate. We give a simple characterization for linear solvability of such networks under the condition that the minimum cut from {s1, s2} to t2equals 3. We develop a region decomposition method for proving this result, which we believe can be an effective approach for non-multicast network coding problem.

In wireless multi-hop networks such as ad-hoc and sensor networks, one node may receive signals from several other nodes simultaneously due to the broadcast nature of the wireless medium. That results in the reduction of bandwidth usage and system efficiency. To deal with this problem, several approaches have been developed to avoid signal collision by appropriate protocols. In this work and in contrast to previous works, we put forward a unicast strategy that can recover the desired signal from the collided signals in wireless multi-hop networks. The proposed strategy is featured as a physical layer network coding scheme that can greatly increase the throughput of unicast in wireless multi-hop networks without synchronization nor power control among the different transmitters, thus, making it ideally fit for distributed networks.

In this correspondence, we investigate the link failure problem based on the recent results of network coding. We propose a concept, named capacity factor of a network, which is the minimum link set that can influence the network capacity, as our basic tool. We define the capacity rank to each link of the network to characterize its criticality and present the concept of the p-stable network. Based on these notions, an upper bound for the capacity factor size is derived and a family of p-stable networks is constructed

A wireless ad-hoc network can be roughly considered as one consisting of a collection of mobile nodes distributed in a finite region, which adopts a non-centralized and self-organized structure. In such networks, messages are transmitted, received and forwarded in a finite geometrical region. In addition, the transmission of messages is highly dependent on the locations of the mobile nodes. As a result, the geometrical relationships between the nodes, and especially the distance between them are of fundamental importance. In this paper, we propose a space decomposition method to analyze the probability distribution of the distance between nodes in an ad-hoc network. In particular, we derive two theoretical expressions for the probability distribution of the distance between nodes under the assumption that the nodes are independently and uniformly distributed in either a rectangular region or hexagonal region. Further results on the node degree distribution and max-flow capacity of the network are then presented based upon these expressions

Capacity Analysis of network coding is a fundamental problem in communication networks. In this paper, we investigate this problem by generalizing the conventional random graph model as G(n, P,C), where the connecting probability between each pair of nodes p obeys an independent and identical distribution and all the links each have an independent and identical transmission capacity distribution. A tight upper bound of the average value of the maximal flow will be derived based on the proposed random graph model. Moreover, the averaged value and the variance of the maximal flow shall be investigated by some simulations, which demonstrate the effectiveness of our theoretical analysis.