A Simple Necessary and Sufficient Condition for the Double Unicast Problem

Abstract-We study the impact of delayed channel state information at the transmitters (CSIT) in two-unicast wireless networks with a layered topology and arbitrary connectivity. We introduce a technique to obtain outer bounds to the degrees-offreedom (DoF) region through the new graph-theoretic notion of bottleneck nodes. Such nodes act as informational bottlenecks only under the assumption of delayed CSIT, and imply asymmetric DoF bounds of the form mD1 + D2 ≤ m. Combining this outer-bound technique with new achievability schemes, we characterize the sum DoF of a class of two-unicast wireless networks, which shows that, unlike in the case of instantaneous CSIT, the DoF of two-unicast networks with delayed CSIT can take an infinite set of values.

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“…Since the received signal at d 1 has the same distribution as the received signal at v in network N , similar to (5), for N BC , we have…”

Section: Bottleneck Nodesmentioning

confidence: 87%

Informational bottlenecks in two-unicast wireless networks with delayed CSIT

Vahid

Shomorony

Calderbank

2015

2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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“…The authors in [8] and [9] explore the case when each source transmits one symbol at a time, or equivalently, R 1 = R 2 = 1 in detail, whereas we allow arbitrary rate pairs. Reference [10], also consider the scenario where the rates are arbitrary.…”

Section: Comparison With Existing Resultsmentioning

confidence: 99%

“…Our setup is somewhat different from the above-mentioned works in that we consider directed acyclic networks with unit capacity edges and assume that we only know certain minimum cut values for the network, e.g., mincut(S i , T j ), where S i ⊆ {s 1 , s 2 } and T j ⊆ {t 1 , t 2 } for different subsets S i and T j . This is related to the work of Wang and Shroff [8] (see also [9]) for two-unicast that presented a necessary and sufficient condition on the network structure for the existence of a network coding solution that supports unit rate transmission for each s i −t i pair. In this work we consider general rates.…”

Section: Introductionmentioning

confidence: 99%

An achievable region for the double unicast problem based on a minimum cut analysis

Huang

Ramamoorthy

2011

2011 IEEE Information Theory Workshop

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Abstract-We consider the multiple unicast problem under network coding over directed acyclic networks when there are two source-terminal pairs, s1 − t1 and s2 − t2. Current characterizations of the multiple unicast capacity region in this setting have a large number of inequalities, which makes them hard to explicitly evaluate. In this work we consider a slightly different problem. We assume that we only know certain minimum cut values for the network, e.g., mincut(Si, Tj), where Si ⊆ {s1, s2} and Tj ⊆ {t1, t2} for different subsets Si and Tj . Based on these values, we propose an achievable rate region for this problem based on linear codes. Towards this end, we begin by defining a base region where both sources are multicast to both the terminals. Following this we enlarge the region by appropriately encoding the information at the source nodes, such that terminal ti is only guaranteed to decode information from the intended source si, while decoding a linear function of the other source. The rate region takes different forms depending upon the relationship of the different cut values in the network.

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“…Unlike the two-unicast network [15], [23], [26], [27], [28], where (a) linear network coding is insufficient for capacity, (b) vector linear codes outperform scalar linear codes, and (c) the generalized network sharing (GNS) cut set bound is not tight in general, the question of whether non-linear network coding, vector linear codes, or bounds stronger than the GNS bound are required to characterize the achievable rate region for two-unicast-Z networks was open. In particular, because two-unicast-Z networks are a special case of two-unicast networks, the results which demonstrate the insufficiency of scalar linear and non-linear codes and the GNS bounds, for two-unicast networks, do not naturally extend to two-unicast-Z networks.…”

Section: Introductionmentioning

confidence: 99%

Linear network coding for two-unicast-Z networks: A commutative algebraic perspective and fundamental limits

Fahim

Cadambe

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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We consider a two-unicast-Z network over a directed acyclic graph of unit capacitated edges; the two-unicast-Z network is a special case of two-unicast networks where one of the destinations has apriori side information of the unwanted (interfering) message. In this paper, we settle open questions on the limits of network coding for two-unicast-Z networks by showing that the generalized network sharing bound is not tight, vector linear codes outperform scalar linear codes, and non-linear codes outperform linear codes in general. We also develop a commutative algebraic approach to deriving linear network coding achievability results, and demonstrate our approach by providing an alternate proof to the previous result of Wang et. al. regarding feasibility of rate (1, 1) in the network. I. INTRODUCTIONThere is significant interest in multiple unicast network coding and index coding in recent times. In addition to capturing the essence of network communication, there are interesting connections between special instances of the multiple unicast network communication problem and several emerging applications including topological interference management in wireless networks [6], codes for caching and content distribution [9] and regenerating and locally recoverable codes for distributed storage [1], [10]. In this paper, we study the most simple multiple unicast communication scenario, in terms of message structure, whose capacity is unknown: the two-unicast-Z network.The two-unicast-Z network, like the two-unicast network, has two independent message sources and two destinations, each destination respectively requiring to decode one of the two message sources. One of the two destinations, say the second destination, has apriori side information of the unintended (first) message source (See Fig. 1). Like the Z-interference channel in wireless communications, the two sources of the network interfere at only one destination. The study of twounicast-Z networks is important, because, like index coding and other simplified variants, insights obtained through code development for two-unicast-Z networks can potentially influence code design for more general multiple unicast networks and its related applications. A low-complexity linear network coding algorithm for two-unicast-Z networks has been developed in [13].It is shown in [12] that the generalized network sharing bound is tight for a specific class of two-unicast-Z networks. However, unlike the two-unicast network [7] where (a) linear network coding is insufficient for capacity, (b) vector linear codes outperform scalar linear codes, and (c) the generalized network sharing (GNS) cut set bound is not tight in general, it is not known whether non-linear network coding, vector linear codes, or bounds stronger than the GNS bound are required to characterize capacity for the two-unicast-Z network. In particular, because twounicast-Z networks are a special case of two-unicast networks, the results of two-unicast networks

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A Simple Necessary and Sufficient Condition for the Double Unicast Problem

Cited by 23 publications

References 9 publications

Informational bottlenecks in two-unicast wireless networks with delayed CSIT

Informational bottlenecks in two-unicast wireless networks with delayed CSIT

An achievable region for the double unicast problem based on a minimum cut analysis

Linear network coding for two-unicast-Z networks: A commutative algebraic perspective and fundamental limits

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