Sreeram Kannan

Approximately Optimal Wireless Broadcasting

We study a wireless broadcast network, where a single source reliably communicates independent messages to multiple destinations, with the potential aid of relays and cooperation between destinations. The wireless nature of the medium is captured by the broadcast nature of transmissions as well as the superposition of transmitted signals plus independent Gaussian noise at the received signal at any radio. We propose a scheme that can achieve rate tuples within a constant gap away from the cut-set bound, where the constant is independent of channel coefficients and power constraints. First, for a deterministic broadcast network, we propose a new coding scheme, constructed by adopting a “receiver-centric” viewpoint, that uses quantize-and-forward relaying as an inner code concatenated with an outer Marton code for the induced deterministic broadcast channel. This scheme is shown to achieve the cut-set bound evaluated with product form distributions. This result is then lifted to the Gaussian network by using a deterministic network called the discrete superposition network as a formal quantization interface. This two-stage construction circumvents the difficulty involved in working with a vector nonlinear non-Gaussian broadcast channel that arises if we construct a similar scheme directly for the Gaussian network.

Multi-Session Function Computation and Multicasting in Undirected Graphs

In the function computation problem, certain nodes of an undirected graph have access to independent data, while some other nodes of the graph require certain functions of the data; this model, motivated by sensor networks and cloud computing, is the focus of this paper. We study the maximum rates at which function computation is possible on a capacitated graph; the capacities on the edges of the graph impose constraints on the communication rate. We consider a simple class of computation strategies based on Steiner-tree packing (so-called computation trees), which does not involve block coding and has minimal delay. With a single terminal requiring function computation, computation trees are known to be optimal when the underlying graph is itself a directed tree, but have arbitrarily poor performance in general directed graphs. Our main result is that computation trees are near optimal for a wide class of function computation requirements even at multiple terminals in undirected graphs. The key technical contribution involves connecting approximation algorithms for Steiner cuts in undirected graphs to the function computation problem. Furthermore, we show that existing algorithms for Steiner tree packings allow us to compute approximately optimal packings of computation trees in polynomial time. We also show a close connection between the function computation problem and a communication problem involving multiple multicasts.

Local phy + global flow: A layering principle for wireless networks

A classical result in undirected wireline networks is the near optimality of routing (flow) for multiple-unicast: the min cut upper bound is within a logarithmic factor of the number of sources of the max flow. Wireless channels differ from wireline ones in two primary ways: the signal out of a transmitting node is broadcast and the signals at a receiving node superpose. In this paper we focus on “extending” the wireline result to the wireless context, by separately considering the broadcast and superposition constraints. Our main result is the approximate optimality of a simple layering principle: local physical-layer schemes combined with global routing. We show this in the context of both Gaussian networks and packet erasure networks. The key technical contribution is an approximation of min cut in a bidirected graph with submodular constraints on the edge capacities by max flow.

Multicommodity flows and cuts in polymatroidal networks

We consider multicommodity flow and cut problems in polymatroidal networks where there are submodular capacity constraints on the edges incident to a node. Polymatroidal networks were introduced by Lawler and Martel [20] and Hassin [15] in the single-commodity setting and are closely related to the submodular flow model of Edmonds and Giles [10]; the well-known maxflow-mincut theorem holds in this more general setting. Polymatroidal networks for the multicommodity case have not, as far as the authors are aware, been previously explored. Our work is primarily motivated by applications to information flow in wireless networks. We also consider the notion of undirected polymatroidal networks and observe that they provide a natural way to generalize flows and cuts in edge and node capacitated undirected networks. We establish poly-logarithmic flow-cut gap results in several scenarios that have been previously considered in the standard network flow models where capacities are on the edges or nodes [21, 22, 13, 19, 12]. Our results from a preliminary version have already found applications in wireless network information flow [16, 7] and we anticipate more in the future. On the technical side our key tools are the formulation and analysis of the dual of the flow relaxations via continuous extensions of submodular functions, in particular the Lovász extension. For directed graphs we rely on a simple yet useful reduction from polymatroidal networks to standard networks. For undirected graphs we rely on the interplay between the Lovász extension of a submodular function and line embeddings with low average distortion introduced by Matoušek and Rabinovich [25]; this connection is inspired by, and generalizes, the work of Feige, Hajiaghayi and Lee [12] on node-capacitated multicommodity flows and cuts. The applicability of embeddings to flow-cut gaps in polymatroidal networks is of independent mathematical interest.

Capacity of Multiple Unicast in Wireless Networks: A Polymatroidal Approach

A classical result in undirected wireline networks is the near optimality of routing (flow) for multiple-unicast traffic (multiple sources communicating independent messages to multiple destinations): the min cut upper bound is within a logarithmic factor of the number of sources of the max flow. In this paper, we extend the wireline result to the wireless context. In particular, we show the following meta-theorem: if for a given channel and its reciprocal channel, the cut-set bound is (approximately) achievable, then for multiple-unicast in a bidirected network comprised of such channels, the cut-set bound is (approximately) achievable within a logarithmic factor of the number of sources. The achievable scheme can be viewed as an instantiation of a simple layering principle: local physical-layer schemes combined with global routing. We use the reciprocity of the wireless channel critically in this result. We prove this result formally as a capacity approximation result for a variety of channel models, including general Gaussian networks under fast fading, networks comprised only of broadcast and MAC channels, and networks comprised of broadcast erasure channels with feedback. The capacity approximations we prove tend to have both an additive gap (power loss) and a multiplicative gap (degrees of freedom loss). The key engineering insight is that layered architectures, common in the engineering-design of wireless networks, can have near-optimal performance if the locality over which physical-layer schemes should operate is carefully designed. Feedback is shown to play a critical role in enabling the separation between the physical and the network layers. The main technical contribution is the usage of polymatroidal network as a graphical model for analyzing the performance of complex wireless networks.

Interactive Interference Alignment

We study interference channels (IFCs) where the interaction among sources and destinations is enabled, e.g., both sources and destinations can talk to each other using full-duplex radios. The interaction can come in two ways. First is through in-band interaction where sources and destinations can transmit and listen in the same channel simultaneously, enabling interaction. Second is through out-of-band interaction where destinations talk back to the sources on an out-of-band channel, which is possible from white-space channels. The flexibility afforded by the interaction among sources and destinations allows for the derivation of interference alignment (IA) strategies that have desirable “engineering properties,” i.e., insensitivity to the rationality or irrationality of channel parameters, small block lengths, and finite SNR operations. We show that, for several classes of IFCs, the interactive IA scheme can achieve the optimal degrees of freedom. In particular, we show a simple scheme (having a finite block length for channels having no diversity) for three-user and four-user IFCs with full-duplex radios to achieve the optimal degrees of freedom even after accounting for the cost of interaction. On the technical side, we show using a Gröbner basis argument that, in a general network potentially utilizing cooperation and feedback, the optimal degrees of freedom under linear schemes of a fixed block length is the same for channel coefficients with a probability of 1. Furthermore, a numerical method to estimate this value is also presented. These tools have potentially wider utility in studying other wireless networks as well.

Multiple-unicast in fading wireless networks: A separation scheme is approximately optimal

A classical result in undirected wireline networks is the approximate optimality of routing (flow) for multiple-unicast: the min-cut upper bound is within a logarithmic factor of the number of sources of the max flow. In this paper we focus on “extending” this result to the wireless context. Our main result is the approximate optimality of a simple layering principle: local physical-layer schemes combined with global routing. We show this in the context of wireless networks, in which links are either absent or undergo i.i.d. fast fading. We also show an approximation result on the degrees-of-freedom, when the channels are fixed, but are chosen from a continuous ensemble. The key technical contribution is an approximation of min-cut in a bidirected graph with submodular constraints on the edge capacities by max flow.

Minimum HGR correlation principle: From marginals to joint distribution

Given low order moment information over the random variables X = (X1, X2, ..., Xp) and Y, what distribution minimizes the Hirschfeld-Gebelein-Rényi (HGR) maximal correlation coefficient between X and Y, while remains faithful to the given moments? The answer to this question is important especially in order to fit models over (X, Y) with minimum dependence among the random variables X and Y. In this paper, we investigate this question first in the continuous setting by showing that the jointly Gaussian distribution achieves the minimum HGR correlation coefficient among distributions with the given first and second order moments. Then, we pose a similar question in the discrete scenario by fixing the pairwise marginals of the random variables X and Y. Subsequently, we derive a lower bound for the HGR correlation coefficient over the class of distributions with fixed pairwise marginals. Then we show that this lower bound is tight if there exists a distribution with certain additive structure satisfying the given pairwise marginals. Moreover, the distribution with the additive structure achieves the minimum HGR correlation coefficient. Finally, we conclude by showing that the event of obtaining pairwise marginals containing an additive structured distribution has a positive Lebesgue measure over the probability simplex.

Multi-terminal function multicasting in undirected graphs

In the function computation problem, certain nodes of an undirected graph have access to independent data, while certain other nodes of the graph require certain functions of the data; this model, motivated by sensor networks and cloud computing, is the focus of this paper. We study the maximum rates at which the function computation is possible on a capacitated graph. We consider a general model of function computation, which we term as multi-session function multicasting. In this general model, there are K independent sessions sharing the communication infrastructure; in each session, a set of D destinations all want the same function of a group of S sources. This traffic model generalizes various well known traffic models like the classical model of function computation (K = 1, D = 1), multiple-unicasting (S = 1, D = 1) and multicasting (K = 1, S = 1). For this general model, we propose a simple achievable strategy in which the function is computed for each session using Steiner trees at a specific destination and then distributed to the other destinations also using Steiner trees. Thus, in our proposed strategy, Steiner trees play the dual role of computation trees and also that of multicasting trees. Our main result is that this achievable strategy is near optimal for multi-session function multicasting of a wide class of functions in undirected graphs. The key technical contribution involves relating algorithmic work on Steiner cuts in undirected graphs to the function computation problem.

Shannon: An Information-Optimal de Novo RNA-Seq Assembler

De novo assembly of short RNA-Seq reads into transcripts is challenging due to sequence similarities in transcriptomes arising from gene duplications and alternative splicing of transcripts. We present Shannon, an RNA-Seq assembler with an optimality guarantee derived from principles of information theory: Shannon reconstructs nearly all information-theoretically reconstructable transcripts. Shannon is based on a theory we develop for de novo RNA-Seq assembly that reveals differing abundances among transcripts to be the key, rather than the barrier, to effective assembly. The assembly problem is formulated as a sparsest-flow problem on a transcript graph, and the heart of Shannon is a novel iterative flow-decomposition algorithm. This algorithm provably solves the information-theoretically reconstructable instances in linear-time even though the general sparsest-flow problem is NP-hard. Shannon also incorporates several additional new algorithmic advances: a new error-correction algorithm based on successive cancelation, a multi-bridging algorithm that carefully utilizes read information in the k-mer de Bruijn graph, and an approximate graph partitioning algorithm to split the transcriptome de Bruijn graph into smaller components. In tests on large RNA-Seq datasets, Shannon obtains significant increases in sensitivity along with improvements in specificity in comparison to state-of-the-art assemblers.