Congduan Li

A new computational approach for determining rate regions and optimal codes for coded networks

A new computational technique is presented for determining rate regions for coded networks. The technique directly manipulates the extreme ray representation of inner and outer bounds for the region of entropic vectors. We use new inner bounds on region of entropic vectors based on conic hull of ranks of representable matroids. In particular, the extreme-ray representations of these inner bounds are obtained via matroid enumeration and minor exclusion. This is followed by a novel use of iterations of the double description method to obtain the desired rate regions. Applications in multilevel diversity coding systems (MDCS) are discussed as an example. The special structure of the problem that makes this technique inherently fast along with being scalable is also discussed. Our results demonstrate that for each of the 31 2-level 3-encoder and the 69 3-level 3-encoder MDCS configurations, if scalar linear codes (over any field) suffice to achieve the rate region, then in fact binary scalar linear codes suffice. For the 31 2-level 3-encoder cases where scalar codes are insufficient we demonstrate that vector linear codes suffice and provide some explicit constructions of these codes.

A computational approach for determining rate regions and codes using entropic vector bounds

A computational technique for determining rate regions for networks and multilevel diversity coding systems based on inner and outer bounds for the region of entropic vectors is discussed. The expression to get rate region in terms of region of entropic vectors is attributed to Yeung and Zhang. An inner bound based on binary representable matroids is discussed that has the added benefit of identifying optimal linear codes. The theorem stated by Hassibi et al. in 2010 ITA is implemented to get H-representation of binary matroid inner bound for more than 4 variables. The computational technique is demonstrated on a series of small examples of multilevel diversity coding systems.

Multilevel Diversity Coding Systems: Rate Regions, Codes, Computation, & Forbidden Minors

The rate regions of multilevel diversity coding systems (MDCSs), a sub-class of the broader family of multi-source multi-sink networks with special structure, are investigated in a systematic way. We enumerate all non-isomorphic MDCS instances with at most three sources and four encoders. Then, the exact rate region of every one of these more than 7000 instances is proven via computations showing that the Shannon outer bound matches with a custom constructed linear code-based inner bound. Results gained from these computations are summarized in key statistics involving aspects, such as the sufficiency of scalar binary codes, the necessary size of vector binary codes, and so on. Also, it is shown how to construct the codes for an achievability proof. Based on this large repository of rate regions, a series of results about general MDCS cases of arbitrary size that they inspired is introduced and proved. In particular, a series of embedding operations that preserve the property of sufficiency of scalar or vector codes is presented. The utility of these operations is demonstrated by boiling the thousands of MDCS instances for which scalar binary (superposition) codes are insufficient down to 12 (26) forbidden the smallest embedded MDCS instances.

Algorithms for computing network coding rate regions via single element extensions of matroids.

We propose algorithms for finding extreme rays of rate regions achievable with vector linear codes over finite fields Fq, q ∈ {2, 3, 4} for which there are known forbidden minors for matroid representability. We use the idea of single element extensions (SEEs) of matroids and enumeration of non-isomorphic matroids using SEEs, to first propose an algorithm to obtain lists of all non-isomorphic matroids representable over a given finite field.We modify this algorithm to produce only the list of all non-isomorphic connected matroids representable over the given finite field. We then integrate the process of testing which matroids in a list of matroids form valid linear network codes for a given network within matroid enumeration. We name this algorithm, which essentially builds all matroids that form valid network codes for a given network from scratch, as network-constrained matroid enumeration.

On Multi-Source Networks: Enumeration, Rate Region Computation, and Hierarchy

Recent algorithmic developments have enabled computers to automatically determine and prove the capacity regions of small hypergraph networks under network coding. A structural theory relating network coding problems of different sizes is developed to make the best use of this newfound computational capability. A formal notion of network minimality is developed, which removes components of a network coding problem that are inessential to its core complexity. Equivalence between different network coding problems under relabeling is formalized via group actions, an algorithm which can directly list single representatives from each equivalence class of minimal networks up to a prescribed network size is presented. This algorithm, together with rate region software, is leveraged to create a database containing the rate regions for all minimal network coding problems with five or fewer sources and edges, a collection of 744119 equivalence classes representing more than 9 million networks. In order to best learn from this database, and to leverage it to infer rate regions and their characteristics of networks at scale, a hierarchy between different network coding problems is created with a new theory of combinations and embedding operators.

Rate region for a class of delay mitigating codes and P2P networks

This paper identifies the relevance of a distributed source coding problem first formulated by Yeung and Zhang in 1999 to two applications in network design: i) the design of delay mitigating codes, and ii) the design of network coded P2P networks. When transmitting time-sensitive frames from a source to a destination over a multipath network using a collection of coded packets, the decoding requirements determine which subsets of packets will be sufficient for decoding which frames. The rate region of packet sizes consistent with these requirements is shown to be an instance of the aforementioned distributed source coding problem. When encoding file chunks into packets in a peer to peer system, the peers wish to receive their chunks as soon as possible while uploading data at as low a rate as possible. It is shown that the region of encoded packet sizes consistent with the decoding constraints is another instance of the aforementioned distributed source coding problem. These rate regions are placed in the larger context of rate-delay tradeoffs in designing delay mitigating codes and efficient P2P systems.

Network embedding operations preserving the insufficiency of linear network codes

Operations for extending/embedding a smaller network into a larger network that preserve the insufficiency of classes of linear network codes are presented. Linear network codes over some finite field are said to be sufficient for a network if and only if for every point in the network coding rate region, there exists a code over that finite field to achieve it. Three operations are defined, and it is proven that they have the desired inheritance property, both for scalar linear network codes and for vector linear network codes, separately. Experimental results on the rate regions of multilevel diversity coding systems (MDCS), a sub-class of the broader family of multi-source multi-sink networks with special structure, are presented for demonstration. These results demonstrate that these notions of embedding operations enable one to investigate the existences of small numbers of forbidden network minors for sufficiency of linear network codes over a given field.

Matroid bounds on the region of entropic vectors

Several properties of the inner bound on the region of entropic vectors obtained from representable matroids are derived. In particular, it is shown that: I) It suffices to check size 2 minors of an integer-valued vector to determine if it is a valid matroid rank; II) the subset of the extreme rays of the Shannon outer bound (the extremal polymatroids) that are matroidal are also the extreme rays of the cone of matroids; III) All matroid ranks are convex independent; and IV) the extreme rays of the conic hull of the binary/ternary/quaternary representable matroid ranks inner bound are a subset of the extreme rays of the conic hull of matroid ranks. These properties are shown to allow for substantial reduction in the complexity of calculating important rate regions in multiterminal information theory, including multiple source multicast network coding capacity regions.

Computer aided proofs for rate regions of independent distributed source coding problems

The rate regions of independent distributed source coding (IDSC) problems, a sub-class of the broader family of multi-source multi-sink networks, are investigated. An IDSC problem consists of multiple sources, multiple encoders, and multiple decoders, where each encoder has access to all sources, and each decoder has access to a certain subset of the encoders and demands a certain subset of the sources. Instead of manually deriving the rate region for a particular problem, computer tools are used to obtain the rate regions for hundreds of nonisomorphic (symmetry-removed) IDSC instances. A method for enumerating all non-isomorphic IDSC instances of a particular size is given. For each non-isomorphic IDSC instance, the Shannon outer bound, superposition coding inner bound, and several achievable inner bounds based on linear codes, are considered and calculated. For all of the hundreds of IDSC instances considered, vector binary inner bounds match the Shannon outer bound, and hence, exact rate regions are proven together with code constructions that achieve them.

Network combination operations preserving the sufficiency of linear network codes

Operations that combine smaller networks into a larger network in a manner such that the rate region of the larger network can be directly obtained from the associated rate regions of the smaller networks are defined and presented. The operations are selected to also have the property that the sufficiency of classes of linear network codes and the tightness of the Shannon (polymatroid) outer bound are preserved in the combination. Four such operations are defined, and the classes of linear codes considered include both scalar and vector codes. It is demonstrated that these operations enable one to obtain rate regions for networks of arbitrary size, and to also determine if some classes of linear codes are sufficient and/or the Shannon outer bound bound is tight for these networks.