Complexity of finding embeddings in a k-tree
SIAM Journal on Algebraic and Discrete Methods
Efficient and constructive algorithms for the pathwidth and treewidth of graphs
Journal of Algorithms
Introduction to Bayesian Networks
Introduction to Bayesian Networks
Handbook of Coding Theory
European Journal of Combinatorics - Special issue on Eurocomb'03 - graphs and combinatorial structures
Addendum to matroid tree-width
European Journal of Combinatorics
On minimal tree realizations of linear codes
IEEE Transactions on Information Theory
Minimal tail-biting trellises: the Golay code and more
IEEE Transactions on Information Theory
The generalized distributive law
IEEE Transactions on Information Theory
Factor graphs and the sum-product algorithm
IEEE Transactions on Information Theory
Codes on graphs: normal realizations
IEEE Transactions on Information Theory
Codes on graphs: constraint complexity of cycle-free realizations of linear codes
IEEE Transactions on Information Theory
The structure of tail-biting trellises: minimality and basic principles
IEEE Transactions on Information Theory
The Extraction and Complexity Limits of Graphical Models for Linear Codes
IEEE Transactions on Information Theory
Asymptotically good codes have infinite trellis complexity
IEEE Transactions on Information Theory
On minimal tree realizations of linear codes
IEEE Transactions on Information Theory
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A graphical realization of a linear code C consists of an assignment of the coordinates of C to the vertices of a graph, along with a specification of linear state spaces and linear "local constraint" codes to be associated with the edges and vertices, respectively, of the graph. The κ-complexity of a graphical realization is defined to be the largest dimension of any of its local constraint codes. κ-complexity is a reasonable measure of the computational complexity of a sum-product decoding algorithm specified by a graphical realization. The main focus of this paper is on the following problem: given a linear code C and a graph G, how small can the κ-complexity of a realization of C on G be? As useful tools for attacking this problem, we introduce the Vertex-Cut Bound, and the notion of "vc-treewidth" for a graph, which is closely related to the well-known graph-theoretic notion of treewidth. Using these tools, we derive tight lower bounds on the κ-complexity of any realization of C on G. Our bounds enable us to conclude that good error-correcting codes can have low-complexity realizations only on graphs with large vc-treewidth. Along the way, we also prove the interesting result that the ratio of the κ-complexity of the best conventional trellis realization of a length-n code C to the κ-complexity of the best cycle-free realization of C grows at most logarithmically with code length n. Such a logarithmic growth rate is, in fact, achievable.