Shortened Array Codes of Large Girth

One approach to designing structured low-density parity-check (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the parity-check matrix contain all the variables involved in short cycles. This approach is especially effective if the parity-check matrix of a code is a matrix composed of blocks of circulant permutation matrices, as is the case for the class of codes known as array codes. We show how to shorten array codes by deleting certain columns of their parity-check matrices so as to increase their girth. The shortening approach is based on the observation that for array codes, and in fact for a slightly more general class of LDPC codes, the cycles in the corresponding Tanner graph are governed by certain homogeneous linear equations with integer coefficients. Consequently, we can selectively eliminate cycles from an array code by only retaining those columns from the parity-check matrix of the original code that are indexed by integer sequences that do not contain solutions to the equations governing those cycles. We provide Ramsey-theoretic estimates for the maximum number of columns that can be retained from the original parity-check matrix with the property that the sequence of their indices avoid solutions to various types of cycle-governing equations. This translates to estimates of the rate penalty incurred in shortening a code to eliminate cycles. Simulation results show that for the codes considered, shortening them to increase the girth can lead to significant gains in signal-to-noise ratio in the case of communication over an additive white Gaussian noise channel.Comment: 16 pages; 8 figures; to appear in IEEE Transactions on Information Theory, Aug 200

Unified bijections for maps with prescribed degrees and girth

This article presents unified bijective constructions for planar maps, with control on the face degrees and on the girth. Recall that the girth is the length of the smallest cycle, so that maps of girth at least $d=1,2,3$ are respectively the general, loopless, and simple maps. For each positive integer $d$, we obtain a bijection for the class of plane maps (maps with one distinguished root-face) of girth $d$ having a root-face of degree $d$. We then obtain more general bijective constructions for annular maps (maps with two distinguished root-faces) of girth at least $d$. Our bijections associate to each map a decorated plane tree, and non-root faces of degree $k$ of the map correspond to vertices of degree $k$ of the tree. As special cases we recover several known bijections for bipartite maps, loopless triangulations, simple triangulations, simple quadrangulations, etc. Our work unifies and greatly extends these bijective constructions. In terms of counting, we obtain for each integer $d$ an expression for the generating function $F_d(x_d,x_{d+1},x_{d+2},...)$ of plane maps of girth $d$ with root-face of degree $d$, where the variable $x_k$ counts the non-root faces of degree $k$. The expression for $F_1$ was already obtained bijectively by Bouttier, Di Francesco and Guitter, but for $d\geq 2$ the expression of $F_d$ is new. We also obtain an expression for the generating function \G_{p,q}^{(d,e)}(x_d,x_{d+1},...) of annular maps with root-faces of degrees $p$ and $q$, such that cycles separating the two root-faces have length at least $e$ while other cycles have length at least $d$. Our strategy is to obtain all the bijections as specializations of a single "master bijection" introduced by the authors in a previous article. In order to use this approach, we exhibit certain "canonical orientations" characterizing maps with prescribed girth constraints

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

A $k$-spanner of a graph $G$ is a sparse subgraph $H$ whose shortest path distances match those of $G$ up to a multiplicative error $k$. In this paper we study spanners that are resistant to faults. A subgraph $H \subseteq G$ is an $f$ vertex fault tolerant (VFT) $k$-spanner if $H \setminus F$ is a $k$-spanner of $G \setminus F$ for any small set $F$ of $f$ vertices that might "fail." One of the main questions in the area is: what is the minimum size of an $f$ fault tolerant $k$-spanner that holds for all $n$ node graphs (as a function of $f$, $k$ and $n$)? This question was first studied in the context of geometric graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more recently been considered in general undirected graphs [Chechik et al. STOC '09, Dinitz and Krauthgamer PODC '11]. In this paper, we settle the question of the optimal size of a VFT spanner, in the setting where the stretch factor $k$ is fixed. Specifically, we prove that every (undirected, possibly weighted) $n$-node graph $G$ has a $(2k-1)$-spanner resilient to $f$ vertex faults with $O_k(f^{1 - 1/k} n^{1 + 1/k})$ edges, and this is fully optimal (unless the famous Erdos Girth Conjecture is false). Our lower bound even generalizes to imply that no data structure capable of approximating $dist_{G \setminus F}(s, t)$ similarly can beat the space usage of our spanner in the worst case. We also consider the edge fault tolerant (EFT) model, defined analogously with edge failures rather than vertex failures. We show that the same spanner upper bound applies in this setting. Our data structure lower bound extends to the case $k=2$ (and hence we close the EFT problem for $3$-approximations), but it falls to $\Omega(f^{1/2 - 1/(2k)} \cdot n^{1 + 1/k})$ for $k \ge 3$. We leave it as an open problem to close this gap.Comment: To appear in SODA 201

From Cages to Trapping Sets and Codewords: A Technique to Derive Tight Upper Bounds on the Minimum Size of Trapping Sets and Minimum Distance of LDPC Codes

Cages, defined as regular graphs with minimum number of nodes for a given girth, are well-studied in graph theory. Trapping sets are graphical structures responsible for error floor of low-density parity-check (LDPC) codes, and are well investigated in coding theory. In this paper, we make connections between cages and trapping sets. In particular, starting from a cage (or a modified cage), we construct a trapping set in multiple steps. Based on the connection between cages and trapping sets, we then use the available results in graph theory on cages and derive tight upper bounds on the size of the smallest trapping sets for variable-regular LDPC codes with a given variable degree and girth. The derived upper bounds in many cases meet the best known lower bounds and thus provide the actual size of the smallest trapping sets. Considering that non-zero codewords are a special case of trapping sets, we also derive tight upper bounds on the minimum weight of such codewords, i.e., the minimum distance, of variable-regular LDPC codes as a function of variable degree and girth