Topic

# Girth (graph theory)

About: Girth (graph theory) is a research topic. Over the lifetime, 1339 publications have been published within this topic receiving 15138 citations.

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IBM

^{1}TL;DR: Simulation results show that the PEG algorithm is a powerful algorithm to generate good short-block-length LDPC codes.

Abstract: We propose a general method for constructing Tanner graphs having a large girth by establishing edges or connections between symbol and check nodes in an edge-by-edge manner, called progressive edge-growth (PEG) algorithm. Lower bounds on the girth of PEG Tanner graphs and on the minimum distance of the resulting low-density parity-check (LDPC) codes are derived in terms of parameters of the graphs. Simple variations of the PEG algorithm can also be applied to generate linear-time encodeable LDPC codes. Regular and irregular LDPC codes using PEG Tanner graphs and allowing symbol nodes to take values over GF(q) (q>2) are investigated. Simulation results show that the PEG algorithm is a powerful algorithm to generate good short-block-length LDPC codes.

1,507 citations

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TL;DR: In this article, the construction of LDPC codes from circulant permutation matrices is investigated, and it is shown that such codes cannot have a Tanner graph representation with girth larger than 12 and a relatively mild necessary and sufficient condition for the code to have a girth of 6, 8, 10, or 12 is derived.

Abstract: In this correspondence, the construction of low-density parity-check (LDPC) codes from circulant permutation matrices is investigated. It is shown that such codes cannot have a Tanner graph representation with girth larger than 12, and a relatively mild necessary and sufficient condition for the code to have a girth of 6, 8,10, or 12 is derived. These results suggest that families of LDPC codes with such girth values are relatively easy to obtain and, consequently, additional parameters such as the minimum distance or the number of redundant check sums should be considered. To this end, a necessary condition for the codes investigated to reach their maximum possible minimum Hamming distance is proposed.

1,310 citations

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TL;DR: The most impressive feature of the data structure is its constant query time, hence the name "oracle", and it provides faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.

Abstract: Let G = (V,E) be an undirected weighted graph with vVv = n and vEv = m. Let k ≥ 1 be an integer. We show that G = (V,E) can be preprocessed in O(kmn1/k) expected time, constructing a data structure of size O(kn1p1/k), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k−1, that is, the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k−1. A 1963 girth conjecture of Erdos, implies that Ω(n1p1/k) space is needed in the worst case for any real stretch strictly smaller than 2kp1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name "oracle". Previously, data structures that used only O(n1p1/k) space had a query time of Ω(n1/k).Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.

618 citations

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TL;DR: Substantial conditions are given, relating the diameter of G with its girth, to assure optimum values of these conditional connectivities of G.

275 citations

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TL;DR: In this paper, the results of over 50 years of searches for cages are presented, along with the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions.

Abstract: A $(k, g)$-cage is a $k$-regular graph of girth $g$ of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions.

205 citations