# DIRECTED ACYCLIC GRAPHS AND DISJOINT CHAINS

### Yangjun Chen

#### Abstract

The problem of decomposing a DAG (directed acyclic graph) into a set of disjoint chains has many applications in data engineering. One of them is the compression of transi¬tive closures to support reachability queries on whether a given node v in a directed graph G is reachable from another node u through a path in G. Recently, an interesting algorithm is proposed by Chen et al. (Y. Chen and Y. Chen, 2008) which claims to be able to decompose G into a minimal set of dis¬joint chains in O(n2 + bn ) time, where n is the number of the nodes of G, and b is G’s width, defined to be the size of a largest node subset U of G such that for every pair of nodes u, v U, there does not exist a path from u to v or from v to u. However, in some cases, it fails to do so. In this paper, we analyze this algorithm and show the problem. More importantly, a new algorithm is discussed, which can always find a minimal set of disjoint chains in the same time complexity as Chen’s.

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#### Paper Citation

#### in Harvard Style

Chen Y. (2009). **DIRECTED ACYCLIC GRAPHS AND DISJOINT CHAINS** . In *Proceedings of the 11th International Conference on Enterprise Information Systems - Volume 1: ICEIS,* ISBN 978-989-8111-84-5, pages 17-24. DOI: 10.5220/0001858300170024

#### in Bibtex Style

@conference{iceis09,

author={Yangjun Chen},

title={DIRECTED ACYCLIC GRAPHS AND DISJOINT CHAINS},

booktitle={Proceedings of the 11th International Conference on Enterprise Information Systems - Volume 1: ICEIS,},

year={2009},

pages={17-24},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0001858300170024},

isbn={978-989-8111-84-5},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 11th International Conference on Enterprise Information Systems - Volume 1: ICEIS,

TI - DIRECTED ACYCLIC GRAPHS AND DISJOINT CHAINS

SN - 978-989-8111-84-5

AU - Chen Y.

PY - 2009

SP - 17

EP - 24

DO - 10.5220/0001858300170024