# DIRECTED ACYCLIC GRAPHS AND DISJOINT CHAINS

### Yangjun Chen

#### 2009

#### 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.

#### References

- H. Alt, N. Blum, K. Mehlhorn, and M. Paul, Computing a maximum cardinality matching in a bipartite graph in time O(n1.5), Information Processing Letters, 37(1991), 237 -240.
- A. S. Asratian, T. Denley, and R. Haggkvist, Bipartite Graphs and their Applications, Cambridge University, 1998.
- J. Banerjee, W. Kim, S. Kim and J.F. Garza, "Clustering a DAG for CAD Databases," IEEE Trans. on Knowledge and Data Engineering, Vol. 14, No. 11, Nov. 1988, pp. 1684-1699.
- K. S. Booth and G.S. Leuker, “Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms,” J. Comput. Sys. Sci., 13(3):335-379, Dec. 1976.
- Y. Chen and Y. Chen, An Efficient Algorithm for Answering Graph Reachability Queries, Proceedings of ICDE, 2008, pp. 893 - 902.
- Y. Chen, “On the Graph Traversal and Linear Binarychain Programs,” IEEE Transactions on Knowledge and Data Engineering, Vol. 15, No. 3, May 2003, pp. 573-596.
- N. H. Cohen, “Type-extension tests can be performed in constant time,” ACM Transactions on Programming Languages and Systems, 13:626-629, 1991.
- E. Cohen, E. Halperin, H. Kaplan, and U. Zwick, Reachability and distance queries via 2-hop labels, SIAM J. Comput, vol. 32, No. 5, pp. 1338-1355, 2003.
- J. Cheng, J.X. Yu, X. Lin, H. Wang, and P.S. Yu, Fast computation of reachability labeling for large graphs, in Proc. EDBT, Munich, Germany, May 26-31, 2006.
- D. Coppersmith, and S. Winograd. Matrix multiplication via arithmetic progression. Journal of Symbolic Computation, vol. 9, pp. 251-280, 1990.
- R. P. Dilworth, A decomposition theorem for partially ordered sets, Ann. Math. 51 (1950), pp. 161-166.
- S. Even, Graph Algorithms, Computer Science Press, Inc., Rockville, Maryland, 1979.
- J. E. Hopcroft, and R.M. Karp, An n2.5 algorithm for maximum matching in bipartite graphs, SIAM J. Comput. 2(1973), 225-231.
- H. V. Jagadish, "A Compression Technique to Materialize Transitive Closure," ACM Trans. Database Systems, Vol. 15, No. 4, 1990, pp. 558 - 598.
- A. V. Karzanov, Determining the Maximal Flow in a Network by the Method of Preflow, Soviet Math. Dokl., Vol. 15, 1974, pp. 434-437.
- T. Keller, G. Graefe and D. Maier, "Efficient Assembly of Complex Objects," Proc. ACM SIGMOD Conf., Denver, Colo., 1991, pp. 148-157.
- H. A. Kuno and E.A. Rundensteiner, "Incremental Maintenance of Materialized Object-Oriented Views in MultiView: Strategies and Performance Evaluation," IEEE Transactions on Knowledge and Data Engineering, vol. 10. No. 5, 1998, pp. 768-792.
- T. Cotman, C. Leiserson, R. Rivest, and C. Stein, Introduction to Algorithms (second edition), McGraw-Hill Book Company, Boston, 2001.
- R. Schenkel, A. Theobald, and G. Weikum, Efficient creation and incrementation maintenance of HOPI index for complex xml document collection, in Proc. ICDE, 2006.
- R. Tarjan: Depth-first Search and Linear Graph Algorithms, SIAM J. Compt. Vol. 1. No. 2. June 1972, pp. 146 -140.
- J. Teuhola, "Path Signatures: A Way to Speed up Recursion in Relational Databases," IEEE Trans. on Knowledge and Data Engineering, Vol. 8, No. 3, June 1996, pp. 446 - 454.
- H. S. Warren, “A Modification of Warshall's Algorithm for the Transitive Closure of Binary Relations,” Commun. ACM 18, 4 (April 1975), 218 - 220.
- H. Wang, H. He, J. Yang, P.S. Yu, and J. X. Yu, Dual Labeling: Answering Graph Reachability Queries in Constant time, in Proc. of Int. Conf. on Data Engineering, Atlanta, USA, April -8 2006.
- S. Warshall, “A Theorem on Boolean Matrices,” JACM, 9. 1(Jan. 1962), 11 - 12.
- Y. Zibin and J. Gil, "Efficient Subtyping Tests with PQEncoding," Proc. of the 2001 ACM SIGPLAN Conf. on Object-Oriented Programming Systems, Languages and Application, Florida, October 14-18, 2001, pp. 96- 107.

#### 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