Yangjun Chen


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.


  1. 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.
  2. A. S. Asratian, T. Denley, and R. Haggkvist, Bipartite Graphs and their Applications, Cambridge University, 1998.
  3. 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.
  4. 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.
  5. Y. Chen and Y. Chen, An Efficient Algorithm for Answering Graph Reachability Queries, Proceedings of ICDE, 2008, pp. 893 - 902.
  6. 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.
  7. N. H. Cohen, “Type-extension tests can be performed in constant time,” ACM Transactions on Programming Languages and Systems, 13:626-629, 1991.
  8. 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.
  9. 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.
  10. D. Coppersmith, and S. Winograd. Matrix multiplication via arithmetic progression. Journal of Symbolic Computation, vol. 9, pp. 251-280, 1990.
  11. R. P. Dilworth, A decomposition theorem for partially ordered sets, Ann. Math. 51 (1950), pp. 161-166.
  12. S. Even, Graph Algorithms, Computer Science Press, Inc., Rockville, Maryland, 1979.
  13. J. E. Hopcroft, and R.M. Karp, An n2.5 algorithm for maximum matching in bipartite graphs, SIAM J. Comput. 2(1973), 225-231.
  14. H. V. Jagadish, "A Compression Technique to Materialize Transitive Closure," ACM Trans. Database Systems, Vol. 15, No. 4, 1990, pp. 558 - 598.
  15. A. V. Karzanov, Determining the Maximal Flow in a Network by the Method of Preflow, Soviet Math. Dokl., Vol. 15, 1974, pp. 434-437.
  16. T. Keller, G. Graefe and D. Maier, "Efficient Assembly of Complex Objects," Proc. ACM SIGMOD Conf., Denver, Colo., 1991, pp. 148-157.
  17. 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.
  18. T. Cotman, C. Leiserson, R. Rivest, and C. Stein, Introduction to Algorithms (second edition), McGraw-Hill Book Company, Boston, 2001.
  19. R. Schenkel, A. Theobald, and G. Weikum, Efficient creation and incrementation maintenance of HOPI index for complex xml document collection, in Proc. ICDE, 2006.
  20. R. Tarjan: Depth-first Search and Linear Graph Algorithms, SIAM J. Compt. Vol. 1. No. 2. June 1972, pp. 146 -140.
  21. 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.
  22. H. S. Warren, “A Modification of Warshall's Algorithm for the Transitive Closure of Binary Relations,” Commun. ACM 18, 4 (April 1975), 218 - 220.
  23. 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.
  24. S. Warshall, “A Theorem on Boolean Matrices,” JACM, 9. 1(Jan. 1962), 11 - 12.
  25. 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

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

in EndNote Style

JO - Proceedings of the 11th International Conference on Enterprise Information Systems - Volume 1: ICEIS,
SN - 978-989-8111-84-5
AU - Chen Y.
PY - 2009
SP - 17
EP - 24
DO - 10.5220/0001858300170024